Complexity in Forecasting and Predictive Modelsdownloads.hindawi.com/journals/specialissues/173202.pdfComplexity Complexity in Forecasting and Predictive Models Lead Guest Editor:

Complexity

Complexity in Forecasting and Predictive Models

Lead Guest Editor Jose L SalmeronGuest Editors Marisol B Correia and Pedro Palos

Complexity in Forecasting andPredictive Models

Complexity



Copyright copy 2019 Hindawi All rights reserved

This is a special issue published in ldquoComplexityrdquo All articles are open access articles distributed under the Creative Commons Attribu-tion License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Editorial Board

Joseacute A Acosta SpainCarlos F Aguilar-Ibaacutentildeez MexicoMojtaba Ahmadieh Khanesar UKTarek Ahmed-Ali FranceAlex Alexandridis GreeceBasil M Al-Hadithi SpainJuan A Almendral SpainDiego R Amancio BrazilDavid Arroyo SpainMohamed Boutayeb FranceAacutetila Bueno BrazilArturo Buscarino ItalyGuido Caldarelli ItalyEric Campos-Canton MexicoMohammed Chadli FranceEacutemile J L Chappin NetherlandsDiyi Chen ChinaYu-Wang Chen UKGiulio Cimini ItalyDanilo Comminiello ItalySara Dadras USASergey Dashkovskiy GermanyManlio De Domenico ItalyPietro De Lellis ItalyAlbert Diaz-Guilera SpainThach Ngoc Dinh FranceJordi Duch SpainMarcio Eisencraft BrazilJoshua Epstein USAMondher Farza FranceThierry Floquet FranceMattia Frasca ItalyJoseacute Manuel Galaacuten SpainLucia Valentina Gambuzza ItalyBernhard C Geiger AustriaCarlos Gershenson Mexico

Peter Giesl UKSergio Goacutemez SpainLingzhong Guo UKXianggui Guo ChinaSigurdur F Hafstein IcelandChittaranjan Hens IndiaGiacomo Innocenti ItalySarangapani Jagannathan USAMahdi Jalili AustraliaJeffrey H Johnson UKM Hassan Khooban DenmarkAbbas Khosravi AustraliaToshikazu Kuniya JapanVincent Labatut FranceLucas Lacasa UKGuang Li UKQingdu Li ChinaChongyang Liu ChinaXiaoping Liu CanadaXinzhi Liu CanadaRosa M Lopez Gutierrez MexicoVittorio Loreto ItalyNoureddine Manamanni FranceDidier Maquin FranceEulalia Martiacutenez SpainMarcelo Messias BrazilAna Meštrović CroatiaLudovico Minati JapanCh P Monterola PhilippinesMarcin Mrugalski PolandRoberto Natella ItalySing Kiong Nguang New ZealandNam-Phong Nguyen USAB M Ombuki-Berman CanadaIrene Otero-Muras SpainYongping Pan Singapore

Daniela Paolotti ItalyCornelio Posadas-Castillo MexicoMahardhika Pratama SingaporeLuis M Rocha USAMiguel Romance SpainAvimanyu Sahoo USAMatilde Santos SpainJosep Sardanyeacutes Cayuela SpainRamaswamy Savitha SingaporeHiroki Sayama USAMichele Scarpiniti ItalyEnzo Pasquale Scilingo ItalyDan Selişteanu RomaniaDehua Shen ChinaDimitrios Stamovlasis GreeceSamuel Stanton USARoberto Tonelli ItalyShahadat Uddin AustraliaGaetano Valenza ItalyAlejandro F Villaverde SpainDimitri Volchenkov USAChristos Volos GreeceQingling Wang ChinaWenqin Wang ChinaZidong Wang UKYan-Ling Wei SingaporeHonglei Xu AustraliaYong Xu ChinaXinggang Yan UKBaris Yuce UKMassimiliano Zanin SpainHassan Zargarzadeh USARongqing Zhang USAXianming Zhang AustraliaXiaopeng Zhao USAQuanmin Zhu UK

Contents

Complexity in Forecasting and Predictive ModelsJose L Salmeron Marisol B Correia and Pedro R Palos-SanchezEditorial (3 pages) Article ID 8160659 Volume 2019 (2019)

An Incremental Learning Ensemble Strategy for Industrial Process Soft SensorsHuixin Tian Minwei Shuai Kun Li and Xiao PengResearch Article (12 pages) Article ID 5353296 Volume 2019 (2019)

Looking for Accurate Forecasting of Copper TCRC Benchmark LevelsFrancisco J Diacuteaz-Borrego Mariacutea del Mar Miras-Rodriacuteguez and Bernabeacute Escobar-PeacuterezResearch Article (16 pages) Article ID 8523748 Volume 2019 (2019)

The Bass Diffusion Model on Finite Barabasi-Albert NetworksM L Bertotti and G ModaneseResearch Article (12 pages) Article ID 6352657 Volume 2019 (2019)

Prediction of Ammunition Storage Reliability Based on Improved Ant Colony Algorithm and BPNeural NetworkFang Liu Hua Gong Ligang Cai and Ke XuResearch Article (13 pages) Article ID 5039097 Volume 2019 (2019)

Green Start-Upsrsquo Attitudes towards Nature When Complying with the Corporate LawRafael Robina-Ramiacuterez Antonio Fernaacutendez-Portillo and Juan Carlos Diacuteaz-CaseroResearch Article (17 pages) Article ID 4164853 Volume 2019 (2019)

A CEEMDAN and XGBOOST-Based Approach to Forecast Crude Oil PricesYingrui Zhou Taiyong Li Jiayi Shi and Zijie QianResearch Article (15 pages) Article ID 4392785 Volume 2019 (2019)

Development of Multidecomposition Hybrid Model for Hydrological Time Series AnalysisHafiza Mamona Nazir Ijaz Hussain Muhammad Faisal Alaa Mohamd Shoukry Showkat Ganiand Ishfaq AhmadResearch Article (14 pages) Article ID 2782715 Volume 2019 (2019)

End-Point Static Control of Basic Oxygen Furnace (BOF) Steelmaking Based onWavelet TransformWeighted Twin Support Vector RegressionChuang Gao Minggang Shen Xiaoping Liu Lidong Wang and Maoxiang ChuResearch Article (16 pages) Article ID 7408725 Volume 2019 (2019)

EditorialComplexity in Forecasting and Predictive Models

Jose L Salmeron 1 Marisol B Correia 2 and Pedro R Palos-Sanchez3

1Universidad Pablo de Olavide de Sevilla Spain2ESGHT University of Algarve amp CiTUR - Centre for Tourism Research Development and Innovation amp CEG-ISTInstituto Superior Tecnico Universidade de Lisboa Portugal3International University of La Rioja Spain

Correspondence should be addressed to Jose L Salmeron salmeronacmorg

Received 20 May 2019 Accepted 21 May 2019 Published 10 June 2019

Copyright copy 2019 Jose L Salmeron et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

1 Introduction

The challenge of this special issue has been to know thestate of the problem related to forecasting modeling andthe creation of a model to forecast the future behaviorthat supports decision making by supporting real-world ap-plications

This issue has been highlighted by the quality of itsresearch work on the critical importance of advanced ana-lytical methods such as neural networks soft computingevolutionary algorithms chaotic models cellular automataagent-based models and finite mixture minimum squares(FIMIX-PLS)

Mainly all the papers are focused on triggering a sub-stantive discussion on how the model predictions can facethe challenges around the complexity field that lie aheadThese works help to better understand the new trends incomputing and statistical techniques that allow us to makebetter forecasts Complexity plays a prominent role in thesetrends given the increasing variety and changing data flowsforcing academics to adopt innovative and hybrid methods

The papers address the recent advances in method-ological issues and practices related to the complexity offorecasting and models All of them are increasingly focusedon heterogeneous sources

Among these techniques FIMIX-PLS has been appliedto study unobserved heterogeneity [1] This technique wasextended by Sarstedt et al [2] FIMIX-PLS is the first latentclass approach for PLS-SEM [3] and is an exploration toolthat allows obtaining the appropriate number of segmentsin which the sample will be divided In this sense the

FIMIX-PLS technique allowed making decisions regardingthe number of segments based on pragmatic reasons andconsidering practical issues related to current research [4]

FIMIX-PLS calculates the probabilities of belonging toa given segment in which each observation is adjusted tothe predetermined numbers of segments by means of theestimation of separate linear regression functions and thebelonging of objects corresponding to several segmentsDifferent cases are assigned to the segment with greaterprobability

2 Static Control Model

A static control model based onWavelet TransformWeightedTwin Support Vector Regression (WTWTSVR) is proposedin the paper ldquoEnd-Point Static Control of Basic OxygenFurnace (BOF) Steelmaking Based on Wavelet TransformWeighted Twin Support Vector Regressionrdquo by C Gao et alThe first approach of the authors was to add a new weightedmatrix and coefficient vector into the objective functionsof Twin Support Vector Regression (TSVR) to improve theperformance of the algorithm and then a static controlmodelwas established based on WTWTSVR and 220 samples inreal plant which consists of prediction models control mod-els regulating units controller and Basic Oxygen Furnace(BOF)

The results indicated that the control error bound with800 Nm3 in the oxygen blowing volume and 55 tons in theweight of auxiliary materials can achieve a hit rate of 90and 88 respectively In conclusion the proposed modelcan provide a significant reference for real BOF applications

HindawiComplexityVolume 2019 Article ID 8160659 3 pageshttpsdoiorg10115520198160659

2 Complexity

and can be extended to the prediction and control of otherindustry applications

3 Hybrid Models

In the paper ldquoDevelopment of Multidecomposition HybridModel for Hydrological Time Series Analysisrdquo byHM Naziret al two hybrid models were developed to improve theprediction precision of hydrological time series data basedon the principal of three stages as denoising decompositionand decomposed component prediction and summationTheperformance of the proposed models was compared with thetraditional single-stage model (without denoised and decom-posed) with the hybrid two-stagemodel (with denoised) andwith the existing three-stage hybrid model (with denoisedand decomposition) Three evaluation measures were used toassess the prediction accuracy of all models (Mean RelativeError (MRE)Mean Absolute Error (MAE) andMean SquareError (MSE))The proposed architecture was applied on dailyrivers inflow time series data of Indus Basin System andthe results showed that the three-stage hybrid models hadshown improvement in prediction accuracy with minimumMRE MAE and MSE for all cases and that the accuracyof prediction was improved by reducing the complexity ofhydrological time series data by incorporating the denoisingand decomposition

4 Complete Ensemble Empirical ModeDecomposition with Adaptive Noise(CEEMDAN)

Y Zhou et al proposed an approach that integrates completeensemble empirical mode decomposition with adaptive noise(CEEMDAN) and extreme gradient boosting (XGBOOST)the so-called CEEMDAN-XGBOOST for forecasting crudeoil prices in the paper ldquoA CEEMDAN and XGBOOST-BasedApproach to Forecast Crude Oil Pricesrdquo To demonstratethe performance of the proposed approach the authorsconducted extensive experiments on the West Texas Inter-mediate (WTI) crude oil prices Finally the experimentalresults showed that the proposed CEEMDAN-XGBOOSToutperforms some state-of-the-art models in terms of severalevaluation metrics

5 Machine Learning (ML) Algorithms

The ability to handle large amounts of data incrementallyand efficiently is indispensable for Machine Learning (ML)algorithms The paper ldquoAn Incremental Learning EnsembleStrategy for Industrial Process Soft Sensorsrdquo by H Tian etal addressed an Incremental Learning Ensemble Strategy(ILES) that incorporates incremental learning to extractinformation efficiently from constantly incoming data TheILES aggregates multiple sublearning machines by differentweights for better accuracy

The weight updating rules were designed by consideringthe prediction accuracy of submachines with new arriveddata so that the update can fit the data change and obtain

new information efficiently The sizing percentage soft sensormodel was established to learn the information from theproduction data in the sizing of industrial processes and totest the performance of ILES where the Extreme LearningMachine (ELM)was selected as the sublearningmachineTheresults of the experiments demonstrated that the soft sensormodel based on the ILES has the best accuracy and ability ofonline updating

6 Bass Innovation Diffusion Model

M L Bertotti and G Modanese in ldquoThe Bass DiffusionModel on Finite Barabasi-Albert Networksrdquo used a het-erogeneous mean-field network formulation of the Bassinnovation diffusion model and exact results by Fotouhi andRabbat [5] on the degree correlations of Barabasi-Albert (BA)networks to compute the times of the diffusion peak andto compare them with those on scale-free networks whichhave the same scale-free exponent but different assortativityproperties

The authors compared their results with those obtainedby DrsquoAgostino et al [6] for the SIS epidemic model with thespectral method applied to adjacency matrices and turnedout that diffusion times on finite BA networks were at aminimumThey believe this may be due to a specific propertyof these networks eg whereas the value of the assortativitycoefficient is close to zero the networks look disassortativeif a bounded range of degrees is solely considered includingthe smallest ones and slightly assortative on the range of thehigher degrees

Finally the authors found that if the trickle-down char-acter of the diffusion process is enhanced by a larger initialstimulus on the hubs (via a inhomogeneous linear term inthe Bass model) the relative difference between the diffusiontimes for BA networks and uncorrelated networks is evenlarger

7 Forecast Copper Treatment Charges(TC)Refining Charges (RC)

In order to overcome the lack of research about the priceat which mines sell copper concentrate to smelters thepaper ldquoLooking for Accurate Forecasting of Copper TCRCBenchmark Levelsrdquo by F J Dıaz-Borrego et al provides a toolto forecast copper Treatment Charges (TC)Refining Charges(RC) annual benchmark levels in which a three-modelcomparison was made by contrasting different measures oferror

The results indicated that the Linear ExponentialSmoothing (LES) model is the one that has the bestpredictive capacity to explain the evolution of TCRC in boththe long and the short term Finally the authors believe thissuggests a certain dependency on the previous levels of TCRC as well as the potential existence of cyclical patterns inthem and that this model enables a more precise estimationof copper TCRC levels which is useful for smelters andmining companies

Complexity 3

8 Ant Colony Optimization Algorithm

F Liu et al in ldquoPrediction of Ammunition Storage ReliabilityBased on Improved Ant Colony Algorithm and BP NeuralNetworkrdquo proposed an Improved Ant Colony Optimization(IACO) algorithmbased on a three-stage and aBackPropaga-tion (BP) Neural Network algorithm to predict ammunitionfailure numbers where the main affecting factors of ammu-nition include temperature humidity and storage periodand where the reliability of ammunition storage is obtainedindirectly by failure numbers Experimental results showthat IACO-BP algorithm has better accuracy and stability inammunition storage reliability prediction than BP networkParticle Swarm Optimization-Back Propagation (PSO-BP)and ACO-BP algorithms

9 Finite Mixture Partial Least Squares(FIMIX-PLS)

Finally the paper ldquoGreen Start-Upsrsquo Attitudes towardsNatureWhen Complying with the Corporate Lawrdquo by R Robina-Ramırez et al examined how Spanish green start-ups developimproved attitudes towards nature having in mind theframework of the new Spanish Criminal Code in relation tocorporate compliance Smart Partial Least Squares (PLS) PathModelling was used to build an interaction model amongvariables and unobserved heterogeneity was analysed usingFinite Mixture Partial Least Squares (FIMIX-PLS)

The model reveals a strong predictive power (R2 =779)with a sampling of one hundred and fifty-two Spanish start-ups This test is performed in a set of four stages In thefirst one it executes FIMIX and in a second it calculates thenumber of optimal segments In the third one we discussedthe latent variables that justify these segments to finallyestimate the model and segments [7]

The results allow concluding that Spanish start-upsshould implement effective monitoring systems and neworganisational standards in order to develop surveillancemeasures to avoid unexpected sanctions

10 Conclusion

An overview of the 8 selected papers for this special issuehas been presented reflecting the most recent progress in theresearch fieldWehope this special issue can further stimulateinterest in the critical importance of advanced analyticalmethods such as neural networks soft computing evolu-tionary algorithms chaotic models cellular automata agent-based models and finite mixture minimum squares (FIMIX-PLS) More research results and practical applications onadvanced analytical methods are expected in the future

Conflicts of Interest

The guest editor team does not have any conflicts of interestor private agreements with companies

Jose L SalmeronMarisol B Correia

Pedro R Palos-Sanchez

References

[1] C Hahn M D Johnson A Herrmann and F Huber ldquoCap-turing customer heterogeneity using a finite mixture PLSapproachrdquo Schmalenbach Business Review vol 54 no 3 pp243ndash269 2002 (Russian)

[2] M Sarstedt J-M Becker C M Ringle and M SchwaigerldquoUncovering and treating unobserved heterogeneity withFIMIX-PLS which model selection criterion provides an ap-propriate number of segmentsrdquo Schmalenbach Business Reviewvol 63 no 1 pp 34ndash62 2011

[3] M Sarstedt ldquoA review of recent approaches for capturingheterogeneity in partial least squares path modellingrdquo Journalof Modelling in Management vol 3 no 2 pp 140ndash161 2008

[4] M Sarstedt M Schwaiger and C M Ringle ldquoDo we fullyunderstand the critical success factors of customer satisfactionwith industrial goods - extending festge and schwaigers modelto account for unobserved heterogeneityrdquo Journal of BusinessMarket Management vol 3 no 3 pp 185ndash206 2009

[5] B Fotouhi and M G Rabbat ldquoDegree correlation in scale-freegraphsrdquoThe European Physical Journal B vol 86 no 12 articleno 510 2013

[6] G DrsquoAgostino A Scala V Zlatic and G Caldarelli ldquoRobust-ness and assortativity for diusion-like processes in scale-freenetworksrdquo EPL (Europhysics Letters) vol 97 no 6 article no68006 2012

[7] P Palos-Sanchez FMartin-Velicia and J R Saura ldquoComplexityin the acceptance of sustainable search engines on the internetan analysis of unobserved heterogeneity with FIMIX-PLSrdquoComplexity Article ID 6561417 19 pages 2018

Research ArticleAn Incremental Learning Ensemble Strategy forIndustrial Process Soft Sensors

Huixin Tian 123 Minwei Shuai1 Kun Li 4 and Xiao Peng1

1 School of Electrical Engineering amp Automation and Key Laboratory of Advanced Electrical Engineering and Energy TechnologyTianjin Polytechnic University Tianjin China

2State Key Laboratory of Process Automation in Mining amp Metallurgy Beijing 100160 China3Beijing Key Laboratory of Process Automation in Mining amp Metallurgy Research Fund Project Beijing 100160 China4School of Economics and Management Tianjin Polytechnic University Tianjin China

Correspondence should be addressed to Kun Li lk neu163com

Received 13 November 2018 Accepted 25 March 2019 Published 2 May 2019

Guest Editor Marisol B Correia

Copyright copy 2019 Huixin Tian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With the continuous improvement of automation in industrial production industrial process data tends to arrive continuouslyin many cases The ability to handle large amounts of data incrementally and efficiently is indispensable for modern machinelearning (ML) algorithms According to the characteristics of industrial production process we address an ILES (incrementallearning ensemble strategy) that incorporates incremental learning to extract information efficiently from constantly incomingdata The ILES aggregates multiple sublearning machines by different weights for better accuracy When new data set arrives anew submachine will be trained and aggregated into ensemble soft sensor model according to its weight The other submachinesweights will be updated at the same time Then a new updated soft sensor ensemble model can be obtained The weight updatingrules are designed by considering the prediction accuracy of submachines with new arrived data So the update can fit the datachange and obtain new information efficientlyThe sizing percentage soft sensor model is established to learn the information fromthe production data in the sizing of industrial processes and to test the performance of ILES where the ELM (Extreme LearningMachine) is selected as the sublearning machineThe comparison is done among newmethod single ELM AdaBoostR ELM andOS-ELM and the test of the extensions is done with three test functions The results of the experiments demonstrate that the softsensor model based on the ILES has the best accuracy and ability of online updating

1 Introduction

During industrial processes plants are usually heavily instru-mented with a large number of sensors for process mon-itoring and control However there are still many processparameters that cannot be measured accurately becauseof high temperature high pressure complex physical andchemical reactions and large delays etc Soft sensor technol-ogy provides an effective way to solve these problems Theoriginal and still the most dominant application area of softsensors is the prediction of process variables which can bedetermined either at low sampling rates or through off-lineanalysis only Because these variables are often related to theprocess product quality they are very important for processcontrol and qualitymanagement Additionally the soft sensor

application field usually refers to online prediction during theprocess of production

Currently with the continuous improvement of automa-tion in industrial production large amounts of industrialprocess data can be measured collected and stored auto-matically It can provide strong support for the establish-ment of data-driven soft sensor models Meanwhile withthe rapid development and wide application of big datatechnology soft sensor technology has already been usedwidely and plays an essential role in the development ofindustrial process detection and control systems in industrialproduction Artificial intelligence and machine learning asthe important core technologies are getting increasinglymore attention The traditional machine learning algorithmgenerally refers to the single learning machine model that


2 Complexity

is trained by training sets and then the unknown sampleswill be predicted based on this model However the singlelearning machine models have to face defects that cannot beovercome by themselves such as unsatisfactory accuracy andgeneralization performance especially for complex industrialprocesses Specifically in the supervised machine learningapproach the modelrsquos hypothesis is produced to predict thenew incoming instances by using predefined label instancesWhen the multiple hypotheses that support the final decisionare aggregated together it is called ensemble learning Com-pared with the single learning machine model the ensemblelearning technique is beneficial for improving quality andaccuracyTherefore increasingly more researchers are study-ing how to improve the speed accuracy and generalizationperformance of integrated algorithms instead of developingstrong learning machines

Ensemble algorithms were originally developed for solv-ing binary classification problems [1] and then AdaBoostM1and AdaBoostM2 were proposed by Freund and Schapire [2]for solving multiclassification problems Thus far there aremany different versions of boosting algorithms for solvingclassification problems [3ndash7] such as boosting by filteringand boosting by subsampling However for regression prob-lems it is not possible to predict the output exactly as that inclassification To solve regression problems using ensembletechniques Freund and Schapire [2] extended AdaBoostM2to AdaBoostR which projects the regression sample into aclassification data set Drucker [8] proposed the AdaBoostR2algorithm which is an ad hoc modification of AdaBoostRAvnimelech and Intrator [9] extended the boosting algorithmfor regression problems by introducing the notion of weakand strong learning as well as an appropriate equivalencetheorem between the two Feely [10] proposed BEM (bigerror margin) boosting method which is quite similar tothe AdaBoostR2 In BEM the prediction error is comparedwith the preset threshold value BEM and the correspondingexample is classified as either well or poorly predictedShrestha and Solomatine [11] proposed an AdaBoostRTwith the idea of filtering out the examples with relativeestimation errors that are higher than the preset thresholdvalue However the value of the threshold is difficult toset without experience For solving this problem Tian andMao [12] present a modified AdaBoostRT algorithm thatcan adaptively modify the threshold value according to thechange in RMSE Although ensemble algorithms have theability to enhance the accuracy of soft sensors they are stillat a loss for the further information in the new coming data

In recent years with the rapid growth of data size afresh research perspective has arisen to face the large amountof unknown important information contained in incomingdata streams in various fields How can we obtain methodsthat can quickly and efficiently extract information fromconstantly incoming dataThe batch learning is meaninglessand the algorithm needs to have the capability of real-timeprocessing because of the demand of real-time updated datain industrial processes Incremental learning idea is helpfulto solve the above problem If a learning machine has thiskind of idea it can learn new knowledge from new data setsand can retain old knowledge without accessing the old data

set Thus the incremental learning strategy can profoundlyincrease the processing speed of new data while also savingthe computerrsquos storage spaceThe ensemble learning methodscan be improved by combining the characteristics of theensemble strategy and incremental learning It is an effectiveand suitable way to solve the problem of stream data mining[13ndash15] Learn++ is a representative ensemble algorithmwith the ability of incremental learning This algorithm isdesigned by Polikar et al based on AdaBoost and supervisedlearning [16] In Learn ++ the new data is also assignedsample weights which update according to the results ofclassification at each iteration Then a newly trained weakclassifier is added to the ensemble classifier Based on theLearn ++ CDS algorithm G Ditzler and Polikar proposedLearn ++ NIE [17] to improve the effect of classification on afew categories Most of research of incremental learning andensemble algorithm focus on the classification field while theresearch in the field of regression is still less Meanwhile thelimitation of ensemble approaches is that they cannot addressthe essential problem of incremental learning well what theessential problem is accumulating experience over time thenadapting and using it to facilitate future learning process [18ndash20]

For the process of industrial production increasinglymore intelligent methods are used in soft sensors withthe fast development of artificial intelligence However thepractical applications of soft sensors in industrial productionare not good The common shortages of soft sensors areunsatisfactory unstable prediction accuracy and poor onlineupdating abilities It is difficult to meet a variety of changes inthe process of industrial production Therefore in this paperwe mainly focus on how to add the incremental learningcapability to the ensemble soft sensor modeling method andhopefully provide useful suggestions to enhance both thegeneration and online application abilities of soft sensors forindustrial process Aiming at the demands of soft sensors forindustrial applications a new detection strategy is proposedwith multiple learning machines ensembles to improve theaccuracy of the soft sensors based on intelligent algorithmsAdditionally in practical production applications acquisi-tion of information in new production data is expensive andtime consuming Consequently it is necessary to update thesoft sensor in an incremental fashion to accommodate newdata without compromising the performance on old dataPractically in most traditional intelligent prediction modelsfor industrial process the updates are often neglected Somemodels use traditional updating methods that retrain themodels by using all production data or using the updateddata and forgo the old data This kind of methods is not goodenough because some good performances have to be lostto learn new information [21] Against this background wepresent a new incremental learning ensemble strategy with abetter incremental learning ability to establish the soft sensormodel which can learn additional information from newdata and preserve previously acquired knowledgeThe updatedoes not require the original data that was used to train theexisting old model

In the rest of this paper we first describe the details ofthe incremental learning ensemble strategy (ILES) which

Complexity 3

involves the strategy of updating the weights the ensemblestrategy and the strategy of incremental learning for real-time updating Then we design experiments to test theperformance of the ILES for industrial process soft sensorsThe sizing percentage of the soft sensor model is built by theILES in the sizing production process The parameters of theILES are discussed We also compare the performance of theILES to those of other methods To verify the universal useof the new algorithm three test functions are used to testthe improvement on the predictive performance of the ILESFinally we summarize our conclusions and highlight futureresearch directions

2 The Incremental LearningEnsemble Strategy

The industrial process needs soft sensors with good accuracyand online updating performance Here we focus on incor-porating the incremental learning idea into the ensembleregression strategy to achieve better performance of softsensors A new ensemble strategy called ILES for industrialprocess soft sensors that combines the ensemble strategywith the incremental learning idea is proposed The ILEShas the ability to enhance soft sensorsrsquo accuracy by aggre-gating the multiple sublearning machines according to theirtraining errors and prediction errors Additionally duringthe iteration process incremental learning is added to obtainthe information from new data by updating the weightsIt is beneficial to enhance the real-time updating ability ofindustrial process online soft sensors The details of the ILESare described as shown in Algorithm 1

21 Strategy of Updating the Weight In each iteration of119896 (119896 = 1 2 119870) the initial 1198631(119894) = 1119898(119896) is distributedto each sample (119909푖 119910푖) with the same values This meansthat the samples have the same chance to be includedin training dataset or tested dataset at the beginning Inthe subsequent iterations the weight will be calculated as119863푡(119894) = 119908푡sum푚푖=1119908푡(119894) for every sublearning machine (ineach iteration of 119905) In contrast to the traditional AdaBoostRhere the testing subdataset is added to test the learningperformance in each iteration It is useful to ensure thegeneralization performance of the ensemble soft sensorsThen the distribution will be changed according to thetraining and testing errors at the end of each iterationHere the training subdataset 119879119877푡 and the testing subdataset119879119864푡 will be randomly chosen according to 119863푡 (for exampleby using the roulette method) The sublearning machine istrained by 119879119877푡 and a hypothesized soft sensor 119891푡 119909 997888rarr 119910will be obtained Then the training error and testing error of119891푡 can be calculated as follows

119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (1)

The error rate of 119891푡 on 119878푘 = 119879119877푡 + 119879119864푡 is defined as follows

120576푡 = sum퐴푅퐸119905(푖)gt훿

119863푡 (119894) (2)

If 120576푡 gt 120593 the submodel is regarded as an unquali-fied and suspicious model The hypothesis 119891푡 is given upOtherwise the power coefficient is calculated as 120573푡 = 120576푛푡(eg linear square or cubic) Here 120593 (0 lt 120593 lt 1) isthe coefficient of determination After 119905 (119905 = 1 2 119879푘)iterations the composite hypothesis 119865푘 can be obtainedaccording to the 119905 hypothesized soft sensors (sublearningmachines) 1198911 1198912 119891푡 The training subdataset error thetesting subdataset error and the error rate of 119865푘 are calculatedsimilarly to those of 119891푡 In the same way if 119864푘 gt 120593the hypothesis 119865푘 is given up At the end of the iterationsaccording to the error rate119864푡 theweight is updated as follows

119908푡+1 = 119908푡 times 119861푘 119860119877119864푘 (119894) lt 1205751 otherwise

(3)

where 119861푘 = 119864푘(1 minus 119864푘) In the next iteration the 119879119877푡 and119879119864푡 will be chosen again according to the new distributionwhich is calculated by the new weight119908푡+1 During the aboveprocess of iterations the updating of the weights dependson the training and testing performance of the sublearningmachines with different data Therefore the data with largeerrors will have a larger distribution for the difficult learningIt means that the ldquodifficultrdquo data will have more chancesto be trained until the information in the data is obtainedConversely the sublearning machines or hypothesized softsensors are reserved selectively based on their performanceTherefore the final hypothesized soft sensors are well qual-ified to aggregate the composite hypothesis This strategy isvery effective for improving the accuracy of ensemble softsensors

22 Strategy of Ensemble with Incremental Learning Aimingat the needs of real-time updates the incremental learningstrategy is integrated into the ensemble process First the sub-datasets 119878푘 = [(1199091 1199101) (1199092 1199102) (119909푚 119910푚)] 119896 = 1 2 119870are selected randomly from the dataset In each iteration119896 = 1 2 119870 the sublearning machines are trained andtested Therefore for each subdataset when the inner loop(119905 = 1 2 119879푘) is finished the 119879푘 hypothesized soft sensors1198911 1198912 119891푇119896 are generated An ensemble soft sensor isobtained based on the combined outputs of the individualhypotheses which constitute the composite hypothesis 119865푘

119865푘 (119909) = sum푡 (lg (1120573푡)) 119891푡 (119909)sum푡 (lg (1120573푡)) (4)

Here the better hypotheses will be aggregated with largerchances Therefore the best performance of ensemble softsensors is ensured based on these sublearning machinesThen the training subdataset error and testing subdataseterror of 119865푘 can be calculated similarly to the error of 119891푡

119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119865푡 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (5)

The error rate of 119865푘 on 119878푘 = 119879119877푡 + 119879119864푡 is defined as follows

119864푘 = sum퐴푅퐸119896(푖)gt훿

119863푡 (119894) (6)

4 Complexity

Input(i) 119878푘 sub datasets are drawn from original data set Here 119896 = 1 2 119870 119878푘 = [(1199091 1199101) (1199092 1199102) (119909푚 119910푚)](ii) The number of sub learning machines is 119879푘(iii) The coefficients of determination are 120575 and 120593

For 119896 = 1 2 119870Initialize 1198631(119894) = 1119898(119896) forall119894 = 1 2 119870 Here119898(119896) is the amount of dataFor 119905 = 1 2 119879푘

(1) Calculate119863푡(119894) = 119908푡sum푚푖=1 119908푡(119894)119863푡 is a distribution and 119908푡(119894) is the weight of (119909푖 119910푖)(2) Randomly choose the training sub dataset 119879119877푡 and the testing sub dataset 119879119864푡 according to119863푡(3) The sub learning machine is trained by 119879119877푡 to obtain a soft sensor model 119891푡 119909 997888rarr 119910(4) Calculate the error of 119891푡 using 119879119877푡 and 119879119864푡

119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (5) Calculate the error rate 120576푡 = sum퐴푅퐸119905(푖)gt훿119863푡(119894) If 120576푡 gt 120593 give up 119891푡 and return to step (2)(6) Calculate 120573푡 = 120576푡푛 where 119899 = 1 2 or 3 Obtain the ensemble soft sensor model according to 120573푡

119865푘 (119909) = sum푡 (lg (1120573푡)) 119891푡 (119909)sum푡 (lg (1120573푡))(7) Calculate the error of 119865푘 using 119878푘 119864푘 = sum퐴푅퐸119896(푖)gt훿119863푡(119894) If 119864푘 gt 120593 give up 119891푡 and return to step (2)(8) Calculate 119861푘 = 119864푘(1 minus 119864푘) to update the weights


Output Obtain the ensemble soft sensor model according to 119861푘

119865푓푖푛 (119909) = sum푘 (lg (1119861푘)) 119865푘 (119909)sum푘 (lg (1119861푘))Algorithm 1 Incremental learning ensemble strategy

After 119870 hypotheses are generated for each subdataset thefinal hypothesis 119865푓푖푛 is obtained by the weighted majorityvoting of all the composite hypotheses

119865푓푖푛 (119909) = sum푘 (lg (1119861푘)) 119865푘 (119909)sum푘 (lg (1119861푘)) (7)

When new data come constantly during the industrialprocess new subdatasets will be generated (they will bethe 119870 + 1 119870 + 2 ) Based on a new subdataset a newhypothesized soft sensor can be trained by a new iterationThe new information in the new data will be obtained andadded to the final ensemble soft sensor according to (7) Asthe added incremental learning strategy the ensemble softsensor is updated based on the old hypothesis Therefore theinformation in the old data is also retained and the incrementof information from new data is achieved

Overall in the above ILES the ensemble strategy is effi-cient to improve the prediction accuracy using the changeddistribution Therefore the ILES will give more attention tothe ldquodifficultrdquo data with big errors in every iteration that areattributable to the new distribution Due to the harder learn-ing for the ldquodifficultrdquo data more information can be obtainedTherefore the soft sensor model is built more completely andthe accuracy of perdition is improvedMoreover the data thatis used to train the sublearning machines is divided into atraining subdataset and a testing subdataset The testing errorwill be used in the following steps the weight update and thecomposite hypothesis ensembleTherefore the generalization

of the soft sensor model based on the ILES can be improvedefficiently especially compared with traditional algorithmsAdditionally when the new data is added the new ILES withincremental learning ability can learn the new data in real-time and does not give up the old information from the olddata The ILES can save the information of old sublearningmachines that have been trained but it does not need to savethe original data In other words only a small amount ofnew production data being saved is enough This strategy isefficient to save space Furthermore the new ILES also maysave time compared with the traditional updating methodThis strategy is attributed to the conservation of the old 119861푘and the sublearning machines in composite hypotheses (7)

3 Experiments

In this chapter the proposed ILES is tested in the sizingproduction process for predicting the sizing percentage Firstthe influence of each parameter on the performance ofthe proposed algorithm is discussed Meanwhile the realindustrial process data is used to establish the soft sensormodel to verify the incremental learning performance of thealgorithm Finally to prove its generalization performancethree test functions are used to verify the improvement of theprediction performanceThemethods are implemented usingMATLAB language and all the experiments are performedon a PC with the Intel Core 7500U CPU (270GHZ for eachsingle core) and the Windows 10 operation system

Complexity 5

31 Sizing Production Process and Sizing Percentage Thedouble-dip and double pressure sizing processes are widelyused in textile mills as shown in Figure 1 The sizing per-centage plays an important role during the process of sizingfor good sizing quality In addition the sizing agent controlof warp sizing is essential for improving both productivityand product quality The sizing percentage online detectionis a key factor for successful sizing control during the sizingprocess The traditional sizing detection methods whichare instruments measurement and indirect calculation haveexpensive prices or unsatisfactory accuracy Soft sensorsprovide an effective way to predict the sizing percentageand to overcome the above shortages According to themechanism analysis of the sizing process the influencingfactors on the sizing percentage are slurry concentrationslurry viscosity slurry temperature the pressure of the firstGrouting roller the pressure of the second Grouting rollerthe position of immersion roller the speed of the sizingmachine the cover coefficient of the yarn the yarn tensionand the drying temperature [22] In the following soft sensormodeling process the inputs of soft sensors are the nineinfluencing factors and the output is the sizing percentage

32 Experiments for the Parameters of the ILES Here weselect ELM as the sublearning machine of the ILES due toits good performance such as fast learning speed and simpleparameter choices [22 23] the appendix reviews the processof ELM Then experiments with different parameters of theILES are done to research the performance trend of the ILESwhen the parameters change

First the experiments to assess the ILES algorithmrsquos per-formance are done with different 119879푘s Here the 119879푘 increasesfrom 1 to 15 Figure 2 shows the results of the training errorsand the testing errors with different 119879푘s Along with theincreasing 119879푘 the training and testing errors decrease When119879푘 increases to 7 the testing error is the smallest Howeverwhen 119879푘 increases to more than 9 the testing error becomeslarger again Furthermore the training errors only slightlydecrease Therefore we can draw the conclusion when theparameter119879푘 is 7 that the performance of ILES is the bestThecomparison is also done between AdaBoostR and the ILESregarding the testing errors with different numbers of ELMsin Figure 3 Although the RMSE means of AdaBoostR andthe ILES are different their performance trends are similarwith the increasing number of ELMs Here the RMSE isdescribed as

RMSE = (1119899푛sum푖=1

1198902 (119894))12 (8)

Second we discuss the impact of parameter (120575 and 120593)changes on the ILES performance The experiments demon-strate that when 120575 is too small the performance of ELM isdifficult to achieve the preset goal and the iteration is difficultto stop 120575 is also not larger than 80 percent of the average of therelative errors of ELMs otherwise the 119865푘 can not be obtainedFurthermore the value of120593 determines the number of ldquobettersamplesrdquo Here the ldquobetter samplesrdquo refer to the samples thatcan reach the expected precision standard of predicted results

Table 1TheRMSE of the ILES with different parametersΔΦ (119879푘 =7)120575 120593

005 01 015 02 025002 mdash mdash mdash 04559 04712003 mdash mdash 04505 04637 04614004 mdash 03484 03829 03888 04125005 03947 03620 03765 03898 03812006 04363 03734 03084 04036 04121007 04591 03342 03323 03909 03954008 04573 03682 03517 04138 04332

by submachines If 120593 is too small the ELM soft sensor model(119891푡) will not be sufficiently obtained If120593 is too large the ldquobadrdquoELM model (119891푡) will be aggregated into the final compositehypothesized 119865푓푖푛(119909) Then the accuracy of the ILES cannotbe improved The relationships among 120575 120593 and RMSE areshown in Table 1 When 120575 = 006 and 120593 = 015 the modelhas the best performance (the RMSE is 03084)

33 Experiments for the Learning Process of the ILES Forestablishing the soft sensor model based on the ILES a totalof 550 observations of real production data are collected fromTianjin Textile Engineering Institute Co Ltd of which 50data are selected randomly as testing dataThe remaining 500data are divided into two data sets according to the time ofproductionThe former 450 data are used as the training dataset and the latter 50 data are used as the update data setThe inputs are 9 factors that affect the sizing percentage Theoutput is the sizing percentage The parameters of the ILESare 119870 = 9 119879푘 = 7 120575 = 006 and 120593 = 015 That is to saythe 450 training data are divided into 9 subdatasets 1198701 sim 1198709and the number of ELMs is 7 According to the needs of thesizing production the predictive accuracy of the soft sensorsis defined as

accuracy = 119873푠119873푤 (9)

where119873푠 is the number of times with an error lt 06 and119873푤is the total number of testing times

Since the learning process is similar to the OS-ELM[24] update process It is an online assessment model thatis capable of updating network parameters based on newarriving datawithout retrains historical dataTherefore whilecomparing the accuracy of IELS learning process it is alsocompared with OS-ELM The two columns on the right sideof Table 2 show the changes in the soft sensor accuracy duringthe learning process of the ILES and OS-ELM It can be seenthat the stability and accuracy of ILES are superior to OS-ELM

34 Comparison In this experiment we used 10-fold crossvalidation to test the modelrsquos performance The first 500 datasets are randomly divided into 10 subdatasets S1 sim S10 Theremaining 50 data sets are used as the updated data set S11The single subdataset from S1 sim S10 will be retained as the

6 Complexity

Table 2 The changes in the soft sensor accuracy during the learning process of the ILES

Dataset Iteration1 2 3 4 5 6 7 8 9 Update OS-ELM

K1 95 95 95 96 95 95 97 95 95 95 56K2 94 94 96 94 95 95 94 94 94 68K3 96 95 96 96 95 95 96 97 64K4 97 96 94 95 95 96 95 62K5 96 94 96 95 95 96 66K6 95 95 97 96 96 70K7 96 96 95 96 80K8 93 94 94 88K9 94 95 88Update set 96 88Testing set 832 85 851 87 90 91 918 92 932 95 90

warpCited yarn roller

Immersion roller Immersion roller

Squeezing rollerSqueezing roller

Drying slurry

size

trough

Figure 1 The double-dip and double pressure sizing processes

2 4 6 8 10 12 1401

02

03

04

05

06

The number of ELMs

TrainingTest

RMSE

Mea

n

Figure 2 The training and testing errors of the ILES with differentparameters 119879푘

validation data for testing the model and the remaining 9subdatasets are used as the training data For comparing thenew method with other soft sensor methods the single ELMand ensemble ELM based on AdaBoostR are also applied tobuild the sizing percentage soft sensor models as traditional

2 4 6 8 10 12 1403

035

04

045

05

055

06

The number of ELMs

RMSE Mean by AdaBoostRRMSE Mean by ILES

RMSE

Mea

n

Figure 3The performance trends of the ILES and AdaBoostR withdifferent parameters 119879푘

methods with the same data set The soft sensor models arelisted as follows

Single ELM model the main factors that affect the sizingpercentage are the inputs of the model The input layer ofthe ELM has 9 nodes The hidden layer of the ELM has 2300nodes and the output layer of the ELM has one node whichis the sizing percentage

AdaBoostR model the parameters and the structure ofthe base ELM are the same as those of the single ELMAdaBoostR has 13 iterations

ILES model the ELMs are same as the single ELMmodeldescribed above The parameters of the ILES are 119879푘 = 7 120575 =006 and 120593 = 015 (time consuming 163s)

Figures 4(a)ndash4(c) shows the predicted sizing percentageof different soft sensor models based on different methodsThe experiments demonstrate that the strategy of the ILEScan improve the accuracy of the soft sensor model Inaddition the training errors of the above three models all

Complexity 7

0 5 10 15 20 25 30 35 40 45

times

real sizing percentageprediction by single ELM

10

15

20

Sizi

ng p

erce

ntag

e

(a) Sizing percentage soft sensor by single ELMmodel

real sizing percentageprediction by AdaBoostR ELM

0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e

(b) Sizing percentage soft sensor by ensemble ELMmodel based on AdaBoostR

real sizing percentageprediction by ILES

0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e

(c) Sizing percentage soft sensor by ILES model

Figure 4 The result of the sizing percentage soft sensor

can achieve 02 However the testing errors of the predictionmodels of AdaBoostR and the ILES are smaller than that ofthe single ELMThis result means that the ensemble methodshave better generalization performance Table 3 shows theperformance of the prediction model based on differentmethods after updatingThe results of the comparison experi-ments show that the soft sensor based on the new ILES has thebest accuracy and the smallest RMSEThis result is attributedto the use of the testing subdataset in the ensemble strategyand the incremental learning strategy during the learningprocess of the ILES algorithm Overall the accuracy of thesoft sensor can fit the needs of actual production processesMoreover the incremental learning performance can ensurethe application of industrial process soft sensors in practicalproduction

35 Experiments for the Performance of the ILES by TestFunctions To verify the universal use of the algorithmthree test functions are used to test the improvement of theprediction performanceThese test functions are Friedman1Friedman2 and Friedman3 Table 4 shows the expressionof each test model and the value range of each variableFriedman1 has a total of 10 input variables of which there arefive input variables associated with the output variable and

the other five input variables are independent of the outputvariables The Friedman2 and Friedman3 test functionsincorporate the impedance phase change of the AC circuit

Through continuous debugging the parameters of eachalgorithm are determined as shown in Table 5 For every testfunction generate a total of 900 data and 78 of the totalsamples were selected as training samples 11 as updatingsamples and 11 as testing samples according to the needfor different test models That is to say the 700 trainingdata are divided into 7 subdatasets 1198701 sim 1198707 Figures 5ndash7show the predicted results of Friedman1 Friedman2 andFriedman 3 with different soft sensor models based ondifferent methods (time consuming 227s) The comparisonof the performances of the different soft sensors is shown inTable 6 It shows the soft sensor model based on ILES has thebest performance

4 Conclusions

An ILES algorithm is proposed for better accuracy and incre-mental learning ability for industrial process soft sensorsThesizing percentage soft sensor model is established to test theperformance of the ILES The main factors that influence thesizing percentage are the inputs of the soft sensor model

8 Complexity

Table 3 The performance of the soft sensor model based on different methods with 10-fold cross validation

Method datasets RMSE ARE Max AREtraining updating testing

ELMS1 sim S9 S11 S10

07987 00698 02265AdaBoostR 05915 00313 01447ILES 03704 00213 00689ELM

S1 sim S8 S10 S11 S907386 00868 03773

AdaBoostR 04740 00394 01846ILES 03309 00182 00542ELM

S1 sim S7 S9 sim S10 S11 S808405 00772 02962


S1 sim S6 S8 sim S10 S11 S70936 00655 04255


S1 sim S5 S7 sim S10 S11 S607342 00618 03451


S1 simS4S6 simS10 S11 S508533 00872 03546


S1 sim S3 S5 sim S10 S11 S407752 00749 03249


S1 sim S2 S4 sim S10 S11 S308430 00823 03281


S1 S3 sim S10 S11 S209127 01104 04302


S2 sim S10 S11 S107905 00910 02245

AdaBoostR 05045 00436 01773ILES 03145 00191 00756

Table 4 Test function expressions

Data set Expression Variable scope

Friedman1 y = 10 sin(120587x1x2) + 20 (x3 minus 05)2 + 10x4 + 5x5 + 120575 x푖 sim 119880 [0 1] 119894 = 1 2 10120575 sim 119880[minus4 4]Friedman2 y = radicx12 + (x2x3 minus 1

x2x4)2 + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]

x3 sim 119880[0 1] 1199094 sim 119880[1 11]120575 sim 119880[minus120 120]Friedman3 y = tanminus1 (x2x3 minus 1x2x4

x1) + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]

x3 sim 119880[0 1] x4 sim 119880[1 11]120575 sim 119880[minus04 04]Table 5 Parameters of the algorithmic performance test

Test function 119879푘 (the number of ELMs) 119870 120575 120593 datasetstraining updating testing

Friedman1 10 7 0012 01050 700 100 100Friedman2 10 7 0012 0510 700 100 100Friedman3 8 7 0010 01810 700 100 100

Complexity 9

Table 6 The performance comparisons of the algorithms

Data set Method RMSE ARE Max ARE

Friedman1ELM 08278 07662 47880

AdaBoostR 07987 06536 38817ILES 03099 03073 27132

Friedman2ELM 06338 09975 179870

AdaBoostR 04190 04495 73544ILES 02079 03683 72275

Friedman3ELM 11409 14314 149348

AdaBoostR 08959 13461 115991ILES 04624 03930 52719

0 10 20 30 40 50 60 70 80 90 100

times

80

60

40

20

minus20

0

y

minus40

Friedman1prediction by single ELMprediction by AdaBoostRprediction by ILES

Figure 5 The comparison of Friedman1 with different models

y


2500

2000

1500

1000

500

0

minus2500

minus2000

minus1500

minus1000

minus500

0 10 20 30 40 50 60 70 80 90 100

times


10 Complexity

y

0 10 20 30 40 50 60 70 80 90 100

times

Friedman3prediction bysingle ELMprediction by AdaBoostRprediction by ILES

15

10

5

0

minus5

minus10

Figure 7The comparison of Friedman3 with different models

Then the ensemble model is trained with different subtrain-ing dataset and a soft sensormodelwith incremental learningperformance is obtained by the ILES strategyWhen new dataadd up to a certain number the model will be updated usingan incremental learning strategy The new sizing percentagesoft sensor model is used in Tianjin Textile EngineeringInstitute Co LtdThe experiments demonstrate that the newsoft sensor model based on the ILES has good performanceThe predictive accuracy of the new soft sensor model couldbe satisfied for sizing production Finally the new ILES is alsotested with three testing functions to verify the performancewith different data sets for universal use Because the sizeof the subdataset is different from the experiment on sizingpercentage it can be conclude from the prediction resultsthat the size of subdataset does not affect the performanceof the algorithm In the future the ILES can also be used inother complex industry processes that require the use of softsensors

Appendix

Review of Extreme Learning Machine

Single Hidden Layer Feedforward Networks (SLFNs) withRandom Hidden Nodes

For 119873 arbitrary distinct samples (119909푗 119905푗)푁푗=1 where 119909푗 =[119909푗1 119909푗2 sdot sdot sdot 119909푗푛]푇 isin 119877푛 and 119905푗 = [119905푗1 119905푗2 sdot sdot sdot 119905푗푚]푇 isin 119877푛standard SLFNs with hidden nodes and the activationfunction 119892(119909) are mathematically modeled as푁sum푖=1

120573푖119892푖 (119909푗) = 푁sum푖=1

120573푖119892 (119908푖 sdot 119909푗 + 119887푖) = 119900푗119895 = 1 sdot sdot sdot 119873

(A1)

where 119908푖 = [119908푖1 119908푖2 sdot sdot sdot 119908푖푛]푇 is the weight vectorconnecting the 119894th hidden node and the input nodes 120573푖 =[120573푖1 120573푖2 sdot sdot sdot 120573푖푚]푇 is the weight vector connecting the 119894thhidden node and the output nodes 119900푗 = [119900푗1 119900푗2 sdot sdot sdot 119900푗푚]푇is the output vector of the SLFN and 119887푖 is the threshold ofthe ith hidden node 119908푖 sdot 119909푗 denotes the inner product of 119908푖and 119909푗 The output nodes are chosen linearly The standardSLFNswith hiddennodeswith the activation function119892(119909)can approximate these119873 samples with zero error means suchthat sum푁푗=1 119900푗 minus 119905푗 = 0 These 119873 equations can be writtencompactly as follows

H120573 = T (A2)

where

H = [[[[[

119892 (1199081 sdot 1199091 + 1198871) sdot sdot sdot 119892 (119908푁 sdot 1199091 + 119887푁) sdot sdot sdot 119892 (1199081 sdot 119909푁 + 1198871) sdot sdot sdot 119892 (119908푁 sdot 119909푁 + 119887푁)

]]]]]푁times푁

(A3)

120573 = [[[[[

120573푇1120573푇푁

]]]]]푁times푚

(A4)

and

T = [[[[[

119905푇1119905푇푁]]]]]푁times푚

(A5)

HereH is the hidden layer output matrix

Complexity 11

ELM Algorithm The parameters of the hidden nodes do notneed to be tuned and can be randomly generated perma-nently according to any continuous probability distributionThe unique smallest norm least squares solution of the abovelinear system is

1006704120573 = HdaggerT (A6)

where H+ is the Moore-Penrose generalized inverse ofmatrix H

Thus a simple learning method for SLFNs called extremelearning machine (ELM) can be summarized as follows

Step 1 Randomly assign input weight 119908푖 and bias 119887푖 119894 =1 sdot sdot sdot

Step 2 Calculate the hidden layer output matrix H

Step 3 Calculate the output weight 120573 120573 = HdaggerT

The universal approximation capability of the ELM hasbeen rigorously proved in an incremental method by Huanget al [23]

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request


The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grants nos 71602143 6140327761573086 and 51607122) Tianjin Natural Science Founda-tion (no 18JCYBJC22000) Tianjin Science and TechnologyCorrespondent Project (no 18JCTPJC62600) the Programfor Innovative Research Team in University of Tianjin (nosTD13-5038 TD13-5036) and State Key Laboratory of ProcessAutomation in Mining ampMetallurgyBeijing Key Laboratoryof Process Automation in Mining amp Metallurgy ResearchFund Project (BGRIMM-KZSKL-2017-01)

References

[1] R E Schapire ldquoThe strength of weak learnabilityrdquo MachineLearning vol 5 no 2 pp 197ndash227 1990

[2] Y Freund and R E Schapire ldquoA decision-theoretic generaliza-tion of on-line learning and an application to boostingrdquo Journalof Computer and System Sciences vol 55 no 1 pp 119ndash139 1997

[3] S Liu S Wang J Chen X Liu and H Zhou ldquoMoving larvalshrimps recognition based on improved principal componentanalysis and AdaBoostrdquo Transactions of the Chinese Society ofAgricultural Engineering (Transactions of the CSAE) vol 33 no1 pp 212ndash218 2017

[4] Y Li H Guo and X Liu ldquoClassification of an integratedalgorithm based on Boosting in unbalanced datardquo Journal ofSystems Engineering and13eory vol 01 pp 189ndash199 2016

[5] L L Wang and Z L Fu ldquoAdaBoost algorithm for multi-tagclassification based on label correlationrdquo Journal of SichuanUniversity (Engineering Science Edition) vol 5 pp 91ndash97 2016

[6] H Tian and A Wang ldquoAdvanced AdaBoost modeling methodbased on incremental learningrdquo Control and Decision vol 09pp 1433ndash1436 2012

[7] MWu J Guo Y Ju Z Lin and Q Zou ldquoParallel algorithm forparallel selection based on hierarchical filtering and dynamicupdatingrdquo Computer Science vol 44 no 1 pp 48ndash52 2017

[8] H Drucker ldquoImproving regressor using boostingrdquo in Proceed-ings of the 14th International Conference on Machine Learningpp 107ndash115 Morgan Kaufmann Publishers Burlington MassUSA 1997

[9] R Avnimelech and N Intrator ldquoBoosting regression estima-torsrdquo Neural Computation vol 11 no 2 pp 499ndash520 1999

[10] R Feely Predicting Stock Market Volatility Using Neural Net-works Trinity College Dublin 2000

[11] D L Shrestha and D P Solomatine ldquoExperiments withAdaBoostRT an improved boosting scheme for regressionrdquoNeural Computation vol 18 no 7 pp 1678ndash1710 2006

[12] H-X Tian and Z-Z Mao ldquoAn ensemble ELM based on mod-ified AdaBoostRT algorithm for predicting the temperature ofmolten steel in ladle furnacerdquo IEEE Transactions on AutomationScience and Engineering vol 7 no 1 pp 73ndash80 2010

[13] L Kao C Chiu C Lu C Chang et al ldquoA hybrid approachby integrating wavelet-based feature extraction with MARS andSVR for stock index forecastingrdquo Decision Support Systems vol54 no 3 pp 1228ndash1244 2013

[14] X Qiu P N Suganthan and G A J Amaratunga ldquoEnsembleincremental learning Random Vector Functional Link networkfor short-term electric load forecastingrdquo Knowledge-Based Sys-tems vol 145 no 4 pp 182ndash196 2018

[15] Z Zhou J Chen and Z Zhu ldquoRegularization incrementalextreme learning machine with random reduced kernel forregressionrdquo Neurocomputing vol 321 no 12 pp 72ndash81 2018

[16] R Polikar L Udpa S S Udpa and V Honavar ldquoLearn++an incremental learning algorithm for supervised neural net-worksrdquo IEEE Transactions on Systems Man and CyberneticsPart C Applications and Reviews vol 31 no 4 pp 497ndash5082001

[17] G Ditzler and R Polikar ldquoIncremental learning of conceptdrift from streaming imbalanced datardquo IEEE Transactions onKnowledge and Data Engineering vol 25 no 10 pp 2283ndash23012013

[18] D Brzezinski and J Stefanowski ldquoReacting to different types ofconcept drift the accuracy updated ensemble algorithmrdquo IEEETransactions on Neural Networks and Learning Systems vol 25no 1 pp 81ndash94 2013

[19] H-J Rong Y-X Jia andG-S Zhao ldquoAircraft recognition usingmodular extreme learning machinerdquoNeurocomputing vol 128pp 166ndash174 2014

[20] Q Lin X Wang and H-J Rong ldquoSelf-organizing fuzzy failurediagnosis of aircraft sensorsrdquoMemetic Computing vol 7 no 4pp 243ndash254 2015

[21] X Xu D Jiang B Li and Q Chen ldquoOptimization of Gaussianmixture model motion detection algorithmrdquo Journal of Chemi-cal Industry and Engineering vol 55 no 6 pp 942ndash946 2004

12 Complexity

[22] H Tian and Y Jia ldquoOnline soft measurement of sizing per-centage based on integrated multiple SVR fusion by BaggingrdquoJournal of Textile Research vol 35 no 1 pp 62ndash66 2014(Chinese)

[23] G Huang Q Zhu and C Siew ldquoExtreme learning machine anew learning scheme of feedforward neural networksrdquo in Pro-ceedings of International Joint Conference on Neural Networksvol 2 pp 985ndash990 2004

[24] N Liang G Huang H Rong et al ldquoOn-line sequential extremelearning machinerdquo in Proceedings of the Iasted InternationalConference on Computational Intelligence pp 232ndash237 DBLPCalgary Alberta Canada 2005

Research ArticleLooking for Accurate Forecasting of Copper TCRCBenchmark Levels

Francisco J D-az-Borrego Mar-a del Mar Miras-Rodr-guez and Bernabeacute Escobar-Peacuterez

Universidad de Sevilla Seville Spain

Correspondence should be addressed to Marıa del Mar Miras-Rodrıguez mmirasuses

Received 22 November 2018 Revised 24 February 2019 Accepted 5 March 2019 Published 1 April 2019


Copyright copy 2019 Francisco J Dıaz-Borrego et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Forecasting copper prices has been the objective of numerous investigations However there is a lack of research about the priceat which mines sell copper concentrate to smelters The market reality is more complex since smelters obtain the copper that theysell from the concentrate that mines produce by processing the ore which they have extracted It therefore becomes necessary tothoroughly analyse the price at which smelters buy the concentrates from the mines besides the price at which they sell the copperIn practice this cost is set by applying discounts to the price of cathodic copper the most relevant being those corresponding tothe smeltersrsquo benefit margin (Treatment Charges-TC and Refining Charges-RC) These discounts are agreed upon annually in themarkets and their correct forecasting will enable makingmore adequate models to estimate the price of copper concentrates whichwould help smelters to duly forecast their benefit margin Hence the aim of this paper is to provide an effective tool to forecastcopper TCRC annual benchmark levels With the annual benchmark data from 2004 to 2017 agreed upon during the LMECopperWeek a three-model comparison is made by contrasting different measures of error The results obtained indicate that the LES(Linear Exponential Smoothing) model is the one that has the best predictive capacity to explain the evolution of TCRC in boththe long and the short term This suggests a certain dependency on the previous levels of TCRC as well as the potential existenceof cyclical patterns in them This model thus allows us to make a more precise estimation of copper TCRC levels which makes ituseful for smelters and mining companies

1 Introduction

11 Background The relevance of copper trading is undeni-able In 2016 exports of copper ores concentrates coppermatte and cement copper increased by 15 reaching 473 b$USD while imports attained 439 b $USD [1] In additionthe global mining capacity is expected to rise by 10 from the235 million tonnes recorded in 2016 to 259 million tonnesin 2020 with smelter production having reached the recordfigure of 190 million tonnes in 2016 [2]

The worldrsquos copper production is essentially achievedthrough alternative processes which depend on the chemicaland physical characteristics of the copper ores extractedAccording to the USGSrsquo 2017 Mineral Commodity Summaryon Copper [3] global identified copper resources contained21 billion tonnes of copper as of 2014 of which about 80

are mainly copper sulphides whose copper content has to beextracted through pyrometallurgical processes [4] In 2010the average grades of ores being mined ranged from 05 to2 Cu which makes direct smelting unfeasible for economicand technical reasons So copper sulphide ores undergoa process known as froth-flotation to obtain concentratescontaining asymp 30 Cu which makes concentrates the mainproducts offered by copper mines [2 5] Concentrates arelater smelted and in most cases electrochemically refinedto produce high-purity copper cathodes (Figure 1) Coppercathodes are generally regarded as pure copper with aminimum copper content of 999935 Cu [6] Cathodesare normally produced by integrated smelters that purchaseconcentrates at a discounted price of the copper marketprice and sell cathodic copper at the market price adding apremium when possible


2 Complexity

Sulfide ores (05 - 20 Cu)

Comminution

Concentrates (20 - 30 Cu)

Other smeltingprocesseslowast

Flotation

Drying

Anodes (995 Cu)

Anode refiningand casting

Drying

Flash smelting Submerged tuyere

smelting

Matte (50-70Cu)

Converting

Blister Cu (99 Cu)

Electrorefining

Cathodes (9999 Cu)

Direct-to-copper smelting

Vertical lance smelting

Figure 1 Industrial processing of copper sulphides ore to obtain copper cathodes (Source Extractive Metallurgic of Copper pp2 [4])

12 TCRC and Their Role in Copper Concentrates TradeThe valuation of these copper concentrates is a recurrenttask undertaken by miners or traders following processes inwhich market prices for copper and other valuable metalssuch as gold and silver are involved as well as relevantdiscounts or coefficients that usually represent a major partof the revenue obtained for concentrate trading smeltingor refining The main deductions are applied to the marketvalue of the metal contained by concentrates such as CopperTreatment Charges (TC) Copper Refining Charges (RC) thePrice Participation Clause (PP) and Penalties for PunishableElements [7] These are fixed by the different parties involvedin a copper concentrates long-term or spot supply contractwhere TCRC are fixed when the concentrates are sold to acopper smelterrefinery The sum of TCRC is often viewedas the main source of revenue for copper smelters along withcopper premiums linked to the selling of copper cathodesFurthermore TCRC deductions pose a concern for coppermines as well as a potential arbitrage opportunity for traderswhose strong financial capacity and indepth knowledge of themarket make them a major player [8]

Due to their nature TCRC are discounts normallyagreed upon taking a reference which is traditionally set onan annual basis at the negotiations conducted by the major

market participants during LME Week every October andmore recently during the Asia Copper Week each Novemberan event that is focused more on Chinese smelters TheTCRC levels set at these events are usually taken as bench-marks for the negotiations of copper concentrate supplycontracts throughout the following year Thus as the yeargoes on TCRC average levels move up and down dependingon supply and demand as well as on concentrate availabilityand smeltersrsquo available capacity Consultants such as PlattsWoodMackenzie andMetal Bulletin regularly carry out theirown market surveys to estimate the current TCRC levelsFurthermoreMetal Bulletinhas created the first TCRC indexfor copper concentrates [9]

13 The Need for Accurate Predictions of TCRC The cur-rent information available for market participants may beregarded as sufficient to draw an accurate assumption ofmar-ket sentiment about current TCRC levels but not enoughto foresee potential market trends regarding these crucialdiscounts far less as a reliable tool which may be ultimatelyapplicable by market participants to their decision-makingframework or their risk-management strategies Hence froman organisational standpoint providing accurate forecastsof copper TCRC benchmark levels as well as an accurate

Complexity 3

mathematical model to render these forecasts is a researchtopic that is yet to be explored in depth This is a questionwith undeniable economic implications for traders minersand smelters alike due to the role of TCRC in the coppertrading revenue stream

Our study seeks to determine an appropriate forecastingtechnique for TCRC benchmark levels for copper concen-trates that meets the need of reliability and accuracy Toperform this research three different and frequently-appliedtechniques have been preselected from among the optionsavailable in the literature Then their forecasting accuracyat different time horizons will be tested and comparedThese techniques (Geometric Brownian Motion -GBM-the Mean Reversion -MR- Linear Exponential Smoothing-LES-) have been chosen primarily because they are commonin modelling commodities prices and their future expectedbehaviour aswell as in stock indicesrsquo predictiveworks amongother practical applications [10ndash13] The selection of thesemodels is further justified by the similarities shared byTCRCwith indices interest rates or some economic variables thatthese models have already been applied to Also in our viewthe predictive ability of these models in commodity pricessuch as copper is a major asset to take them into consider-ation The models have been simulated using historical dataof TCRC annual benchmark levels from 2004 to 2017 agreedupon during the LME Copper Week The dataset employedhas been split into two parts with two-thirds as the in-sampledataset and one-third as the out-of-sample one

The main contribution of this paper is to provide auseful and applicable tool to all parties involved in thecopper trading business to forecast potential levels of criticaldiscounts to be applied to the copper concentrates valuationprocess To carry out our research we have based ourselves onthe following premises (1)GBMwould deliver good forecastsif copper TCRC benchmark levels vary randomly over theyears (2) a mean-reverting model such as the OUP woulddeliver the best forecasts if TCRC levels were affected bymarket factors and consequently they move around a long-term trend and (3) a moving average model would givea better forecast than the other two models if there werea predominant factor related to precedent values affectingthe futures ones of benchmark TCRC In addition we havealso studied the possibility that a combination of the modelscould deliver the most accurate forecast as the time horizonconsidered is increased since there might thus be a limitedeffect of past values or of sudden shocks on future levels ofbenchmark TCRC So after some time TCRC levels couldbe ldquonormalizedrdquo towards a long-term average

The remainder of this article is structured as followsSection 2 revises the related work on commodity discountsforecasting and commodity prices forecasting techniques aswell as different forecasting methods Section 3 presents thereasoning behind the choice of each of the models employedas well as the historic datasets used to conduct the studyand the methodology followed Section 4 shows the resultsof simulations of different methods Section 5 indicates errorcomparisons to evaluate the best forecasting alternative forTCRC among all those presented Section 6 contains thestudyrsquos conclusions and proposes further lines of research

2 Related Work

The absence of any specific method in the specialised litera-ture in relation to copper TCRC leads us to revisit previousliterature in order to determine the most appropriate modelto employ for our purpose considering those that havealready been used in commodity price forecasting as thelogical starting point due to the application similarities thatthey share with ours

Commodity prices and their forecasting have been atopic intensively analysed in much research Hence there aremultiple examples in literature with an application to oneor several commodities such as Xiong et al [14] where theaccuracy of differentmodelswas tested to forecast the intervalof agricultural commodity future prices Shao and Dai [15]whose work employs the Autoregressive Integrated MovingAverage (ARIMA) methodology to forecast food crop pricessuch as wheat rice and corn and Heaney [16] who tests thecapacity of commodities future prices to forecast their cashprice were the cost of carry to be included in considerationsusing the LME Lead contract as an example study As wasdone in other research [17ndash19] Table 1 summarizes similarresearch in the field of commodity price behaviours

From the summary shown in Table 1 it is seen that mov-ing average methods have become increasingly popular incommodity price forecastingThemost broadly implementedmoving average techniques are ARIMA and ExponentialSmoothing They consider the entire time series data anddo not assign the same weight to past values than thosecloser to the present as they are seen as affecting greater tofuture valuesThe Exponential Smoothingmodelsrsquo predictiveaccuracy has been tested [20 21] concluding that there aresmall differences between them (Exponential Smoothing)and ARIMA models

Also GBM andMRmodels have been intensively appliedto commodity price forecasting Nonetheless MR modelspresent a significant advantage over GBM models whichallows them to consider the underlying price trend Thisadvantage is of particular interest for commodities thataccording to Dixit and Pindyck [22]mdashpp 74 regarding theprice of oil ldquoin the short run it might fluctuate randomly upand down (in responses to wars or revolutions in oil producingcountries or in response to the strengthening or weakeningof the OPEC cartel) in the longer run it ought to be drawnback towards the marginal cost of producing oil Thus onemight argue that the price of oil should be modelled as a mean-reverting processrdquo

3 Methodology

This section presents both the justification of the modelschosen in this research and the core reference dataset used asinputs for each model compared in the methodology as wellas the steps followed during the latter to carry out an effectivecomparison in terms of the TCRC forecasting ability of eachof these models

31 Models in Methodology GBM (see Appendix A for mod-els definition) has been used in much earlier research as a

4 Complexity

Table 1 Summary of previous literature research

AUTHOR RESEARCH

Shafiee amp Topal [17] Validates a modified version of the long-term trend reverting jump and dip diffusion model for forecastingcommodity prices and estimates the gold price for the next 10 years using historical monthly data

Li et al [18] Proposes an ARIMA-Markov Chain method to accurately forecast mineral commodity prices testing themethod using mineral molybdenum prices

Issler et al [19]Investigates several commoditiesrsquo co-movements such as Aluminium Copper Lead Nickel Tin and Zinc atdifferent time frequencies and uses a bias-corrected average forecast method proposed by Issler and Lima [43]to give combined forecasts of these metal commodities employing RMSE as a measure of forecasting accuracy

Hamid amp Shabri [44]Models palm oil prices using the Autoregressive Distributed Lag (ARDL) model and compares its forecastingaccuracy with the benchmark model ARIMA It uses an ARDL bound-testing approach to co-integration in

order to analyse the relationship between the price of palm oil and its determinant factorsDuan et al [45] Predicts Chinarsquos crude oil consumption for 2015-2020 using the fractional-order FSIGMmodel

Brennan amp Schwartz [46] Employs the Geometric Brownian Motion (GBM) to analyse a mining projectrsquos expected returns assuming itproduces a single commodity

McDonald amp Siegel [47] Uses GBM to model the random evolution of the present value of an undefined asset in an investment decisionmodel

Zhang et al [48] Models gold prices using the Ornstein-Uhlenbeck Process (OUP) to account for a potentially existent long-termtrend in a Real Option Valuation of a mining project

Sharma [49] Forecasts gold prices in India with the Box Jenkins ARIMA method

way of modelling prices that are believed not to follow anyspecific rule or pattern and hence seen as random Blackand Scholes [23] first used GBM to model stock prices andsince then others have used it to model asset prices as well ascommodities these being perhaps themost common of all inwhich prices are expected to increase over time as does theirvariance [11] Hence following our first premise concerningwhether TCRC might vary randomly there should not exista main driving factor that would determine TCRC futurebenchmark levels and therefore GBM could to a certainextent be a feasible model for them

However although GBM or ldquorandom walkrdquo may be wellsuited to modelling immediate or short-term price paths forcommodities or for TCRC in our case it lacks the abilityto include the underlying long-term price trend should weassume that there is oneThus in accordance with our secondpremise on benchmark TCRC behaviour levels would movein line with copper concentrate supply and demand as wellas the smeltersrsquo and refineriesrsquo available capacity to trans-form concentrates into metal copper Hence a relationshipbetween TCRC levels and copper supply and demand isknown to exist and therefore is linked to its market priceso to some extent they move together coherently Thereforein that case as related works on commodity price behavioursuch as Foo Bloch and Salim [24] do we have opted for theMR model particularly the OUP model

Both GBM and MR are Markov processes which meansthat future values depend exclusively on the current valuewhile the remaining previous time series data are not consid-eredOn the other handmoving averagemethods employ theaverage of a pre-established number of past values in differentways evolving over time so future values do not rely exclu-sively on the present hence behaving as though they had onlya limitedmemory of the pastThis trait of themoving averagemodel is particularly interestingwhen past prices are believedto have a certain though limited effect on present values

which is another of the premises for this research Existingalternatives ofmoving average techniques pursue consideringthis ldquomemoryrdquo with different approaches As explained byKalekar [25] Exponential Smoothing is suitable only for thebehaviours of a specific time series thus Single ExponentialSmoothing (SES) is reasonable for short-term forecastingwith no specific trend in the observed data whereas DoubleExponential Smoothing or Linear Exponential Smoothing(LES) is appropriate when data shows a cyclical pattern ora trend In addition seasonality in observed data can becomputed and forecasted through the usage of ExponentialSmoothing by the Holt-Winters method which adds an extraparameter to the model to handle this characteristic

32 TCRC Benchmark Levels and Sample Dataset TCRClevels for copper concentrates continuously vary through-out the year relying on private and individual agreementsbetween miners traders and smelters worldwide Nonethe-less the TCRC benchmark fixed during the LME week eachOctober is used by market participants as the main referenceto set actual TCRC levels for each supply agreed upon for thefollowing year Hence the yearrsquos benchmark TCRC is takenhere as a good indicator of a yearrsquos TCRC average levelsAnalysed time series of benchmark TCRC span from 2004through to 2017 as shown in Table 2 as well as the sourceeach valuewas obtained fromWe have not intended to reflectthe continuous variation of TCRC for the course of anygiven year though we have however considered benchmarkprices alone as we intuitively assume that annual variations ofactual TCRC in contracts will eventually be reflected in thebenchmark level that is set at the end of the year for the yearto come

33 TCRC Relation TC and RC normally maintain a 101relation with different units with TC being expressed in US

Complexity 5

GBM RC First 20 pathsGBM TC First 20 paths

0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012

GBM TC First 20 paths - Year Averaged

05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012

Observed TC Benchmark Levels

2013 2014 2015 2016 20172012

Observed RC Benchmark Levels

2013 2014 2015 2016 20172012

GBM RC First 20 paths - Year Averaged

Figure 2 The upper illustrations show the first 20 Monte Carlo paths for either TC or RC levels using monthly steps The illustrations belowshow the annual averages of monthly step forecasts for TC and RC levels

MR TC First 20 paths

2012 2013 2014 2015 2016 2017

MR RC First 20 paths

MR RC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012

MR TC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)

Figure 3 The first 20 Monte Carlo paths following the OUP model using monthly steps are shown on the upper illustrations for either TCor RC The annual averaged simulated paths every 12 values are shown below

dollars per metric tonne and RC in US cents per pound ofpayable copper content in concentrates In fact benchmarkdata show that of the last 14 years it was only in 2010that values of TC and RC did not conform to this relation

though it did remain close to it (46547) Nonetheless thisrelation has not been observed in the methodology hereinthus treating TC and RC independently developing separateunrelated forecasts for TC and RC to understand whether

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017


Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013




Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018


2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)



the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability



The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

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[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 15

[6] The London Metal Exchange ldquoSpecial contract rules for cop-per grade Ardquo httpswwwlmecomen-GBTradingBrandsComposition

[7] U Soderstrom ldquoCopper smelter revenue streamrdquo 2008[8] F K Crundwell Finance for Engineers (Evaluation and Funding

of Capital Projects) 2008[9] Metal Bulletin TCRC Index ldquoMeteoritical Bulletinrdquo https

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[10] J Savolainen ldquoReal options in metal mining project valuationreview of literaturerdquo Resources Policy vol 50 pp 49ndash65 2016

[11] W J Hahn J A DiLellio and J S Dyer ldquoRisk premia incommodity price forecasts and their impact on valuationrdquoEnergy Econ vol 7 p 28 2018

[12] C Alberto C Pinheiro and V Senna ldquoPrice forecastingthrough multivariate spectral analysis evidence for commodi-ties of BMampFbovespardquo Brazilian Business Review vol 13 no 5pp 129ndash157 2016

[13] Z Hlouskova P Zenıskova and M Prasilova ldquoComparison ofagricultural costs prediction approachesrdquo AGRIS on-line Papersin Economics and Informatics vol 10 no 1 pp 3ndash13 2018

[14] T Xiong C Li Y Bao Z Hu and L Zhang ldquoA combinationmethod for interval forecasting of agricultural commodityfutures pricesrdquo Knowledge-Based Systems vol 77 pp 92ndash1022015

[15] Y E Shao and J-T Dai ldquoIntegrated feature selection of ARIMAwith computational intelligence approaches for food crop pricepredictionrdquo Complexity vol 2018 Article ID 1910520 17 pages2018

[16] R Heaney ldquoDoes knowledge of the cost of carrymodel improvecommodity futures price forecasting ability A case study usingthe london metal exchange lead contractrdquo International Journalof Forecasting vol 18 no 1 pp 45ndash65 2002

[17] S Shafiee and E Topal ldquoAn overview of global gold market andgold price forecastingrdquo Resources Policy vol 35 no 3 pp 178ndash189 2010

[18] Y Li N Hu G Li and X Yao ldquoForecastingmineral commodityprices with ARIMA-Markov chainrdquo in Proceedings of the 20124th International Conference on Intelligent Human-MachineSystems andCybernetics (IHMSC) pp 49ndash52Nanchang ChinaAugust 2012

[19] J V Issler C Rodrigues and R Burjack ldquoUsing commonfeatures to understand the behavior of metal-commodity pricesand forecast them at different horizonsrdquo Journal of InternationalMoney and Finance vol 42 pp 310ndash335 2014

[20] S Makridakis M Hibon and C Moser ldquoAccuracy of forecast-ing an empirical investigationrdquo Journal of the Royal StatisticalSociety Series A (General) vol 142 no 2 p 97 1979

[21] S Makridakis A Andersen R Carbone et al ldquoThe accuracyof extrapolation (time series) methods Results of a forecastingcompetitionrdquo Journal of Forecasting vol 1 no 2 pp 111ndash1531982

[22] A K Dixit and R S Pindyck Investment under uncertainty vol256 1994

[23] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[24] N Foo H Bloch and R Salim ldquoThe optimisation rule forinvestment in mining projectsrdquo Resources Policy vol 55 pp123ndash132 2018

[25] P Kalekar ldquoTime series forecasting using Holt-Wintersexponential smoothingrdquo Kanwal Rekhi School of InformationTechnology Article ID 04329008 pp 1ndash13 2004 httpwwwitiitbacinsimprajacadsseminar04329008 Exponential-Smoothingpdf

[26] L Gomez-Valle and J Martınez-Rodrıguez ldquoAdvances in pric-ing commodity futures multifactor modelsrdquoMathematical andComputer Modelling vol 57 no 7-8 pp 1722ndash1731 2013

[27] M Khashei and M Bijari ldquoA novel hybridization of artificialneural networks and ARIMA models for time series forecast-ingrdquo Applied Soft Computing vol 11 no 2 pp 2664ndash2675 2011

[28] M Khashei and M Bijari ldquoAn artificial neural network (pd q) model for timeseries forecastingrdquo Expert Systems withApplications vol 37 no 1 pp 479ndash489 2010

[29] A Choudhury and J Jones ldquoCrop yield prediction using timeseries modelsrdquo The Journal of Economic Education vol 15 pp53ndash68 2014

[30] R R Marathe and S M Ryan ldquoOn the validity of the geometricbrownian motion assumptionrdquoThe Engineering Economist vol50 no 2 pp 159ndash192 2005

[31] G E Uhlenbeck and L S Ornstein ldquoOn the theory of theBrownian motionrdquo Physical Review A Atomic Molecular andOptical Physics vol 36 no 5 pp 823ndash841 1930

[32] A Tresierra Tanaka andCM CarrascoMontero ldquoValorizacionde opciones reales modelo ornstein-uhlenbeckrdquo Journal ofEconomics Finance and Administrative Science vol 21 no 41pp 56ndash62 2016

[33] R Nau Forecasting with Moving Averages Dukersquos FuquaSchool of Business Duke University Durham NC USA 2014httppeopledukeedu128simrnauNotes on forecasting withmoving averagesndashRobert Naupdf

[34] J M Woolridge Introductory Econometrics 2011[35] E S Gardner Jr ldquoExponential smoothing the state of the art-

part IIrdquo International Journal of Forecasting vol 22 no 4 pp637ndash666 2006

[36] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Dover Phoenix Editions Dover Publications 2ndedition 1963

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[38] P J Harrison ldquoExponential smoothing and short-term salesforecastingrdquo Management Science vol 13 no 11 pp 821ndash8421967

[39] D C Montgomery and L A Johnson Forecasting and TimeSeries Analysis New York NY USA 1976

[40] C Chatfield ldquoThe holt-winters forecasting procedurerdquo Journalof Applied Statistics vol 27 no 3 p 264 1978

[41] J O McClain and L JThomas ldquoResponse-variance tradeoffs inadaptive forecastingrdquo Operations Research vol 21 pp 554ndash5681973

[42] J O McClain ldquoDynamics of exponential smoothing with trendand seasonal termsrdquo Management Science vol 20 no 9 pp1300ndash1304 1974

[43] J V Issler and L R Lima ldquoA panel data approach to economicforecasting the bias-corrected average forecastrdquo Journal ofEconometrics vol 152 no 2 pp 153ndash164 2009

[44] M F Hamid and A Shabri ldquoPalm oil price forecasting modelan autoregressive distributed lag (ARDL) approachrdquo in Pro-ceedings of the 3rd ISM International Statistical Conference 2016(ISM-III) Bringing Professionalism and Prestige in StatisticsKuala Lumpur Malaysia 2017

16 Complexity

[45] H Duan G R Lei and K Shao ldquoForecasting crude oilconsumption in China using a grey prediction model with anoptimal fractional-order accumulating operatorrdquo Complexityvol 2018 12 pages 2018

[46] M J Brennan and E S Schwartz ldquoEvaluating natural resourceinvestmentsrdquoThe Journal of Business vol 58 no 2 pp 135ndash1571985

[47] RMcDonald and D Siegel ldquoThe value of waiting to investrdquoTheQuarterly Journal of Economics vol 101 no 4 pp 707ndash727 1986

[48] K Zhang A Nieto and A N Kleit ldquoThe real option valueof mining operations using mean-reverting commodity pricesrdquoMineral Economics vol 28 no 1-2 pp 11ndash22 2015

[49] R K Sharma ldquoForecasting gold price with box jenkins autore-gressive integratedmoving average methodrdquo Journal of Interna-tional Economics vol 7 pp 32ndash60 2016

[50] J Johansson ldquoBoliden capital markets day 2007rdquo 2007[51] N Svante ldquoBoliden capital markets day September 2011rdquo 2011[52] Teck ldquoTeck modelling workshop 2012rdquo 2012[53] A Shaw ldquoAngloAmerican copper market outlookrdquo in Proceed-

ings of the Copper Conference Brisbane vol 24 Nanning China2015 httpwwwminexconsultingcompublicationsCRU

[54] P Willbrandt and E Faust ldquoAurubis DVFA analyst conferencereport 2013rdquo 2013


[56] A G Aurubis ldquoAurubis copper mail December 2015rdquo 2015[57] D B Drouven and E Faust ldquoAurubis DVFA analyst conference

report 2015rdquo 2015[58] A G Aurubis ldquoAurubis copper mail February 2016rdquo 2016[59] EY ldquoEY mining and metals commodity briefcaserdquo 2017[60] J Schachler Aurubis AG interim report 2017 Frankfurt Ger-

many 2017[61] Y Nakazato ldquoAurubis AG interim reportrdquo 2017

Research ArticleThe Bass Diffusion Model on Finite Barabasi-Albert Networks

M L Bertotti and G Modanese

Free University of Bolzano-Bozen Faculty of Science and Technology 39100 Bolzano Italy

Correspondence should be addressed to G Modanese GiovanniModaneseunibzit

Received 14 November 2018 Accepted 3 March 2019 Published 1 April 2019


Copyright copy 2019 M L Bertotti and G Modanese This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Using a heterogeneous mean-field network formulation of the Bass innovation diffusion model and recent exact results on thedegree correlations of Barabasi-Albert networks we compute the times of the diffusion peak and compare them with those onscale-free networks which have the same scale-free exponent but different assortativity properties We compare our results withthose obtained for the SIS epidemic model with the spectral method applied to adjacency matrices It turns out that diffusiontimes on finite Barabasi-Albert networks are at a minimumThis may be due to a little-known property of these networks whereasthe value of the assortativity coefficient is close to zero they look disassortative if one considers only a bounded range of degreesincluding the smallest ones and slightly assortative on the range of the higher degreesWe also find that if the trickle-down characterof the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model)the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger reaching for instancethe 34 in a typical case on a network with 104 nodes

1 Introduction

The study of epidemic diffusion is one of the most importantapplications of network theory [1ndash3] the absence of anepidemic threshold on scale-free networks being perhapsthe best known result [4] This result essentially also holdsfor networks with degree correlations [5] although someexceptions have been pointed out [6 7] In [8] the depen-dence of the epidemic threshold and diffusion time onthe network assortativity was investigated using a degree-preserving rewiring procedure which starts from a Barabasi-Albert network and analysing the spectral properties of theresulting adjacency matrices In this paper we mainly focuson Barabasi-Albert (BA) networks [9] using the exact resultsby Fotouhi and Rabbat on the degree correlations [10] Weemploy the mean-field method [11] and our network formu-lation of the Bass diffusion model for the description of theinnovation diffusion process [12 13] We solve numericallythe equations and find the time of the diffusion peak animportant turning point in the life cycle of an innovationfor values of the maximum network degree 119899 of the orderof 102 which correspond to medium-size real networks with

119873 ≃ 104 nodes Then we compare these diffusion times withthose of networks with different kinds of degree correlations

In Section 2 as a preliminary to the analysis of diffusionwe provide the mathematical expressions of the averagenearest neighbor degree function 119896119899119899(119896) and the Newmanassortativity coefficient 119903 [14] of BA networks

In Section 3 we write the network Bass equations and givethe diffusion times found from the numerical solutions Wecompare these times with those for uncorrelated scale-freenetworks with exponent 120574 = 3 and with those for assortativenetworks whose correlation matrices are mathematicallyconstructed and studied in another work Then we apply amethod proposed by Newman in [15] to build disassortativecorrelation matrices and evaluate the corresponding diffu-sion timesOur results are in qualitative agreementwith thoseby DrsquoAgostino et al [8] for the SIS model The minimumdiffusion time is obtained for the BA networks which havewith 119899 ≃ 102 (corresponding to 119873 ≃ 104 nodes seeSection 4) 119903 ≃ minus010 The minimum value of 119903 obtained forthe disassortative networks (with 120574 = 3) is 119903 ≃ minus009 also notfar from the values obtained in [8] For assortative networkson the contrary values of 119903 much closer to the maximum119903 = 1 are easily obtained


2 Complexity

Section 4 contains a discussion of the results obtained forthe 119903 coefficient and the 119896119899119899 function of BA networks

In Section 5 we further reexamine the validity of themean-field approximation also by comparison to a versionof the Bass model based on the adjacency matrix of BAnetworks

In Section 6 we compute diffusion times with heteroge-neous 119901 coefficients (ldquotrickle-downrdquo modified Bass model)

In summary our research objectives in this work havebeen the following (a) Analyse the assortativity propertiesof BA networks using the correlation functions recentlypublished These properties are deduced from the Newmancoefficient 119903 and from the average nearest neighbor degreefunction 119896119899119899(119896) (b) Compute the peak diffusion times of thenetwork Bass model in the mean-field approximation on BAnetworks and compare these times with those obtained forother kinds of networks (c) Briefly discuss the validity ofthe mean-field approximation compared to a first-momentclosure of the Bass model on BA networks described throughthe adjacency matrix

Section 7 contains our conclusions and outlook

2 The Average Nearest Neighbor DegreeFunction of Barabasi-Albert (BA) Networks

We emphasize that for the networks considered in this papera maximal value 119899 isin N is supposed to exist for the admissiblenumber of links emanating from each node Notice that asshown later in the paper (see Section 41 4th paragraph) thisis definitely compatible with a high number119873 of nodes in thenetwork (eg with a number 119873 differing from 119899 by variousmagnitude orders)

Together with the degree distribution 119875(119896) whichexpresses the probability that a randomly chosen node has119896 links other important statistical quantities providinginformation on the network structure are the degreecorrelations 119875(ℎ | 119896) Each coefficient 119875(ℎ | 119896) expresses theconditional probability that a node with 119896 links is connectedto one with ℎ links In particular an increasing character ofthe average nearest neighbor degree function

119896119899119899 (119896) = 119899sumℎ=1

ℎ119875 (ℎ | 119896) (1)

is a hallmark of assortative networks (ie of networksin which high degree nodes tend to be linked to otherhigh degree nodes) whereas a decreasing character of thisfunction is to be associated with disassortative networks(networks in which high degree nodes tend to be linked tolow degree nodes) We recall that the 119875(119896) and 119875(ℎ | 119896)mustsatisfy besides the positivity requirements

119875 (119896) ge 0and 119875 (ℎ | 119896) ge 0 (2)

both the normalizations

119899sum119896=1

119875 (119896) = 1and119899sumℎ=1

119875 (ℎ | 119896) = 1(3)

and the Network Closure Condition (NCC)

ℎ119875 (119896 | ℎ) 119875 (ℎ) = 119896119875 (ℎ | 119896) 119875 (119896)forallℎ 119896 = 119894 = 1 119899 (4)

A different tool usually employed to investigate structuralproperties of networks is theNewman assortativity coefficient119903 also known as the Pearson correlation coefficient [14]To define it we need to introduce the quantities 119890119895119896 whichexpress the probability that a randomly chosen edge linksnodes with excess degree 119895 and 119896 Here the excess degree ofa node is meant as its total degree minus one namely as thenumber of all edges emanating from the node except the oneunder considerationThe distribution of the excess degrees iseasily found to be given [14] by

119902119896 = (119896 + 1)sum119899119895=1 119895119875 (119895)119875 (119896 + 1) (5)

The assortativity coefficient 119903 is then defined as

119903 = 11205902119902119899minus1sum119896ℎ=0

119896ℎ (119890119896ℎ minus 119902119896119902ℎ) (6)

where 120590119902 denotes the standard deviation of the distribution119902(119896) ie1205902119902 = 119899sum119896=1

1198962119902119896 minus ( 119899sum119896=1

119896119902119896)2 (7)

The coefficient 119903 takes values in [minus1 1] and it owes its nameto the fact that if 119903 lt 0 the network is disassortative if119903 = 0 the network is neutral and if 119903 gt 0 the network isassortative A formula expressing the 119890119896ℎ in terms of knownquantities is necessary if one wants to calculate 119903 This can beobtained as discussed next Let us move from the elements119864119896ℎ expressing the number of edges which link nodes withdegree 119896 and ℎ with the only exception that edges linkingnodes with the same degree have to be counted twice [16 17]Define now 119890119896ℎ = 119864119896ℎ(sum119896ℎ 119864119896ℎ) Each 119890119896ℎ corresponds thento the fraction of edges linking nodes with degree 119896 andℎ (with the mentioned interpretation of the 119890119896119896) We alsoobserve that 119890119896ℎ = 119890119896+1ℎ+1 holds trueThedegree correlationscan be related to the 119890119896ℎ through the formula

119875 (ℎ | 119896) = 119890119896ℎsum119899119895=1 119890119896119895 forallℎ 119896 = 119894 = 1 119899 (8)

What is of interest for us here is the following ldquoinverserdquoformula

119890ℎ119896 = 119875 (ℎ | 119896) 119896119875 (119896)sum119899119895=1 119895119875 (119895) (9)

Complexity 3

In the rest of this section we discuss the quantities whichallow us to calculate the average nearest neighbor degreefunction 119896119899119899(119896) and the coefficient 119903 for finite Barabasi-Albert(BA) networks

The degree distribution of the Barabasi-Albert networksis known to be given by

119875 (119896) = 2120573 (120573 + 1)119896 (119896 + 1) (119896 + 2) (10)

where 120573 ge 1 is the parameter in the preferential attachmentprocedure characterizing them [9 17] In particular (10)yields 119875(119896) sim 1198881198963 with a suitable constant 119888 for large 119896

An explicit expression for the degree correlations119875(ℎ | 119896)was given by Fotouhi and Rabbat in [10] They showed thatfor a growing network in the asymptotic limit as 119905 997888rarr infin

119875 (ℎ | 119896) = 120573119896ℎ (119896 + 2ℎ + 1 minus 1198612120573+2120573+1119861119896+ℎminus2120573ℎminus120573119861119896+ℎ+2ℎ

) (11)

with 119861119898119895 denoting the binomial coefficient

119861119898119895 = 119898119895 (119898 minus 119895) (12)

Since the networks we consider have a maximal number oflinks 119899 [18] wemust normalize thematrixwith elements119875(ℎ |119896) We calculate 119862119896 = sum119899ℎ=1 119875(ℎ | 119896) and take as a new degreecorrelation matrix the matrix whose (ℎ 119896)-element is

119875119899 (ℎ | 119896) = 119875 (ℎ | 119896)119862119896 (13)

with 119875(ℎ | 119896) as in (11)The average nearest neighbor degree function 119896119899119899(119896) and

the coefficient 119903 can be now easily calculated with softwarelike Mathematica by using (1) (6) (10) and (13) Resultsare reported and discussed in Section 4

3 The Bass Diffusion Equation onComplex Networks

In [12 13] we have reformulated the well-known Bass equa-tion of innovation diffusion

119889119865 (119905)119889119905 = [1 minus 119865 (119905)] [119901 + 119902119865 (119905)] (14)

(where 119865(119905) is the cumulative adopter fraction at the time 119905and 119901 and 119902 are the innovation and the imitation coefficientrespectively) providing versions suitable for the case inwhichthe innovation diffusion process occurs on a network Themodel can be expressed in such a case by a system of 119899ordinary differential equations

119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (119905)]119894 = 1 119899

(15)

The quantity 119866119894(119905) in (15) represents for any 119894 = 1 119899the fraction of potential adopters with 119894 links that at thetime 119905 have adopted the innovation More precisely denotingby 119865119894(119905) the fraction of the total population composed byindividuals with 119894 links who at the time 119905 have adopted theinnovation we set 119866119894(119905) = 119865119894(119905)119875(119894) Further heterogeneitycan be introduced allowing also the innovation coefficient(sometimes called publicity coefficient) to be dependent on119894 In this case the equations take the form

119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901119894 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (t)]119894 = 1 119899

(16)

and for example the 119901119894 can be chosen to be inversely propor-tional to 119875(119894) or to have a linear dependence decreasing in 119894in the first case more publicity is delivered to the hubs withensuing ldquotrickle-downrdquo diffusion while in the second case aldquotrickle-uprdquo diffusion from the periphery of the network canbe simulated

The function 119891119894(119905) = 119894(119905) gives the fraction of newadoptions per unit time in the ldquolink class 119894rdquo (or ldquodegree class119894rdquo) ie in the subset of individuals having 119894 links The leftpanel in Figure 1 shows an example of a numerical solutionwith plots of all the 119891119894rsquos in a case where for graphical reasonswe have taken 119899 small (119899 = 15) The underlying networkis a BA with 120573 = 1 As is clear from the plot the largestfraction of new adopters belongs at all times to the link classwith 119894 = 1 which reaches its adoption peak later than theothers In general the more connected individuals are theearlier they adoptThis phenomenon is quite intuitive and hasbeen evidenced in previous works on mean-field epidemicmodels see for instance [19] As discussed in our paper[13] in applications of the Bass model to marketing this mayallow to estimate the 119902 coefficient when it is not known inadvance by monitoring the early occurrence of adoption inthe most connected classes The right panel in Figure 1 showsthe total adoption rate119891(119905) corresponding to the same case asin the left panelThe simple Bass curve (homogeneousmodelwithout underlying network) is also shown for comparison

In the Bass model unlike in other epidemic modelswhere infected individuals can return to the susceptible statethe diffusion process always reaches all the population Thefunction119891(119905) = sum119899119894=1 119894(119905) which represents the total numberof new adoptions per unit time usually has a peak as we haveseen in the previous exampleWe choose the time of this peakas a measure of the diffusion time it is computed for eachnumerical solution of the diffusion equations by samplingthe function 119891(119905) For fixed coefficients of publicity 119901 andimitation 119902 the time depends on the features of the networkIn this paper we consider only scale-free networks with 120574 = 3for comparison with BA networks

Figure 2 shows the peak times obtained for differentnetworks with maximum degree 119899 = 100 as a function ofthe imitation parameter 119902 This value of 119899 has been chosenbecause it corresponds to a number 119873 of nodes of the orderof 104 this allows a comparisonwith the results of DrsquoAgostinoet al (see below) and displays finite-size effects as discussed

4 Complexity

t t

ftot

002

004

006

008

002

004

006

008

010

012

20 30 40 5010 20 30 40 5010

fi

Figure 1 Left panel fraction 119891119894 of new adoptions per unit time in the ldquolink class 119894rdquo (the subset of all individuals having 119894 links) as a functionof time in the Bass innovation diffusion model on a BA network The parameter 120573 of the network (number of child nodes in the preferentialattachment scheme) is 120573 = 1 The maximum number of links in this example is 119899 = 15 while in the comprehensive numerical solutionswhose results are summarized in Figure 2 it is 119899 = 100 The adoption peak occurs later for the least connected (and most populated) classwith 119894 = 1 Right panel cumulative new adoptions 119891119905119900119905 = sum119899119894=1 119891119894 per unit time as a function of time compared with the same quantity forthe homogeneous Bass model without an underlying network The peak of the homogeneous Bass model is slightly lower and shifted to theright For the values of the model parameters 119901 and 119902 and the measuring unit of time see Figure 2

q = 03 q = 048

= 1

= 2

= 3

= 5

Assortative

DisassortativeUncorrelated

= 4

DIFFUSION TIME

BASS q PARAMETER

2 4 6 8 10

40

45

50

55

60

Figure 2 Time of the diffusion peak (in years) for the Bass model on different kinds of scale-free networks with exponent 120574 = 3 as afunction of the imitation coefficient 119902 All networks have maximum degree 119899 = 100 (119873 ≃ 104)The 119902 coefficient varies in the range 03minus048corresponding to a typical set of realistic values in innovation diffusion theory [20] The publicity coefficient is set to the value 119901 = 003 (seeFigure 9 for results with an inhomogeneous 119901119896 depending on the link class 119896 = 1 119899) The lines with 120573 = 1 2 3 4 5 correspond to BAnetworks with those values of120573Their assortativity coefficients are respectively 119903 = minus0104 minus0089 minus0078 minus0071 minus0065The disassortativenetwork is built with the Newman recipe (Section 34) with 119889 = 4 and has 119903 = minus0084 The assortative network is built with our recipe(Section 33) with 120572 = 12 and has 119903 = 0863

in Section 4 (note however that such effects are still presentwith119873 ≃ 106 and larger)

It is clear from Figure 2 that diffusion is faster on the BAnetworks with 120573 = 1 2 3 than on an uncorrelated networkFor 120573 = 4 the diffusion time is almost the same and for 120573 = 5(and 120573 gt 5 not shown) diffusion on the BA network is slowerthan on the uncorrelated network On purely disassortativenetworks diffusion is slightly slower than on an uncorrelatednetwork and much slower than on assortative networks

31 Comparison with the SIS Model In [8] DrsquoAgostino et alhave studied the dependence of the epidemic threshold anddiffusion time for the SIS epidemic model on the assortativeor disassortative character of the underlying network Boththe epidemic threshold and the diffusion time were evaluatedfrom the eigenvalues of the adjacency matrix The networksemployed had a number of nodes119873 = 104 a scale-free degree

distribution with exponent 120574 = 3 and were obtained from aBA seed network through a Monte Carlo rewiring procedurewhich preserves the degree distribution but changes thedegree correlations The Monte Carlo algorithm employsan ldquoHamiltonianrdquo function which is related to the Newmancorrelation coefficient 119903 The values of 119903 explored in this waylie in the range from -015 to 05 and are therefore comparablewith those obtained for our assortative and disassortativematrices

Although the epidemic model considered and the defini-tion of the diffusion time adopted in [8] are different fromours there is a qualitative agreement in the conclusions thediffusion time increases with the assortativity of the networkand is at a minimum for values of 119903 approximately equalto -010 This value of 119903 corresponds to those of finite BAnetworks with 120573 = 1 and 119873 ≃ 104 and is slightly smallerthan the minimum value of 119903 obtained for disassortativenetworks with a matrix 119890119895119896 built according to Newmanrsquos

Complexity 5

recipe (Section 34) Note that for those networks the function119896119899119899(119896) is decreasing for any 119896 while for BA networks it isdecreasing at small values of 119896 and increasing at large 119896(Section 4) nevertheless their 119903 coefficient never becomessignificantly less than ≃ minus01 even with other choices in therecipe as long as 120574 = 3 We also recall that the clusteringcoefficient of the BA networks is exactly zero for 120573 = 1 foran analysis of the role of the clustering coefficient in epidemicspreading on networks see for instance [21]

A rewiring procedure comparable to that of [8] (withoutevaluation of the epidemic threshold and diffusion time) hasbeen mathematically described by Van Mieghem et al [22]

In the next subsections we provide information on thedegree correlation matrices and other features of the net-works used above for comparison with the BA networks Tocompare diffusion peak times on networks which althoughcharacterized by different assortativitydisassortativity prop-erties share some similarity we consider networks whosedegree distribution 119875(119896) obeys a power-law with exponentthree ie is of the form

119875 (119896) = 1198881198963 (17)

where 119888 is the normalization constant

32 Uncorrelated Networks Let us start with uncorrelatednetworks As their name suggests in these networks thedegree correlations 119875(ℎ | 119896) do not depend on 119896 They canbe easily seen to be given by

119875 (ℎ | 119896) = ℎ119875 (ℎ)⟨ℎ⟩ (18)

with ⟨ℎ⟩ = sum119899ℎ=1 ℎ119875(ℎ) and hence their average nearestneighbor degree function (1) is

119896119899119899 (119896) = ⟨ℎ2⟩⟨ℎ⟩ (19)

a constant The coefficient 119903 is trivially found to be equal tozero

33 A Family of Assortative Networks We consider now afamily of assortative networks we have introduced in [12] Togive here the expressions of their degree correlations119875(ℎ | 119896)we need to recall their construction We start defining theelements of a 119899 times 119899matrix 1198750 as

1198750 (ℎ | 119896) = |ℎ minus 119896|minus120572 if ℎ lt 119896and 1198750 (ℎ | 119896) = 1 if ℎ = 119896 (20)

for some parameter 120572 gt 0 and we define the elements 1198750(ℎ |119896)with ℎ gt 119896 in such a way that formula (4) is satisfied by the1198750(ℎ | 119896) Hence since the normalization sum119899ℎ=1 1198750(ℎ | 119896) = 1has to hold true we compute for any 119896 = 1 119899 the sum119862119896 = sum119899ℎ=1 1198750(ℎ | 119896) and call 119862119898119886119909 = max119896=1119899119862119896 Thenwe introduce a new matrix 1198751 requiring that its diagonalelements be given by 1198751(119896 | 119896) = 119862119898119886119909 minus 119862119896 for any 119896 =

1 119899 whereas the nondiagonal elements are the same asthose of the matrix 1198750 1198751(ℎ | 119896) = 1198750(ℎ | 119896) for ℎ = 119896 Forany 119896 = 1 119899 the column sum sum119899ℎ=1 1198751(ℎ | 119896) is then equalto 119862119896 minus 1 + 119862119898119886119909 minus 119862119896 = 119862119898119886119909 minus 1 Finally we normalize theentire matrix by setting

119875 (ℎ | 119896) = 1(119862119898119886119909 minus 1)1198751 (ℎ | 119896) for ℎ 119896 = 1 119899 (21)

Again the average nearest neighbor degree function 119896119899119899(119896)and the coefficient 119903 can be calculated with a software Theincreasing character of 119896119899119899(119896) for a network of the family inthis subsection with 120572 = 12 and 119899 = 101 is shown forexample in Figure 3

34 A Family of Disassortative Networks Different modelsof disassortative networks can be constructed based on asuggestion byNewman contained in [15] According to it onecan set 119890119896ℎ = 119902119896119909ℎ + 119909119896119902ℎ minus 119909119896119909ℎ for ℎ 119896 = 0 119899 minus 1where 119902119896 is the distribution of the excess degrees and 119909119896 isany distribution satisfyingsum119899minus1119896=0 119909119896 = 1 and with 119909119896 decayingfaster than 119902119896 Choosing to fix ideas 119909119896 = (119896+1)minus120574sum119899minus1119895=0 (119895+1)minus120574 with a parameter 120574 gt 2 we denote 119878 = sum119899minus1119895=0 (119895+1)minus120574 and119879 = sum119899minus1119895=0 (119895 + 1)minus2 and then set for all ℎ 119896 = 0 119899 minus 1119890119896ℎ = ( 1119878119879 ((119896 + 1)minus2 (ℎ + 1)minus120574 + (ℎ + 1)minus2 (119896 + 1)minus120574)minus 11198782 (119896 + 1)minus120574 (ℎ + 1)minus120574)

(22)

We show in Appendix that the inequalities 0 le 119890119896ℎ le 1 holdtrue for any ℎ 119896 = 0 119899 minus 1 In view of (8) the degreecorrelations are then obtained as

119875 (ℎ | 119896) = 119890119896minus1ℎminus1sum119899119895=1 119890119896minus1119895minus1 forallℎ 119896 = 1 119899 (23)

with the 119890119896ℎ as in (22) It is immediate to check that thecoefficient 119903 is negative see eg [15] As for the averagenearest neighbor degree function 119896119899119899(119896) it can be calculatedwith a software The decreasing character of 119896119899119899(119896) for anetwork of the family in this subsection with 120574 = 4 and119899 = 101 is shown for example in Figure 4

4 Discussion

41 The Newman Assortativity Coefficient for BA NetworksIn [14] Newman reported in a brief table the values of 119903 forsome real networks More data are given in his book [23]Table 81 Focusing on disassortative scale-free networks forwhich the scale-free exponent 120574 is available one realizes thattheir negative 119903 coefficient is generally small in absolute valueespecially when 120574 is equal or close to 3 for instance

(i) wwwndedu 120574 from 21 to 24 119903 = minus0067(ii) Internet 120574 = 25 119903 = minus0189(iii) Electronic circuits 120574 = 30 119903 = minus0154

6 Complexity

k_nn

k20 40 60 80 100

10

20

30

40

50

60

70

Figure 3 Function 119896119899119899 for an assortative network as in Section 33 with 120572 = 12 119899 = 101 (119873 ≃ 104)

k

k_nn

20 40 60 80 100

1115

1120

1125

1130

Figure 4 Function 119896119899119899 for a disassortative network as in Section 34 with 120574 = 4 119899 = 101 (119873 ≃ 104)

The size of these networks varies being of the magnitudeorder of119873 asymp 104 to119873 asymp 106The data mentioned above havebeen probably updated and extended but in general it seemsthat scale-free disassortative networks with these values of 120574tend to have an 119903 coefficient that is much closer to 0 than tominus1 As we have seen this also happens with disassortativescale-free networks mathematically defined whose degreecorrelations are given by a procedure also introduced byNewman

For ideal BA networks Newman gave in [14] an asymp-totic estimate of the 119903 coefficient based on the correlationscomputed in [24] and concluded that 119903 is negative butvanishes as (log119873)2119873 in the large-119873 limit Other authorsshare the view that BA networks are essentially uncorrelated[19 25] The smallness of their 119903 coefficient is also con-firmed by numerical simulations in which the network isgrown according to the preferential attachment scheme eventhough the results of such simulations are affected by sizablestatistical errors when119873 varies between asymp 104 and asymp 106

In the recent paper [26] Fotouhi and Rabbat use theirown expressions for the 119901(119897 119896) correlations (expressing thefraction of links whose incident nodes have degrees 119897 and119896) to compute the asymptotic behavior of 119903 according toan alternative expression given by Dorogovtsev and Mendes[27] They conclude that the estimate 119903 ≃ (log119873)2119873 givenby Newman is not correct They find |119903| asymp (log 119899)2119899 Inorder to relate the network size 119873 to the largest degree 119899

they use the relation 119899 ≃ radic119873 based on the continuum-like criterium intinfin

119899119875(119896)119889119896 = 119873minus1 [28] So in the end |119903| asymp(log119873)2radic119873 They check that this is indeed the correct

behavior by performing new large-scale simulationsWe have computed 119903 using the original definition by

Newman [14] based on the matrix 119890119895119896 and the exact 119875(ℎ | 119896)coefficients with 119899 up to 15000 which corresponds to 119873 ≃225000000 (Figure 5) We found a good agreement with theestimate of Fotouhi and Rabbat Note however that althoughthe ldquothermodynamicrdquo limit of 119903 for infinite119873 is zero for finite119873 the value of 119903 cannot be regarded as vanishingly small inconsideration of what we have seen above for disassortativescale-free networks with 120574 = 342 The Function 119896119899119899(119896) An early estimate of the function119896119899119899(119896) valid also for BA networks has been given by Vespig-nani and Pastor-Satorras in an Appendix of their book on thestructure of the Internet [29] Their formula is based in turnon estimates of the behavior of the conditional probability119875(1198961015840 | 119896) which hold for a growing scale-free network witha degree distribution 119875(119896) prop 119896minus120574 This formula reads

119875 (1198961015840 | 119896) prop 1198961015840minus(120574minus1)119896minus(3minus120574) (24)

and holds under the condition 1 ≪ 1198961015840 ≪ 119896 Using the exact119875(1198961015840 | 119896) coefficients of Fotouhi and Rabbat it is possible tocheck the approximate validity of this formula

Complexity 7

2 4 6 8 10n1000

minus0025

minus0020

minus0015

minus0010

minus0005

r

Figure 5 Newman assortativity coefficient 119903 for BA networks with 120573 = 1 2 3 4 and largest degree 119899 = 1000 2000 10000 The uppercurve is for 120573 = 1 and the curves for the values 120573 = 2 3 4 follow from the top to the bottom

0 20 40 60 80 100k

36

37

38

39

40

k_nn

Figure 6 Function 119896119899119899 for a BA network with 120573 = 1 119899 = 100 (largest degree the corresponding number of nodes is119873 ≃ 104)

In [29] the expression (24) is then used to estimate 119896119899119899(119896)Since for 120574 = 3 119875(1198961015840 | 119896) does not depend on 119896 theconclusion is that 119896119899119899 is also independent from 119896 It is notentirely clear however how the condition 1 ≪ 1198961015840 ≪ 119896 canbe fulfilled when the sum over 1198961015840 is performed and how thediverging factor sum1198991198961015840=1(11198961015840) ≃ ln(119899) ≃ (12) ln(119873) should betreated

In their recent work [26] Fotouhi and Rabbat use theirown results for the 119875(1198961015840 | 119896) coefficients in order to estimate119896119899119899(119896) in the limit of large 119896 They find 119896119899119899(119896) asymp 120573 ln(119899) andcite a previous work [30] which gives the same result in theform 119896119899119899(119896) asymp (12)120573 ln(119873) (we recall that 119899 ≃ radic119873) Inthis estimate we can observe as compared to [29] the explicitpresence of the parameter 120573 and the diverging factor ln(119899)

Concerning the Newman assortativity coefficient 119903Fotouhi and Rabbat also make clear that even though 119903 997888rarr 0when119873 997888rarrinfin this does not imply that the BA networks areuncorrelated and in fact the relation 119896119899119899(119896) = ⟨1198962⟩⟨119896⟩ validfor uncorrelated networks does not apply to BA networks

A direct numerical evaluation for finite 119899 of the function119896119899119899(119896) based on the exact 119875(1198961015840 | 119896) shows further interestingfeatures As can be seen in Figures 6 and 7 the functionis decreasing at small 119896 and slightly and almost linearlyincreasing at large 119896 Note that this happens for networks ofmedium size (119899 = 100 119873 ≃ 104) like those employed in

our numerical solution of the Bass diffusion equations andemployed in [8] but also for larger networks (for instancewith 119899 = 1000 119873 ≃ 106 compare Figure 7) It seems thatthe ldquoperipheryrdquo of the network made of the least connectednodes has a markedly disassortative character while thehubs are slightly assortative (On general grounds one wouldinstead predict for finite scale-free networks a small structuraldisassortativity at large 119896 [17])

Since evidence on epidemic diffusion obtained so farindicates that generally the assortative character of a networklowers the epidemic threshold and the disassortative char-acter tends to make diffusion faster once it has started thisldquomixedrdquo character of the finite BA networks appears to facil-itate spreading phenomena and is consistent with our dataon diffusion time (Section 3) Note in this connection thatsome real networks also turn out to be both assortative anddisassortative in different ranges of the degree 119896 compare theexamples in [17] Ch 7

Finally we would like to relate our numerical findingsfor the function 119896119899119899 to the general property (compare forinstance [5])

⟨1198962⟩ = 119899sum119896=1

119896119875 (119896)119870 (119896 119899) (25)

8 Complexity

k

0 200 400 600 800 1000

56

58

60

62

64knn

Figure 7 Function 119896119899119899 for a BA network with 120573 = 1 119899 = 1000 (119873 ≃ 106) A linear fit for large 119896 is also shown The slope is approximatelyequal to 5 sdot 10minus4 a comparison with Figure 6 shows that the slope decreases with increasing 119899 but is still not negligible

where for simplicity 119870 denotes the function 119896119899119899 and thedependence of this function on the maximum degree 119899 isexplicitly shown For BAnetworks with120573 = 1we have at large119899 on the lhs of (25) from the definition of ⟨1198962⟩⟨1198962⟩ = 119899sum

119896=1

1198962 4119896 (119896 + 1) (119896 + 2) = 4 ln (119899) + 119900 (119899) (26)

where the symbol 119900(119899) denotes terms which are constant ordo not diverge in 119899

For the expression on the rhs of (25) we obtain119899sum119896=1

119896119875 (119896)119870 (119896 119899) = 119899sum119896=1

4(119896 + 1) (119896 + 2)119870 (119896 119899) (27)

Equations (26) and (27) are compatible in the sense thattheir diverging part in 119899 is the same in two cases (1) if forlarge 119896 we have 119870(119896 119899) ≃ 119886 ln (119899) where 119886 is a constant thisis true because in that case the sum on 119896 is convergent (2)if more generally 119870(119896 119899) ≃ 119886 ln (119899) + 119887(119899)119896 this is still truebecause also for the term 119887(119899)119896 the sum in 119896 leads to a resultproportional to ln(119899) Case (2) appears to be what happensaccording to Figures 6 and 7

5 Validity of the Mean-Field Approximation

The validity of the heterogeneous mean-field approximationin epidemic diffusionmodels on networks has been discussedin [11 31] and references As far as the form of the equation isconcerned the Bass model is very similar to other epidemicmodels A network which is completely characterized by thefunctions 119875(119896) 119875(ℎ | 119896) is called in [31] a ldquoMarkoviannetworkrdquoMean-field statistical mechanics on these networkshas been treated in [32] Recent applications can be foundfor instance in [33] It has in a certain sense an axiomaticvalue and it is well defined in the statistical limit of infinitenetworks Also the BA networks studied by Fotouhi andRabbat are defined in this limit In our work we are mainlyinterested into the properties of the networks therefore whenwe speak of their properties with respect to diffusion wemake reference to this kind of idealized diffusion Notethat the mean-field approximation employed has a quite

rich structure (whence the denomination ldquoheterogeneousrdquo)since the population can be divided into a large number ofconnectivity classes and the functions 119875(119896) 119875(ℎ | 119896) containa lot of networking information about these classes

Itmay therefore be of interest to check at least numericallyif the mean-field approximation is a good approximation ofthe full Bass model in this case For this purpose one mayconsider a network formulation of the model which employsthe adjacency matrix 119886119894119895 The coupled differential equationsfor a network with119873 nodes are in this formulation

119889119865119894 (119905)119889119905 = [1 minus 119865119894 (119905)](119901 + 119902 119873sum119895=1

119886119894119895119865119895 (119905)) 119894 = 1 119873

(28)

The solution 119865119894(119905) defines for each node an adoptionlevel which grows as a function of time from 119865119894 = 0 at119905 = 0 to 119865119894 = 1 when 119905 goes to infinity The ldquobarerdquo119902 parameter is renormalized with the average connectivity⟨119896⟩ This model can be seen as a first-moment closure of astochasticmodel [23] and is actuallymore suitable to describeinnovation diffusion among firms or organizations under theassumption that inside each organization innovations diffusein a gradual way In fact in forthcoming work we apply themodel to real data concerning networks of enterprises Itis also useful however in order to illustrate the differencebetween a mean-field approach and a completely realisticapproach For the case of a BA network it is possible togenerate the adjacency matrix with a random preferentialattachment algorithmThismatrix is then fed to a code whichsolves the differential equations (28) and computes for eachnode the characteristic diffusion times The results clearlyshow that in general different nodes with the same degreeadopt at different rates (unlike in the mean-field approxima-tion) It happens for instance as intuitively expected that anode with degree 1 which is connected to a big central hubadopts more quickly than a node with the same degree thathas been attached to the network periphery in the final phaseof the growth However these fluctuations can be estimatednumerically and it turns out that mean-field approximationsfor the diffusion times agree with the average times within

Complexity 9

Figure 8 An example of a BA network with 400 nodes and 120573 = 1 to which (28) has been applied for comparison to the correspondingmean-field approximation

standard errors For instance in a realization of a BA networkwith 120573 = 1 119873 = 400 (Figure 8) we obtain (consideringthe most numerous nodes for which statistics is reliable) thefollowing

Degree 1 266 nodes with average diffusion peak time52 120590 = 13Degree 2 67 nodes with average diffusion peak time48 120590 = 11Degree 3 33 nodes with average diffusion peak time43 120590 = 09

With the mean-field model with the same bare 119901 = 003and 119902 = 035 we obtain the following

Degree-1 nodes diffusion peak time 52Degree-2 nodes diffusion peak time 49Degree-3 nodes diffusion peak time 46

The agreement is reasonable also in consideration of therelatively small value of119873 Concerning this a final remark isin order while random generation of BA networks mainlyyields networks with maximum degree close to the mostprobable value 119899 = radic119873 sizable fluctuations may occur aswell The network in Figure 8 for instance has 119873 = 400and a hub with degree 35 We intentionally included thisnetwork in our checks in order to make sure that the mean-field approximation can handle also these cases This turnsout to be true provided the real value of themaximumdegreeis used (the mean-field peak times given above are obtainedwith 119899 = 35) In fact the full numerical solutions of (28)routinely show as expected that for fixed values of 119873 thepresence of large hubs speeds up the overall adoption For anldquoaveragerdquo network the mean-field approximation works welljust with 119899 = radic119873

6 Diffusion Times withHeterogeneous 119901 Coefficients

In the Bass model the publicity coefficient 119901 gives theadoption probability per unit time of an individual whohas not yet adopted the innovation independently fromthe fraction of current adopters Therefore it is not dueto the word-of-mouth effect but rather to an externalstimulus (advertising) which is received in the same wayby all individuals The intensity of this stimulus is in turnproportional to the advertising expenses of the produceror seller of the innovation One can therefore imagine thefollowing alternative to the uniform dissemination of adsto all the population the producer or seller invests foradvertising to each individual in inverse proportion to thepopulation of the individualrsquos link class so as to speed upadoption in the most connected classes and keep the totalexpenses unchanged In terms of the 119901119894 coefficients in (16)this implies 119901119894 prop 1119875(119894) with normalization sum119899119894=1 119901119894119875(119894) = 119901[12]

As can be seen also from Figure 9 this has the effectof making the total diffusion faster or slower dependingon the kind of network The differences observed in thepresence of a homogeneous 119901 coefficient (Figure 2) are nowamplified We observe that diffusion on BA networks is nowalways faster than on uncorrelated networks independentlyfrom 120573 It is also remarkable how slow diffusion becomes onassortative networks in this case A possible interpretation isthe following due to the low epidemic threshold of assortativenetworks the targeted advertising on the hubs (which arewell connected to each other) causes them to adopt veryquickly but this is not followed by easy diffusion on the wholenetwork The BA networks appear instead to take advantagein the presence of advertising targeted on the hubs bothfrom their low threshold and from a stronger linking to theperiphery

10 Complexity

BASS q PARAMETER

DIFFUSION TIME (TRICKLE-DOWN)

Assortative


BA

q = 048q = 03

2 4 6 8 10

4

5

6

7

8

9

10

Figure 9 Time of the diffusion peak (in years) for the ldquotrickle-downrdquo modified Bass model on different kinds of scale-free networks withexponent 120574 = 3 as a function of the imitation coefficient 119902 All networks have maximum degree 119899 = 100 The 119902 coefficient varies in therange 03 minus 048 corresponding to a typical set of realistic values in innovation diffusion theory [20] The publicity coefficient 119901119896 of the linkclass 119896 with 119896 = 1 119899 varies in inverse proportion to 119875(119896) (Section 6) The low-lying group of lines corresponds to BA networks with120573 = 1 2 3 4 5 and the same assortativity coefficients as in Figure 2The disassortative network is built with the Newman recipe (Section 34)with 119889 = 4 and has 119903 = minus0084 The assortative network is built with our recipe (Section 33) with 120572 = 14 and has 119903 = 0827

7 Conclusions

In this workwe have studied the assortativity properties of BAnetworks with maximum degree 119899 which is finite but large(up to 119899 ≃ 104 corresponding to a number of nodes 119873 ≃108)These properties were not known in detail until recentlya new decisive input has come from the exact calculationof the conditional probabilities 119875(ℎ | 119896) by Fotouhi andRabbat [10] We have done an explicit numerical evaluationof the average nearest neighbor degree function 119896119899119899(119896) whosebehavior turns out to be peculiar and unexpected exhibitinga coexistence of assortative and disassortative correlationsfor different intervals of the node degree 119896These results havebeen compared with previous estimates both concerning thefunction 119896119899119899(119896) and the Newman assortativity index 119903

The next step has been to write the Bass innovationdiffusion model on BA networks following the mean-fieldscheme we have recently introduced and tested on genericscale-free networks This allows computing numerically thedependence of the diffusion peak time (for 119899 ≃ 102) fromthe modelrsquos parameters and especially from the features ofthe network We have thus compared the diffusion timeson BA networks with those on uncorrelated assortative anddisassortative networks (the latter built respectively withour mathematical recipes (20) and (21) and with Newmansrsquosrecipe (22))

The BA networks with small values of 120573 (120573 is the numberof child nodes in the preferential attachment scheme) turnout to have the shortest diffusion time probably due to theirmixed assortativedisassortative character diffusion appearsto start quite easily among the (slightly assortative) hubsand then to proceed quickly in the (disassortative) peripheryof the network This interpretation is confirmed by the factthat in a modified ldquotrickle-downrdquo version of the model withenhanced publicity on the hubs the anticipation effect ofBA networks compared to the others is stronger and almostindependent from 120573

Concerning the dependence of the diffusion time onthe values of the 119903 coefficient we have found a qualitativeagreement with previous results by DrsquoAgostino et al [8] Westress however that the use of two-point correlations in thisanalysis is entirely new

In forthcoming work we shall analyse mathematically theconstruction and the properties of the mentioned families ofassortative networks from which we only have chosen herea few samples for comparison purposes because the focus inthis paper has been on BA networks

Appendix

We show here that 119890119896ℎ in (22) satisfies 0 le 119890119896ℎ le 1 for allℎ 119896 = 0 119899 minus 1 If 120574 = 2 + 119889 with 119889 gt 0 119890119896minus1ℎminus1 can berewritten for any ℎ 119896 = 1 119899 as

119890119896minus1ℎminus1 = (119896119889 + ℎ119889)sum119899119895=1 (1119895120574) minus sum119899119895=1 (11198952)119896120574ℎ120574sum119899119895=1 (11198952) (sum119899119895=1 (1119895120574))2 (A1)

The nonnegativity of 119890119896minus1ℎminus1 is hence equivalent to that of thenumerator of (A1) This is for all ℎ 119896 = 1 119899 greater thanor equal to

2 119899sum119895=1

1119895120574 minusinfinsum119895=1

11198952 ge 2 minus 1205876 gt 0 (A2)

To show that 119890119896minus1ℎminus1 le 1 we distinguish two cases(i) if 119896 = 1 and ℎ = 1 the expression on the rhs in (A1)

is equal to

Complexity 11

2sum119899119895=1 (1119895120574) minus sum119899119895=1 (11198952)sum119899119895=1 (11198952) (sum119899119895=1 (1119895120574))2

le 2sum119899119895=1 (11198952) minus sum119899119895=1 (11198952)sum119899119895=1 (11198952) (sum119899119895=1 (1119895120574))2 le1

(sum119899119895=1 (1119895120574))2le 1

(A3)

(ii) otherwise ie if at least one among 119896 and ℎ is greaterthan 1 the expression on the rhs in (A1) is no greater than

(119896119889 + ℎ119889)sum119899119895=1 (11198952+119889)1198962+119889ℎ2+119889sum119899119895=1 (11198952) (sum119899119895=1 (11198952+119889))2

le (119896119889 + ℎ119889)119896119889ℎ119889 11198962ℎ2 le 1(A4)

Data Availability

No data were used to support this study



Acknowledgments

This work was supported by the Open Access PublishingFund of the Free University of Bozen-Bolzano

References

[1] M J Keeling and K T D Eames ldquoNetworks and epidemicmodelsrdquo Journal of The Royal Society Interface vol 2 no 4 pp295ndash307 2005

[2] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[3] A M Porter and P J Gleeson ldquoDynamical systems on net-worksrdquo in Frontiers in Applied Dynamical Systems Reviews andTutorials vol 4 2016

[4] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics andendemic states in complex networksrdquo Physical Review E vol 63no 6 Article ID 066117 2001

[5] M Boguna R Pastor-Satorras and A Vespignani ldquoAbsenceof Epidemic Threshold in Scale-Free Networks with DegreeCorrelationsrdquo Physical Review Letters vol 90 no 2 2003

[6] VM Eguıluz andK Klemm ldquoEpidemic threshold in structuredscale-free networksrdquo Physical Review Letters vol 89 no 10Article ID 108701 2002

[7] P Blanchard C Chang and T Kruger ldquoEpidemic Thresholdson Scale-Free Graphs the Interplay between Exponent andPreferential Choicerdquo Annales Henri Poincare vol 4 no S2 pp957ndash970 2003

[8] G DrsquoAgostino A Scala V Zlatic and G Caldarelli ldquoRobust-ness and assortativity for diffusion-like processes in scale-freenetworksrdquo Europhys Lett vol 97 no 6 p 68006 2012

[9] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[10] B Fotouhi and M G Rabbat ldquoDegree correlation in scale-freegraphsrdquo European Physical Journal B vol 86 no 12 p 510 2013

[11] A Vespignani ldquoModelling dynamical processes in complexsocio-technical systemsrdquoNature Physics vol 8 no 1 pp 32ndash392012

[12] M Bertotti J Brunner and G Modanese ldquoThe Bass diffusionmodel on networks with correlations and inhomogeneousadvertisingrdquo Chaos Solitons amp Fractals vol 90 pp 55ndash63 2016

[13] M L Bertotti J Brunner and G Modanese ldquoInnovationdiffusion equations on correlated scale-free networksrdquo PhysicsLetters A vol 380 no 33 pp 2475ndash2479 2016

[14] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 2002

[15] M E Newman ldquoMixing patterns in networksrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 67 no 22003

[16] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[17] A-L Barabasi Network Science Cambridge University PressCambridge UK 2016

[18] R Pastor-Satorras and A Vespignani ldquoEpidemic dynamics infinite size scale-free networksrdquoPhys Rev E vol 65 no 3 ArticleID 035108 2002

[19] MBarthelemyA Barrat R Pastor-Satorras andAVespignanildquoDynamical patterns of epidemic outbreaks in complex hetero-geneous networksrdquo Journal of Theoretical Biology vol 235 no2 pp 275ndash288 2005

[20] Z Jiang F M Bass and P I Bass ldquoVirtual Bass Model andthe left-hand data-truncation bias in diffusion of innovationstudiesrdquo International Journal of Research in Marketing vol 23no 1 pp 93ndash106 2006

[21] V Isham J Kaczmarska and M Nekovee ldquoSpread of informa-tion and infection on finite random networksrdquo Physical ReviewE vol 83 no 4 Article ID 046128 2011

[22] P V Mieghem H Wang X Ge S Tang and F A KuipersldquoInfluence of assortativity and degree-preserving rewiring onthe spectra of networksrdquo The European Physical Journal B vol76 no 4 pp 643ndash652 2010

[23] M Newman Networks An Introduction Oxford UniversityPress Oxford UK 2010

[24] P LKrapivsky and S Redner ldquoOrganization of growing randomnetworksrdquo Physical Review E vol 63 no 6 Article ID 0661232001

[25] R Noldus and P Van Mieghem ldquoAssortativity in complexnetworksrdquo Journal of Complex Networks vol 3 no 4 pp 507ndash542 2015

[26] B FOTOUHI and M RABBAT ldquoTemporal evolution of thedegree distribution of alters in growing networksrdquo NetworkScience vol 6 no 01 pp 97ndash155 2018

[27] S N Dorogovtsev and J F F Mendes ldquoEvolution of networksrdquoAdvance in Physics vol 51 no 4 pp 1079ndash1187 2002

[28] M Boguna R Pastor-Satorras and A Vespignani ldquoCut-offsand finite size effects in scale-free networksrdquo The EuropeanPhysical Journal B vol 38 no 2 pp 205ndash209 2004

[29] R Pastor-Satorras and A Vespignani Evolution and Structureof the Internet A Statistical Physics Approach CambridgeUniversity Press Cambridge UK 2004

12 Complexity

[30] A Barrat and R Pastor-Satorras ldquoRate equation approach forcorrelations in growing networkmodelsrdquoPhysical ReviewE vol71 no 3 Article ID 036127 2005

[31] M Boguna and R Pastor-Satorras ldquoEpidemic spreading incorrelated complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 4 Article ID047104 2002

[32] J Marro and R Dickman Nonequilibrium Phase Transitions inLattice Models Cambridge University Press Cambridge UK1999

[33] C Osorio and C Wang ldquoOn the analytical approximation ofjoint aggregate queue-length distributions for traffic networksA stationary finite capacity Markovian network approachrdquoTransportationResearch Part BMethodological vol 95 pp 305ndash339 2017

Research ArticlePrediction of Ammunition Storage Reliability Based onImproved Ant Colony Algorithm and BP Neural Network

Fang Liu 1 Hua Gong 1 Ligang Cai2 and Ke Xu1

1College of Science Shenyang Ligong University Shenyang 110159 China2College of Science Shenyang University of Technology Shenyang 110178 China

Correspondence should be addressed to Hua Gong gonghuasylueducn

Received 30 November 2018 Accepted 24 February 2019 Published 18 March 2019

Guest Editor Pedro Palos

Copyright copy 2019 Fang Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Storage reliability is an important index of ammunition product quality It is the core guarantee for the safe use of ammunitionand the completion of tasks In this paper we develop a prediction model of ammunition storage reliability in the naturalstorage state where the main affecting factors of ammunition reliability include temperature humidity and storage period A newimproved algorithm based on three-stage ant colony optimization (IACO) and BP neural network algorithm is proposed to predictammunition failure numbersThe reliability of ammunition storage is obtained indirectly by failure numbersThe improved three-stage pheromone updating strategies solve two problems of ant colony algorithm local minimum and slow convergence Aiming atthe incompleteness of field data ldquozero failurerdquo data pretreatment ldquoinverted hangingrdquo data pretreatment normalization of data andsmall sample data augmentation are carried out A homogenization sampling method is proposed to extract training and testingsamples Experimental results show that IACO-BP algorithm has better accuracy and stability in ammunition storage reliabilityprediction than BP network PSO-BP and ACO-BP algorithm

1 Introduction

Reliability is the primary quality index of military productsOn the battlefield the unreliable ammunition products willlead to the failure of military tasks or endanger the lives ofour soldiers In the weapon system the reliability is the coreof the design development and maintenance of ammunitionproducts Sincemost of the life cycle of ammunition productsis the storage period the storage reliability of ammunitiondirectly affects the field reliability of ammunition Quantita-tive prediction of ammunition products storage reliability isthe key link of ammunition reliability engineering

In recent years scholars have carried out extensiveresearch on reliability in their respective fields Many modelsandmethods are devised to predict and evaluate the reliabilityof different products and systems (see [1ndash6]) On the reliabil-ity of ammunition storage the traditional methods are basedon natural storage condition data and accelerated testingdata respectively The reliability of ammunition storage ispredicted by mathematical statistical methods Based on nat-ural storage condition data a Poisson reliabilitymathematical

model is established in [7] to predict ammunition storagelife Binary Logic Regression (BLR) model is obtained in[8] to evaluate recent system failures in non-operationalstorage Under the assumption of exponential distribution astorage reliability prediction model based on time-truncateddata is established in [9] In [10] Gamma distribution isused to fit the temperature distribution in the missile duringstorage and proportional risk model is created to predictthe effect of temperature on product reliability In order toforecast the residual storage life second-order continuous-time homogeneous Markov model and stochastic filteringmodel are utilized in [11] and [12] respectively In [13] an E-Bayes statistical model is proposed to predict storage life byusing the initial failure number

Based on the accelerated test data the storage life offuze is evaluated in [14] by applying step stress acceleratedtest The ammunition storage reliability is predicted in [15]based on ammunition failure mechanism and accelerated lifemodel In [16] the storage reliability function expression ofammunition is established based on the state monitoringdata of accelerated testTheArrhenius acceleration algorithm


2 Complexity

is used in [17] to design a new model to estimate theacceleration factor In [18] a prediction method combiningaccelerated degradation test with accelerated life test isproposed Monte Carlo method is used to generate pseudo-failure life data under various stress levels to evaluate reli-ability and predict life of products Since the traditionalstatistical methods usually need the original life distributioninformation in solving the reliability prediction problemsthey are conservative in solving the uncertain life distri-bution highly nonlinear problems and small sample dataproblems

Many scholars recently consider intelligent algorithmsin reliability prediction Specifically a comprehensive pre-diction model which combines SVM with Ayers methodsis given to predict the reliability of small samples in [19]The wavelet neural network is established in [20] to predictthe data block of hard disk storage system Two kinds ofmissile storage reliability prediction models are designed andcompared in [21] Two models are BP network and RBFnetwork Artificial neural network equipment degradationmodel is proposed in [22 23] to predict the remaininglife of equipment A particle swarm optimization model isemployed in [24ndash26] to solve the reliability optimizationproblem with uncertainties The intelligent algorithms basedon the original data keep the information of data to thegreatest extent It is noted that the intelligent predictionalgorithms do not depend on the prior distribution ofammunition life The exploration process of prior distribu-tion of raw data is eliminated The intelligent algorithmsprovide a new way to predict the reliability of ammu-nition storage However there are still many problemsneed to be solved when applying intelligent algorithmsto reliability prediction such as the local minimum slowconvergence speed and high sensitivity of SVM to datamissing

In this paper a prediction model of ammunition fail-ure number is proposed A new three-stage ACO and BPneural network is created in the model This predictionmodel excavates the mathematical relationship between thestorage temperature humidity and period of ammunitionunder natural conditions and the number of ammuni-tion failure The reliability of ammunition can be obtainedindirectly by the failure number In the aspect of sam-ple selection ldquozero failurerdquo data pretreatment ldquoinvertedhangingrdquo data pretreatment normalization of small sam-ple data augmentation and homogenization sampling areadopted to achieve sample integrity Experimental resultsshow that our IACO-BP out-performs BP ACO-BP andPSO-BP

The rest of this paper is organized as follows Section 2introduces the basic theory of ammunition storage reliabilitySection 3 presents IACO-BP algorithm and describes theimplementation details Section 4 demonstrates data pre-treatment methods and training sample and testing sam-ple extraction strategy Section 5 shows the experimen-tal results and compares algorithm performance with BPPSO-BP and ACO-BP The conclusions are denoted inSection 6

2 Basic Theory of AmmunitionStorage Reliability

The storage reliability of ammunition is the ability of ammu-nition to fulfill the specified functions within the specifiedstorage time and under the storage conditions The storagereliability of ammunition 119877(119905) is the probability of keepingthe specified function Here 119877(119905) = 119875(120578 gt 119905) where 120578 is thestorage time before ammunition failure and t is the requiredstorage time

According to the definition of reliability we can obtainthe following

119877 (119905) = 119873119905=0 minus 119891 (119905)119873119905=0 (1)

where119873119905=0 is the initial ammunition number and f (t) is thecumulative failure number of ammunitions from the initialtime to the time t

The storage of modern ammunitions is a complex systemwhich composes of chemical mechanical and photoelectricmaterials Ammunitions are easy to be affected by differentenvironmental factors For example nonmetallic mold leadsto the insulation ability of insulation materials decline thecharge is moisture to reduce the explosive force of ammuni-tion and plastic hardening results in self-ignition of ammu-nition Researches show that temperature and humidity arethe main two factors that affect ammunition reliability andstorage period is an important parameter of ammunitionreliability

3 IACO and BP Neural Network

31 Traditional Ant Colony Optimization Algorithm (ACO)Ant Colony Optimization (ACO) proposed by Dorigo isa global strategy intelligent optimization algorithm whichsimulates the foraging behaviors of ants in [27] Ants thatset out to find food always leave the secretion which iscalled pheromone on the path The following ants adaptivelydecide the shortest path to the food position by the residualpheromone concentration The overall cooperative behaviorsof ant colony foraging constitute a positive feedback phe-nomenon Ant colony algorithm simulates the optimizationmechanism of information exchange and cooperation amongindividuals in this process It searches the global optimal solu-tion through information transmission and accumulation inthe solution space

The main steps of ant colony algorithm are described asfollows

Step 1 (setting the initial parameters value) The number ofants is set to Q 120578119894 (1 le 119894 le 119899) is n parameters to beoptimized Each parameter has M values in the range ofvalues These values form set 119860120578119894 At the initial time eachelement carries the same concentration pheromone that isrecorded as 120585119895(119860120578119894)(0) = 119886 (1 le 119895 le 119872) The maximumnumber of iterations is119873max

Complexity 3

Step 2 (choosing paths according to strategy) Start the antsEach ant starts from set 119860120578119894 and chooses the foraging pathaccording to (2) route selection strategy

119875 (120585119896119895 (119860120578119894)) =120585119895 (119860120578119894)

sum119872119895=1 120585119895 (119860120578119894)(2)

Equation (2) describes the probability that ant k selects j valueof parameter i Ants determine the next foraging positionaccording to the probability maximum selection rule

Step 3 (updating pheromone) According to selection strat-egy in Step 2 each ant selects a location element in each set119860120578119894 120591 is the time of this process When ants finish a cycleof parameter selection pheromones are updated Updatingstrategies are adopted as follows

120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (3)

where 120588 (0 le 120588 le 1) is pheromone volatilization coefficient1 minus 120588 is pheromone residual degrees

120585119895 (119860120578119894) =119876

sum119896=1

120585119895119896 (119860120578119894) (4)

where 120585119895119896(119860120578119894) is the increment of the pheromone of ant kduring the time interval 120591120585119895119896 (119860120578119894)

=

119862119871119896 ant 119896 selects the 119895 element of the set 1198601205781198940 otherwise

(5)

where C is a constant and it represents the sum of thepheromones released by the ants after the completion ofa cycle 119871119896 indicates the total path length of ant k in thisforaging cycle

Step 4 (repeating step 2 to step 3) When all the ants select thesame path or reach the maximum number of iterations119873maxthe algorithm ends

32 Improved Ant ColonyOptimizationAlgorithm (IACO) Intraditional ACO algorithm the initial pheromone concentra-tion of each location is the same value These ants randomlychoose the initial path to forage with equal probability strat-egy Since the descendant ants choose the paths accordingto the cumulative pheromone concentration of the previousgeneration ant colony it may appear that the number ofants on the non-global optimal paths is much more thanthat on the optimal path at the initial selection This initialrandom selection strategy can increase the possibility of non-optimal paths selectionThe algorithm is easy to fall into localoptimum

Here we are focusing on revising the pheromone updatestrategy of traditional ant colony algorithm In this paper thetraditional route selection process of ant colony is divided

into three stages pheromone pure increase stage pheromonevolatilization and accumulation stage and pheromone dou-bling stage The path selection strategies of three stages areshown as (6) (3) and (7)

The strategy of pheromone pure increase stage is formedas follows

120585119895 (119860120578119894) (119905 + 120591) = 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (6)

where the formula for calculating 120585119895(119860120578119895) is like (4)The strategy of pheromone volatilization and accumula-

tion stage is the same as the traditional ant colony routingstrategy The strategy is described in (3)

The strategy of pheromone doubling stage is expressed asfollows

120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120574120585119895 (119860120578119894) (7)

where parameter 120574 is the doubling factor of processpheromone

The procedure of IACO algorithm is as shown in Figure 1Thepheromones are only accumulated andnot volatilized

in the previous N1 iterations of algorithm This stage isdefined as the stage of pure pheromone addition The totalamount of pheromone increases 120585119895(119860120578119895 ) during the time120591 Furthermore this value is inversely proportional to thelength of ants walking through the path This means thatthe concentration of pheromones of shorter paths increasesmore during this process The difference of cumulativepheromone concentration between the non-optimal pathsand the optimal path is reduced This strategy improves thepossibility of optimal paths selection Since the pheromonevolatilization factor is eliminated in the pure increase stageof pheromone the convergence speed of the algorithm slowsdown In the following two stages we adopt two strategies toaccelerate the convergence of this algorithm In the middle ofthe iterations the traditional pheromone updating strategyis adopted The strategy combines pheromone volatilizationand accumulation While speeding up the volatilization ofpheromones the possibility of non-optimal paths can betaken into account In the last stage of the iteration theprocess pheromone doubling factor is introduced whenthe number of iterations reaches N2 The growth rate ofpheromone is further improved and the search speed of theoptimal solution is also accelerated This stage is called thepheromone doubling stage

33 BP Neural Network BP neural network [28] is a multi-layer feedforward artificial neural network This networksimulates the cognitive process of human brain neurons toknowledge BP algorithm updates the weights and thresholdsof the network continuously by forward and backwardinformation transmission and error back propagation Thenetwork is trained several times by both positive and negativeprocesses until the minimum error is satisfied BP networkhas strong self-learning and fault tolerance properties Fig-ure 2 shows the topology structure of three layers BPnetwork

4 Complexity

Set initial pheromone concentration

Ant i chooses paths based onpheromone concentration

All ants search onceNo

Yes

Satisfy ACO termination condition

Output current best results

i=i+1

i=1

Update pheromones according to

pheromone pure increase strategy

Update pheromones according to pheromone

accumulation and volatilization strategy


doubling strategy

End

Start

Select pheromone update strategy

No

Yes

NltN1 NgtN2

N1⩽N⩽N2

Figure 1 Flowchart of IACO algorithm

Input layer (m) Output layer(g)Hidden layer(h)

Rout1

Rout1

Rout

x1

x2

xm

Figure 2 The topology structure of BP neural network

BP network fitness function can be described as follows

119864 (120578119894119895 1205781015840119902119896 120572119895 120573119896) = 12119875

sum119901=1

119892

sum119896=1

(119877119901119896minus 119877119901119900119906119905119896)

120578119894119895 isin 119877119898timesℎ 1205781015840119902119896 isin 119877ℎtimes119897 120572119895 isin 119877ℎtimes1 120573119896 isin 119877119892times1(8)

where m is the number of nodes in the input layer h is thenumber of hidden layer nodes and 119892 is the number of nodesin the output layer P is the total number of training samples120578119894119895 and 1205781015840119902119896 are the network weight from node i in the inputlayer to node j in the hidden layer and the network weightfrom node q in the hidden layer to node k in the outputlayer respectively 120572119895 and 120573119896 are the threshold of node j in the

hidden layer and the threshold of node k in the output layerrespectively 119877119901119900119906119905119896 is the network output of training sample 119901119877119901119896is the expected value of sample 119901 BP network adjusts

the error function 119864(120578119894119895 1205781015840119902119896 120572119895 120573119896) through bidirectionaltraining to meet the error requirements determines thenetwork parameters and obtains a stable network structure

34 IACO-BP Algorithm When BP neural network is usedto predict the initial weights and thresholds of each layerare given randomly This method brings a great fluctuationin forecasting the same data IACO-BP algorithm maps theoptimal solution generated by IACO algorithm to the weightsand thresholds of BP neural network This strategy solves

Complexity 5

ldquo Zero failurerdquo pretreatment

ldquo Inverted hangingrdquo pretreatmentRaw data

Data normalization

Data augment Homogenizationsampling

Figure 3 The process of data pretreatment and extraction

largely the randomness of initial weights and thresholds ofBP network IACO-BP algorithm improves effectively theaccuracy and stability of BP network prediction The specificsteps of the algorithm are demonstrated as follows

Step 1 (normalizing the sample data)

119909119894 = 119909119894 minus 119909min05 (119909max minus 119909min) minus 1 (9)

where 119909max and 119909min are the maximum value and theminimum value in the sample data respectively 119909119894 is the ithinput variable value The range of 119909119894 is in the interval [minus1 1]Step 2 (establishing three-layer network topology119898times ℎ times 119892)Determine the value ofm h 119892Step 3 (initializing ant colony parameters) Set the parame-tersM Q 119886 C 12058811987311198732119873max and dimension 119898 times ℎ + ℎ times119892 + ℎ + 119892 of BP neural network parameters

Step 4 (starting ants) Each ant starts from a location elementin the set 1198601205781 to find the shortest path The pheromone isupdated according to the improved ant colony three-stagepheromone update strategy

Step 5 Repeat Step 4 until all ants choose the same path orreach the maximum number of iterations and turn to Step 6

Step 6 Map the global optimal value in Step 5 to the initialweights and thresholds of BP neural network Output thepredicted value of the network Calculate the fitness function119864(120578119894119895 1205781015840119902119896 120572119895 120573119896) Propagate the error back to the output layerand adjust the weights and thresholds Repeat the aboveprocess until the terminating condition of BP algorithm 120576119861119875is satisfied or reaches BP maximum iteration1198731015840max

Step 7 Reverse the test data and restore the form of the testdata

119910 = 05 (119910119894 + 1) (119910max minus 119910min) + 119910min (10)

where 119910max and 119910min are the maximum value and theminimum value of the network output data respectively 119910119894is the ith output value

The framework of IACO-BP algorithm is as shown inAlgorithm 1

4 Data Collection and Pretreatment

41 Data Set In this paper the statistical data of ammunitionfailure numbers are selected as sample data at differenttemperatures humidity and storage period under naturalstorage conditions The data are shown in Table 1 We employammunition failure data to train ammunition failure predic-tion model Ammunition reliability is indirectly predicted byammunition failure numbers Each sample capacity in dataset is 10

In Table 1 the unit of temperature is the internationalunit Kelvin It is represented by the symbol K Relativehumidity RH record is used to indicate the percentageof saturated water vapor content in the air under the samecondition

42 Data Pretreatment The experimental samples collectedat the scene usually contain samples with ldquozero failurerdquoand ldquoinverted hangingrdquo samples These samples affect theaccuracy of ammunition reliability prediction It is necessaryto pre-treat the data before the simulation experiments Bayesestimation method is applied to treat ldquoZero failurerdquo andldquoinverted hangingrdquo data Method of artificially filling noisesignal is used to augment small sample size Homogenizationsampling is introduced to extract training samples and testsamples Figure 3 shows the process of data pretreatment andextraction

(1) ldquoZero Failurerdquo Data Pretreatment In the random sam-pling of small sample ammunition life test it may appear thatall the samples are valid This means that the ammunitionfailure number is zero In reliability data prediction sincezero failure data provides less effective information it needsto be pretreated Aiming at the ldquozero failurerdquo problem ofsamples No 1 No 9 and No 29 in the data set the incrementfunction of reliability p is selected as the prior distributionof p that is 120587(119901) prop 1199012 The posterior distribution of pis calculated based on Bayes method Zero failure data aretreated

When zero failure occurs the likelihood function of p is119871(0119901) = 1199012 By Bayes estimation method the reliability 119901 isobtained under the condition of square loss as shown in (11)The zero failure 119891119894 is transformed into 119891119894 as described in (12)

119901 = int10119901ℎ( 119901119899119894) 119889119901 (11)

119891119894 = 119899119894 (1 minus 119901) (12)

6 Complexity

Table 1 Raw data set

Number Temperature Humidity Period Failure number(K) (RH) (Year)

1 293 40 3 02 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 010 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 116 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 128 308 50 20 329 308 55 2 030 308 55 5 131 308 55 15 232 308 55 20 2

After ldquozero failurerdquo processing the failure numbers ofsamples No 1 No 9 and No 29 in the data set are allconverted to 0714286

(2) ldquoInverted Hangingrdquo Data Pretreatment Theoreticallyunder the same storage conditions the number ofammunition failures with longer storage time is relativelylarge During the same storage time the number ofammunition failures with poor storage conditions isrelatively large The failure numbers of samples No14and No15 and samples No26 and No27 are arrangedinversely with the storage time that is the ldquoinvertedhangingrdquo phenomenon It needs to be ldquoinverted hangingrdquotreated

Two sets of inverted data in the sample can be used tomodify 119901119894 to 119905119894 time by selecting 119905119895 time failure data 119901119895 as thereference data in the corresponding period (119905119895 119905119894) Since bothinverted samples are not at the end of the experiment at 119905119894

time the uniform distribution U(119901119895 119901119894) is taken as a priordistribution of 119901119894 119901119894 is revised to 119901119894

119901119894 =int119901119894+1119901119895

119901119891119895+1+1 (1 minus 119901)119904119895+1 119889119901int119901119894+1119901119895

119901119891119895+1 (1 minus 119901)119904119895+1 119889119901 (13)

where 119904119895 is the unfailing number in the sample119901119894 is the failureprobability corrected after 119905119894 times The corrected number offailures is 119891119894 = 119899119894119901119894 After ldquoinverted hangingrdquo pretreatmentthe failure number of samples No15 and No27 in the dataset is converted to 2 The ldquoinverted hangingrdquo phenomenon iseliminated The data after zero failure and inverted hangingpretreatment are shown in Table 2

(3) Normalization of Data Temperature humidity and stor-age period in the data set are corresponding to different quan-tity rigidity and quantity scale It is necessary to normalize

Complexity 7

Initialize the ant colony parametersbegin

while(ants iltants num)ants i chooses paths based on pheromone concentrationants i ++ Select different strategies in stages

if(iterations numltN1)Update pheromones according to pheromone pure increase strategy

if(N1ltiterations numltN2)Update pheromones according to pheromone accumulation and

volatilization strategy if(iterations numgtN2)Update pheromones according to pheromone doubling strategy

if (current errorgtmargin error)

ants i=1goto begin

else Input ant colony result is the initial solution of neural networkneural network trainingresulte = neural network training results

return resulte

Algorithm 1 IACO-BP (improved ant colony optimization Bp)

the data The normalized data can eliminate the influence ofvariable dimension and improve the accuracy of evaluationNormalization method is denoted as (9)

(4) Small Sample Data Augment Since the particularity ofammunition samples the problem of small sample size isusually encountered in the study of ammunition reliabilityprediction In this paper we add noise signals into theinput signals after normalization The method is applied tosimulate the sample data and increase the sample size Thegeneralization ability of BP network is enhanced

The number of samples is set to ns and the sample kis recorded as 119901119896 = (119909119896 119877119896119900119906119905) 119896 = 1 2 sdot sdot sdot 119899119904 119909119896 is theinput signal vector for the sample k and 119877119896119900119906119905 is the outputsignal of the sample k The noise vector 120575119896 is added to theinput signal The output signal remains unchanged The newsample becomes 119901119896 = (119909119896 + 120575119896 119877119896119900119906119905) Set 120575119896 = 0001Firstly three components of input signal vector are addedwith noise120575119896 respectivelyThen twoof the three componentsare grouped together respectively They are added with noise120575119896 Finally these three components are added with noise 120575119896respectively 32 data sets are extended to 256 data sets Theincrease of sample size provides data support for BP neuralnetwork to mine the potential nonlinear laws in ammunitionsample data The smoothness of network training curve andthe stability of network structure are improved

43 Homogenization Sampling In practical application sce-narios data have aggregation phenomenon Aggregationphenomenon refers that a certain type of data set is dis-tributed in a specific location of the data set During analyzingdata it may occur that the sample data cannot represent

the overall data In this paper a homogenization samplingdata extraction method is proposed Training samples andtest samples are selected in the whole range The bias ofsample characteristics caused by aggregation sampling isremoved Ensure that the sample can maximize the overalldata characteristics

The implementation of homogenization sampling is asfollows Assuming that there are N sets of sample data Xsets of data are constructed as training samples and Y setsare constructed as test data sets Every NY-1 (if NY is adecimal take a whole downward) data is test data until itis countlessly desirable The remaining N-Y data are usedas training samples The homogenization sampling rule isshown in Figure 4

It is known that the data pretreated in Section 42 arehomogenized InN=256 group samples Y=10 group samplesare taken as test samples According to the homogenizationsampling rule one sample is taken from every [NY]-1=24groups as the test sample and the rest groups are as thetraining sample Table 3 demonstrates four ammunitionstoragemodel precisionsThese precisions are obtained basedon homogenization sampling data and non-homogenizationsampling data respectively

Table 3 shows the model accuracy of ammunition storagereliability prediction based on homogenization samplingmuch higher than that based on non-homogenization sam-pling All algorithms in this paper are modeled and analyzedon the basis of homogenization sampling

5 The Simulation Experiments

In this section we show the specific parameter settings andthe results of the simulation experiment The mean square

8 Complexity

Table 2 Data after zero failure and inverted hanging treatment

Number Temperature Humidity Period Failure number(k) (RH) (Year)

1 293 40 3 07142862 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 071428610 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 216 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 228 308 50 20 329 308 55 2 071428630 308 55 5 131 308 55 15 232 308 55 20 2

Table 3 Precision comparison

Method IACO-BP ACO-BP PSO-BP BPNon-homogenization sampling 08075 08201 03259 12656Homogenization sampling 688e-04 429e-03 00377 00811

error (MSE) and mean absolute percentage error (MAPE) ofperformance indicators are used to evaluate the predictioneffect of themodelThe IACO-BPmodel shows its implementperformance in comparison with BP PSO-BP and ACO-BPmodels

119872119878119864 = 1119873119873

sum119896=1

10038161003816100381610038161003816119877119896 minus 119900119906119905119896 10038161003816100381610038161003816 (14)

119872119860119875119864 = 1119873119873

sum119896=1

1003816100381610038161003816100381610038161003816100381610038161003816119877119896 minus 119900119906119905119896

1198771198961003816100381610038161003816100381610038161003816100381610038161003816times 100 (15)

where119877119896 and 119900119906119905119896 are expected reliability and actual reliabil-ityN is the total number of data used for reliability predictionand comparison

51 Parameter Settings In this paper the storage temper-ature humidity and storage period of ammunition undernatural storage are used as input variables of neural networkto predict the number of ammunition failure The storagereliability of ammunition is indirectly predicted by thenumber of ammunition failureThenumber of input nodes inBP network ism=3 and the number of network output nodesis 119892=1 According to empirical formula ℎ = radic119898 + 119892 + 119888 the

Complexity 9

Table 4 The results of trial and errorHidden layer numbers 2 3 4 5 6 7 8 9 10 11 12 13The mean of MSE 0226 0165 0198 0269 0357 0266 0413 0205 0288 0026 0378 0566

Group [NY]- 1

Group [NY]-1

Group [NY]-1

Group [NY]-1

Group 2[NY]

Group (Y-1) [NY]

Group [NY]

Group Y[NY]

1

N

Train Test

Group (YY

Figure 4 Schematic diagram of homogenization sampling

value range of hidden layer nodes is [2 12] c is the constantin the interval [1 10] In order to avoid the contingency ofthe result caused by the instability of the neural network 10experiments are conducted on the same number of hiddenlayer nodesThe value ofMSE of the predicted and true valuesis observed The experimental results of trial and error areshown in Table 4

Table 4 indicates that the mean value of MSE is thesmallest when the number of hidden layer nodes is 11Therefore the neural network topology in this paper is 3 times11 times 1

The activation function of the hidden layer of BP networkis ldquotansigrdquo the activation function of the output layer is ldquopure-linrdquo and the training function of the network is ldquotrainlmrdquoThelearning rate is set to 01 The expected error is 120576119861119875 = 0001The weights and thresholds parameters dimension of BPneural network are 56 Each parameter is randomly assignedto 20 values in the internal [minus1 1] The number of ants is119876 = 80 Pheromone initial value is set to119886 = 1The coefficientof pheromone volatilization is 120588 = 01 The increment ofpheromone is 119862 = 1 The number of terminations of thepheromone pure increase stage is set to 200 the number ofinitial iterations of the pheromone doubling stage is set to 500and the number of iterations between 200 and 500 is the stageof pheromone volatilization and accumulation The doublingfactor of process pheromone is 120574 = 2 Maximum iteration ofant colony is set to119873max = 800

The algorithm is coded by MATLAB 2016a The failurenumbers of ammunition storage are generated by BP neuralnetwork algorithm ACO-BP algorithm PSO-BP algorithmand IACO-BP algorithm respectively Then the results arecompared and analyzed

52 Results Analysis In order to evaluate performance ofthe four algorithms 20 simulation tests are conducted on

each algorithm to obtain the value of the prediction errorevaluation index MSE and MAPE The arithmetic meanvalues of MSE and MAPE in 20 tests are used to characterizethe accuracy of four algorithms The variance standarddeviation and range of MSE and MAPE are applied toevaluate the stability of algorithms The results are shown inTables 5 and 6

Themean value of MSE of IACO-BP algorithm is 12e-03which is 92 95 and 99 lower than that of ACO-BP PSO-BP and BP algorithm respectively Themean value of MAPEof IACO-BP algorithm is 21e+00 which is 66 79 and83 lower than that of ACO-BP PSO-BP and BP algorithmrespectively It can be seen that in the four intelligentalgorithms the IACO-BP algorithm has the highest accuracyFor IACO-BP algorithm the variance of MSE is 46e-07the mean standard deviation of MSE is 68e-04 and therange of MSE is 22e-03 Each stability index of IACO-BPis the minimum value of corresponding index in the fouralgorithms Hence IACO-BP algorithm has better stabilityThe MSE value of the BP algorithm fluctuates significantlyhigher than the other three optimized BP algorithms asshown in Figure 5 IACO-BP algorithm is the best stabilityamong these three intelligent BP optimization algorithms asshown in Figure 6 Table 6 shows the statistical value of thenumber of iterations obtained from 10 random trainings offour networks The maximum number of iterations in thenetworks is 10000 and the network accuracy is 0001 Asshown in Table 6 the 10 simulations of PSO-BP network failto satisfy the accuracy requirement of the network during10000 iterations and terminate the algorithm in the way ofachieving the maximum number of iterations The averagenumber of iterations in IACO-BP network is 3398 whichis the smallest among these four algorithms The MSE of

10 Complexity

Table 5 Simulation results

Serial ID IACO-BP ACO-BP PSO-BP BPMSE MAPE MSE MAPE MSE MAPE MSE MAPE

1 23e-03 30e+00 48e-04 16e+00 38e-02 13e+01 38e-02 98e+002 43e-04 14e+00 33e-03 37e+00 26e-02 12e+01 38e-02 10e+013 94e-04 21e+00 32e-03 41e+00 79e-03 54e+00 10e-02 61e+004 11e-03 22e+00 34e-03 43e+00 21e-02 88e+00 10e-02 69e+005 21e-03 27e+00 91e-03 62e+00 12e-02 90e+00 77e-03 51e+006 11e-03 23e+00 26e-02 83e+00 40e-02 13e+01 12e-02 78e+007 13e-03 20e+00 46e-02 13e+01 11e-02 70e+00 11e+00 41e+018 20e-03 27e+00 40e-02 11e+01 17e-02 78e+00 18e-03 26e+009 67e-04 20e+00 21e-03 26e+00 37e-02 11e+01 27e-03 38e+0010 25e-03 24e+00 20e-03 26e+00 11e-02 87e+00 61e-02 14e+0111 10e-03 18e+00 43e-02 12e+01 43e-02 13e+01 11e+00 41e+0112 43e-04 16e+00 14e-03 25e+00 90e-03 71e+00 54e-03 46e+0013 34e-04 97e-01 37e-03 36e+00 22e-02 90e+00 26e-02 11e+0114 20e-03 27e+00 13e-03 21e+00 21e-02 85e+00 41e-02 11e+0115 17e-03 30e+00 45e-04 14e+00 78e-02 19e+01 16e-03 24e+0016 11e-03 21e+00 21e-03 27e+00 76e-02 17e+01 44e-03 43e+0017 49e-04 16e+00 40e-03 44e+00 26e-02 92e+00 89e-03 61e+0018 87e-04 18e+00 12e-02 64e+00 34e-03 37e+00 11e+00 41e+0119 49e-04 15e+00 46e-02 12e+01 88e-03 64e+00 83e-03 44e+0020 13e-03 22e+00 20e-02 84e+00 23e-02 94e+00 35e-02 12e+01mean 12e-03 21e+00 15e-02 61e+00 27e-02 99e+00 17e-01 12e+01Variance 46e-07 31e-01 29e-04 14e+01 43e-04 15e+01 15e-01 17e+02Standard deviation 68e-04 56e-01 17e-02 37e+00 21e-02 38e+00 38e-01 13e+01Range 22e-03 20e+00 46e-02 11e+01 75e-02 16e+01 11e+00 39e+01

Table 6 Comparison of iterations

Number of iterations IACO-BP ACO-BP PSO-BP BPMean 3398 4387 10000 7083MSE 14488 90117 0 111008

Table 7 Comparison of reliability prediction

Actual reliability 08 09 07 09 09 09286IACO-BP 08012 08956 06984 09041 09059 09256ACO-BP 07967 09112 07020 08949 09000 09293PSO-BP 08128 08924 07018 08656 08926 09341BP 08025 08985 07019 08470 09012 09286

iteration is 14488 which is also the smallest value amongthese three algorithms which satisfy the iterative precisiontermination procedure

Six failure data samples are randomly selected fromthe field records and the failure numbers are predictedusing by the four trained networks The storage reliability ofammunition is indirectly predicted by using (1) The resultsare described in Table 7 and Figure 7The indirect predictionof ammunition storage reliability model based on IACO-BP

algorithm has the best fitting degree and the highest accuracyamong the four algorithms

6 Conclusion

In this paper a prediction model of the ammunition storagereliability is considered under natural storage conditionsIACO-BP neural network algorithm is proposed to solve thisproblem Reliability is indirectly predicted by the number

Complexity 11

IACO-BPACO-BP

PSO-BPBP

0

02

04

06

08

1

12

14

MSE

2 4 6 8 10 12 14 16 18 200Number of experiments

Figure 5 MSE curves of four algorithms

IACO-BPACO-BPPSO-BP

0001002003004005006007008

MSE


Figure 6 MSE curves of three optimizing BP algorithms

065

07

08

09

1

Relia

bilit

y

IACO-BPACO-BPPSO-BP

BPActual reliability

2 3 4 5 61Sample sequence

Figure 7 Reliability prediction curves of four algorithms

12 Complexity

of ammunition failures obtained by IACO-BP ammunitionfailure number prediction model In order to improve theaccuracy and stability of network prediction the data pre-treatment and algorithm are promoted In data pretreatmentaspect the standardization and rationality of the data sampleare derived by using the methods of ldquozero failurerdquo pretreat-ment ldquoinverted hangingrdquo pretreatment small sample dataaugmentation and homogenization sampling In algorithmaspect we improve a pheromone updating strategy from thetraditional ant colony to a three-stage updating strategy ofpheromone pure increment volatilization and accumulationand doubling Compared with BP PSO-BP and ACO-BPmodel the experimental results prove that IACO-BP modelhas great advantages in precision stability and iterationtimes

Data Availability

The raw data used to support the findings of study areincluded within the article



Acknowledgments

This work was supported in part by Open Foundationof Science and Technology on Electro-Optical Informa-tion Security Control Laboratory (Grant no 61421070104)Natural Science Foundation of Liaoning Province (Grantno 20170540790) and Science and Technology Project ofEducational Department of Liaoning Province (Grant noLG201715)

References

[1] Z Xiaonan Y Junfeng D Siliang and H Shudong ldquoA NewMethod on Software Reliability PredictionrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 385372 8 pages 2013

[2] P Gao L Xie and W Hu ldquoReliability and random lifetimemodels of planetary gear systemsrdquo Shock and Vibration vol2018 Article ID 9106404 12 pages 2018

[3] S Wang ldquoReliability model of mechanical components withdependent failure modesrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 828407 6 pages 2013

[4] X Chen Q Chen X Bian and J Fan ldquoReliability evaluationof bridges based on nonprobabilistic response surface limitmethodrdquo Mathematical Problems in Engineering vol 2017Article ID 1964165 10 pages 2017

[5] G Peter OrsquoHara ldquoDynamic analysis of a 155 mm cannonbreechrdquo Shock and Vibration vol 8 no 3-4 pp 215ndash221 2001

[6] Z Z Muhammad I B Shahid I Tauqeer Z E Syed and FP Zhang ldquoNonlinear material behavior analysis under highcompression pressure in dynamic conditionsrdquo InternationalJournal of Aerospace Engineering vol 2017 Article ID 361693215 pages 2017

[7] B Zheng H G Xu and Z B Jiang ldquoAn estimation methodof ammunition storage life based on poisson processrdquo ActaArmamentarli vol 26 no 4 pp 528ndash530 2005

[8] L J Gullo A T Mense J L Thomas and P E ShedlockldquoModels and methods for determining storage reliabilityrdquo inProceedings of the 2013 Annual Reliability and MaintainabilitySymposium (RAMS) pp 1ndash6 Orlando Fla USA 2013

[9] C Su Y-J Zhang and B-X Cao ldquoForecast model for real timereliability of storage system based on periodic inspection andmaintenance datardquo Eksploatacja I Niezawodnosc-Maintenanceand Reliability vol 14 no 4 pp 342ndash348 2012

[10] Z Liu X Ma and Y Zhao ldquoStorage reliability assessment formissile component with degradation failure mode in a temper-ature varying environmentrdquo Acta Aeronautica Et AstronauticaSinica vol 33 no 9 pp 1671ndash1678 2012

[11] X S Si C H Hu X Y Kong and D H Zhou ldquo A residualstorage life prediction approach for systemswith operation stateswitchesrdquo IEEE Transactions on Industrial Electronics vol 61no 11 pp 6304ndash6315 2014

[12] Z Wang C Hu W Wang Z Zhou and X Si ldquoA case study ofremaining storage life prediction using stochastic filtering withthe influence of condition monitoringrdquo Reliability Engineeringamp System Safety vol 132 pp 186ndash195 2014

[13] Y Zhang M Zhao S Zhang J Wang and Y Zhang ldquoAnintegrated approach to estimate storage reliability with initialfailures based on E-Bayesian estimatesrdquo Reliability Engineeringamp System Safety vol 159 pp 24ndash36 2017

[14] B Zheng and G-P Ge ldquoEstimation of fuze storage life basedon stepped stress accelerated life testingrdquoTransactions of BeijingInstitute of Technology vol 23 no 5 pp 545ndash547 2003

[15] W B Nelson ldquoA bibliography of accelerated test plansrdquo IEEETransactions on Reliability vol 54 no 2 pp 194ndash197 2005

[16] J L Cook ldquoApplications of service life prediction for US armyammunitionrdquo Safety and Reliability vol 30 no 3 pp 58ndash742010

[17] Z G Shen J C Yuan J Y Dong and L Zhu ldquoResearch onacceleration factor estimation method of accelerated life test ofmissile-borne equipmentrdquoEngineering ampElectronics vol 37 no8 pp 1948ndash1952 2015

[18] Z-C Zhao B-W Song and X-Z Zhao ldquoProduct reliabilityassessment by combining accelerated degradation test andaccelerated life testrdquo System Engineering Theory and Practicevol 34 no 7 pp 1916ndash1920 2014

[19] X Y Zou and R H Yao ldquoSmall sample statistical theory and ICreliability assessmentrdquo Control and Decision vol 23 no 3 pp241ndash245 2008

[20] M Zhang and W Chen ldquoHot spot data prediction modelbased on wavelet neural networkrdquo Mathematical Problems inEngineering vol 2018 Article ID 3719564 10 pages 2018

[21] H J Chen K N Teng B Li and J Y Gu ldquoApplication of neuralnetwork on missile storage reliability forecastingrdquo Journal ofProjectiles Rockets Missiles and Guidance vol 30 no 6 pp 78ndash81 2010

[22] J Liu D Ling and S Wang ldquoAmmunition storage reliabilityforecasting based on radial basis function neural networkrdquo inProceedings of the 2012 International Conference on QualityReliability Risk Maintenance and Safety Engineering pp 599ndash602 Chengdu China 2012

[23] Z G Tian L N Wong and N M Safaei ldquoA neural networkapproach for remaining useful life prediction utilizing bothfailure and suspension historiesrdquoMechanical Systems and SignalProcessing vol 24 no 5 pp 1542ndash1555 2010

[24] E-Z Zhang and Q-W Chen ldquoMulti-objective particle swarmoptimization for uncertain reliability optimization problemsrdquoControl and Decision vol 30 no 9 pp 1701ndash1705 2015

Complexity 13

[25] H Gong E M Zhang and J Yao ldquoBP neural networkoptimized by PSO algorithm on ammunition storage reliabilitypredictionrdquo in Proceedings of the 2017 Chinese AutomationCongress (CAC) pp 692ndash696 Jinan China 2017

[26] X Liu L Fan L Wang and S Meng ldquoMultiobjective reliablecloud storage with its particle swarm optimization algorithmrdquoMathematical Problems in Engineering vol 2016 Article ID9529526 14 pages 2016

[27] LinYuan Chang-AnYuan andDe-ShuangHuang ldquoFAACOSEa fast adaptive ant colony optimization algorithm for detectingSNP epistasisrdquo Complexity vol 2017 Article ID 5024867 10pages 2017

[28] F P Wang ldquoResearch on application of big data in internetfinancial credit investigation based on improved GA-BP neuralnetworkrdquo Complexity vol 2018 Article ID 7616537 16 pages2018

Research ArticleGreen Start-Upsrsquo Attitudes towards Nature WhenComplying with the Corporate Law

Rafael Robina-Ram-rez 1 Antonio Fernaacutendez-Portillo 2 and Juan Carlos D-az-Casero1

1Business and Sociology Department University of Extremadura Avda de la Universidad sn 10071 Caceres (Extremadura) Spain2Finance and Accounting Department University of Extremadura Avda de la Universidad sn 10071 Caceres (Extremadura) Spain

Correspondence should be addressed to Rafael Robina-Ramırez rrobinaunexes

Received 1 December 2018 Revised 25 January 2019 Accepted 10 February 2019 Published 21 February 2019

Academic Editor Rongqing Zhang

Copyright copy 2019 Rafael Robina-Ramırez et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper examines how Spanish green start-ups develop improved attitudes towards nature Despite the prevalence of theoriesbased on green entrepreneurs very little research has been conducted on how green attributes influence nature in the frameworkof the new Spanish Criminal Code in what concerns corporate compliance One hundred and fifty-two start-ups were interviewedin 2018 Smart PLS Path Modelling has been used to build an interaction model among variables The results obtained have asignificant theoretical and practical implication since they add new findings to the current literature on how start-ups incorporatethe recent Criminal Code in their environmental decisionsThemodel reveals a strong predictive power (R-squared = 779)Thereare grounds to say that start-ups should implement effective monitoring systems and new organisational standards in addition todeveloping surveillance measures to avoid unexpected sanctions

1 Introduction

The paper delves into green entrepreneurs attitudes in start-ups when complying with the environmental law in the con-text of the recent Spanish Criminal Code in what concernscorporate compliance Regulative compliance explains theaim that organisations want to reach in their efforts to complywith environmental regulations

It is estimated that 999 of Spanish organisations areSMEs which means that there are 2498870 SMEs in Spaincapable of delivering more sustainable goods and servicesthrough organic food production fair trade natural andhandmade craft sustainable tourism services environmentalconsulting green energy firms etc [1]

Spain has implemented several policies to enhanceentrepreneurship in the past decades Businesses are requiredto develop rapidly reduce unemployment rates and improvethe deteriorated economy without harming the environment[1]

Since Brundtland wrote Our Common Future (1987)[2] companies have been aided to delve into the needs ofthe current time It is proposed to look after the future

generations in order to meet their own demands in theirdevelopment efforts The sustainable argument in reducedcompanies has gradually expanded [3] and is supported byempirical studies [4]

For decades the emerging literature on green entrepre-neurship in small companies has focused on environmentallyoriented entrepreneurship achieving sustainable develop-ment [5] The evolution of green entrepreneurs has recentlytranslated into the appearance of many green start-ups in awide variety of sectors business strategies andmarketing tar-gets These entrepreneurs have created products and servicesto meet Green Opportunities and benefit nature at the sametime [6] Ivanko [7] added the social dimension linked tothe environment to ascertain the problems suffered by thecommunity Social entrepreneurship brings a range of newsocial values to profits environment and fair trade [8]

Even though there is no agreement on the definitionof green entrepreneurs [9] they are viewed by scholarsas companies with the predisposition to pursue potentialopportunities that produce both economic and ecologicalbenefits through green activities


2 Complexity

Their green market orientation has defined their dualidentity [10] Thus profit and environment may competeon equal terms As for the way green companies oper-ate in the market their environmental orientation reflectsgreen innovativeness market proactiveness and risk-taking[11 12]

Considering these features Walley and Taylor [13] high-lighted three pillars of green entrepreneurs as three keyfeatures of sustainable development economic prosperitybased on the balance between economy and environment thepursuit of environmental quality to protect nature throughinnovative and competitive processes and the social andethical consideration of the business culture focused onpromoting Green Opportunities among companies

Based on the literature review four attributes wereextracted from these pillars to define a model for creat-ing environmental value for green start-ups [14 15] Theseattributes are BalancedGreen Investment [16] environmentalimpact of green entrepreneurs [4 17 18] innovation andcompetitiveness [19ndash21] and Green Opportunities [22]

The attributes had to be assessed since a new regula-tory compliance emerged in 2015 in Spain [23] The newlegal perspective relies on regulation and public concern toovercome substantial damage to the quality of air soil orwater animals or plants In the context of the regulatoryframework effective monitoring systems not only improvethe organisational management but also avoid unexpectedsanctions [24]

The contribution of this paper is threefold (1) it triesto ascertain whether the Environmental Value Model isempirically reflected in these attributes and validate thedifferent opportunities for green start-ups (2) it analyseswhether the recent Spanish regulatory compliance influencesgreen start-ups not only by imposing coercive rules but alsoby allowing them to design organisational norms of surveil-lance [25] and (3) it explores whether the new regulatorycompliance scenario affects green entrepreneurs attitudestowards nature

The relation between green entrepreneurs attitudes andthe recent pressure exerted by the Spanish regulatory com-pliance appears not to have been considered by greenentrepreneurs As a result the paper examines what rolegreen start-ups play in the new legal framework whencomplying with the Corporate Law to protect nature

One hundred and fifty-two environmental green start-ups were involved in the study The data collected was pro-cessed using SEM-PLS PathModellingTheoretical and prac-tical implications were provided not only for environmentalauthorities but also for green entrepreneurs when avoidingunlawful environmental activity that damages nature Theresearch conducted further adds content to the currentliterature particularly about the findings of green start-upswhen incorporating related measures into their organisa-tional decisions

The paper is structured as follows First greenentrepreneurship is studied in the context of Spanishregulatory compliance and then an Environmental ValueModel for green entrepreneurs is proposed In third place themethodology used for this paper is explained Fourth results

are drawn from the data collected Finally the discussionconclusions and limitations are presented

2 Literature Review

21 Green Entrepreneurship in the Context of SpanishRegulatory Compliance

H1 Environmental Corporate Compliance (ECC) PositivelyInfluences Green Attitudes towards Nature (GAN) amongEntrepreneurs Since the industrial degradation occurredenvironmental entrepreneurs emerged quite rapidly as newcompanies in the environmental sector and more recentlyas green start-ups [26] Sustainable research in the fieldof entrepreneurship becomes crucial to avoid threats tonature [27] In this framework entrepreneurs have constantlydesired to minimise their impact on the environment [28]seeking solutions to align economy with ecology [16] Theproposal is structured in ongoing process of improved actionsand attitudes connected to the emerging environmentalopportunities through organisational discoveries [9]

However since the Spanish regulatory complianceemerged in 2015 a new perspective to deter environmentaldamage has been developedThe first environmental concernfor nature came with the Spanish Constitution [29] andlater the Spanish Criminal Code of 1983

Administrative sanctions were established for individualswho committed unlawful environmental activities such ascausing emissions discharges extractions or excavationsvibration injections or deposits in the atmosphere soil orterrestrial and marine waters (Article 1731) [29]

Fortunately the rules and regulations have graduallyincreased their protection over environment The purposehas been to make an inhabitable environment place to liveas well as to raise green awareness of sustainable principlesamong decision-makers [30]

To date most ethical endeavour to defend the environ-ment has not been consistent enough to adequately addressenvironmental threats [31]

With the reform of the Spanish Corporate GovernanceCode [23] judges were allowed to hear cases not only aboutindividuals having committed unlawful acts but also aboutcompanies The legal reform was based on transferring tocompanies the same rights and obligations as individualsAlthough individuals are typically liable for committingenvironmental crimes towards nature liability can now betransmitted to the enterprise (vicarious liability) [32]

On the one hand the new legal framework is moresevere than the previous one due to the new obligationfor companies to comply with the Criminal Code and onthe other hand the legal reform allows companies to avoidsanctions by developing surveillance and control measures[23] Such organisational standardsmust prevent informationbias and foster green attitudes towards nature [33]

The strategy used to comply with this law should bedesigned through innovation and the creation of a specificdepartment to deliver organisational measures [34 35]

The process however requires developing surveillanceprotocols through innovation and creation procedures [36]

Complexity 3

4 GreenOpportunities

(GO)

Environmental Value Model

(EVM)

2 Environmental Impact

(EI)

3 Innovation andCompetitiveness

(IC)

1 Balanced Green Investment

(BGI)

Figure 1 Environmental Value Model (EVM)

to balance companies economic and ecologic activities [16]Protocols alert green entrepreneurs about the risks thatcompanies could be facing when their workers behave irre-sponsibly towards nature [37] which entails the companybeing seen as carrying out nonenvironmental and illegalactivities [38]

H2 Environmental Corporate Compliance (ECC) PositivelyInfluences the Environmental Value Model (EVM) This newculture promoted by law has led to a new behavioural modelthat adds value to companies The EVM prevents companiesnot only from committing unlawful environmental activity[39] but also from performing incorrect environmental pro-cedures and the underlying uncertainty of reducing the riskof negative implications for nature [4]

Likewise the preventive activities of the EVM for regula-tory compliance require frequent assessment by establishingstandards of procedures to ensure being up to date in theevent of irresponsible behaviours leading to environmentalcrimes [40] Moreover it is also essential to settle disciplinaryprocedures through sanctions and prevent illegal conductalong the creation and innovation process [41]

Organisational standards of procedures are key factorsof the environmental model They have to be establishedby the compliance officer [42] Such rules not only have tobe supervised periodically [43] but also have to be com-municated to the employees eg by training programmesor spreading publications to explain these items and detectpossible unlawful conduct empirically

The following organisational decisions must be made incompanies in their endeavour to comply with legal require-ments according to the SpanishCorporate Governance Code[23]

(i) A department must be created to supervise the com-pliance of green innovations procedures and services[34]This department should establish amanagementmodel to avert unlawful environmental actions andallocate a corporate budget for surveillance practiceswithin the company [44]

(ii) Training programmes on the protocols or proceduresmust be taught to appropriately implement surveil-lance measures in companies [36] Protocols mustbe designed to report any prior unlawful behaviourwithin the company [38] and reveal any necessarychanges to maintain legality in every business activity[40]

(iii) A disciplinary system must be implemented toaddress unlawful environmental actions and also thefinancial managerial and social obligations with thecompany and society [41]

For the three steps to be put into effect recentcorporate laws have enhanced several transparencyand cooperation models within companies that haveadded value to environmental companies

22 Environmental Value Model for GreenEntrepreneurs (EVM)

H3 The Environmental Value Model (EVM) Positively Influ-ences GreenAttitudes towardsNature (GAN) Innovations andbusiness models have added increasing value to companiesin the past decades Emerging sustainable approaches such asfair trade circular economy or lowsumerism have set up newtrends [45] in response to demanding green consumers [46]

These models require defining key attributes to ascer-tain practical consequences in green entrepreneurs Fourattributes have been extracted from the literature review inthis respect (1) Balanced Green Investment (2) environ-mental impact of green entrepreneurs (3) innovation andcompetitiveness and (4) Green Opportunities

The contribution of this model is twofold first to finda predictive green model to delve into the empirical impli-cations for Spanish green start-ups and second to analysewhether the balance between economy and sustainability isempirically relevant for Spanish green start-ups Figure 1shows the attributes of the model

4 Complexity

H4 The Environmental Value Model (EVM) Positively Influ-ences Balanced Green Investment (BGI) According to theOECD [47] sustainability creates conditions in which indi-viduals and environment interact harmoniously Sustain-ability allows reaching environmental economic and socialstandards to respect future generations

In the context of environmental deterioration sustain-able management has become a pivotal issue [48] Eventhough literature about damaging environmental outcomesis vast there is a widespread ignorance towards the positiveinitiatives taken by green entrepreneurs to address suchdamage Such ignorance is due to failures in coordinat-ing expectations and preferences throughout the businessmodels to reduce the actual high cost for companies [49]Green models have introduced potential opportunities toproduce both economic and ecological benefits througheco-friendly products and services [50] How these greenmodels influence environmental and financial performanceremains unclear While some studies have found a nega-tive relationship between tangible-external green strategiesand entrepreneurial performance [51] the impact of greenentrepreneurship performance has also been positive in somecases [16]

The balance between investment for profit and to protectnature largely depends on the responsible performance ofgreen entrepreneurs [52] and the voluntary disclosure of envi-ronmental standards [53] Developing managerial standardsto have a positive impact on the environment is precisely oneof the core concepts of the Spanish regulatory compliance

H5 The Environmental Value Model (EVM) Positively Influ-ences Innovation and Competitiveness (IC) The Environmen-tal Value Model (EVM) is also associated with three featuresof entrepreneurs innovativeness proactiveness and risk-taking [19]

First innovativeness describes a tendency to delivernew ideas engage in experimentation and support creativeprocesses Innovation enables entrepreneurs to combineresources to launch new products or processes [20] togain advantages over competitors [54] and to differentiatethemselves from others [21] Competitive advantages can alsoenhance companies absorptive capacity by developing theirability to imitate advanced green technologies [55]

Second proactiveness refers to the capacity to respond tocustomer needs by introducing green products services ortechnology [21] Companies are facing growing pressure fromcustomers with raising awareness of environmental issuesProactive companies are likely to respondmore quickly to theneeds of customers than their competitors Under the trendof customers attitude towards green marketing companiescan reap the financial benefits of becoming a pioneer in greeninnovation practices

Third risk-taking is one of the primary personalattributes of entrepreneurs and it reflects the tendency toadopt an active stance when investing in projects [56]Although the propensity of risk may bring fresh revenue itis often associated with complex situations and uncertaintiesthat can entail companies becoming trapped in changingcircumstances [57]

H6 The Environmental Value Model (EVM) Positively Influ-ences Environmental Impact (EI) Green entrepreneurialactivity can simultaneously foster economic and ecologicalbenefits for society by creating market opportunities andpreventing environmental degradation [58] in two differentways First the entrepreneurial action may reduce environ-mental degradation and capture economic value by allevi-ating market failure [4] and using green technologies todecrease the consumption ofwater electricity coal or oil [17]Second green entrepreneurs can reduce unhealthy damageto employees at work by decreasing the consumption of toxicsubstances [18] and protecting nature by applying corporatetax incentives to green companies [59]

H7 The Environmental Value Model (EVM) Positively Influ-ences Green Opportunities (GO) One way of building oppor-tunities to expand environmental values within the companyis by developing employees green skills [22]

According to TECCes argument Green Opportunitiesfacilitate the generation of new product processes [21]Assessing potential opportunities and adopting eco-friendlytechnologies could verymuch help to overcome environmen-tal market failures [60] The electrical utility industry forinstance has the possibility of taking more advantage outof wind power [10] and to make more green use of naturalresources [61]

After describing the four green attributes that form theEnvironmental Value Model the paper turns to the analysisof the methodology results discussion and conclusions

3 Research Methodology

31 Data Collection and Sample The lack of an officiallist of Spanish environmental start-ups has hampered ourendeavour to list and collect the data of SMEs The researchteam chose green start-ups for two reasons (1)The number ofgreen start-ups has recently increased in Spain and they arebecoming a new phenomenon of environmental awarenessand (2) given the lack of studies in the context of the recentSpanish regulatory compliance this paper helps to shed lighton the role green start-ups play and how the new regulationaffects them

242 online environmental green start-ups were initiallyidentified in Spain Following the literature review the focushas been on those who met the four attributes Thus the firststep was to ensure that green start-ups indeed complied withthose attributes and for this the answers to four questionswere required

(i) Balanced Green Investment (BGI) does your com-pany seek a balance between economic and environ-mental aims

(ii) Innovation and Competitiveness (IC) has your com-pany implemented innovative green ideas and prac-tices in the market

(iii) Environmental Impact (EI) has your company imple-mented green technologies to prevent contributing tothe environmental degradation of nature

Complexity 5

Table 1 Online environmental startups in Spain

Sector Population Sample Sector Population SampleAgriculture 2 1 Fashion 3 3Art 4 4 Finance 4 3Automotive 4 4 Food 9 7Clean Technology 15 11 Funeral Industry 1 1Construction 3 3 Health and Wellness 10 9Consulting 11 12 Health Care 2 2Cosmetic 4 4 Human Resources 1 1Design 10 9 Information Technology 15 10Digital Marketing 2 2 Internet of things 5 4E-commerce 8 8 Life Sciences 1 1Education 6 5 Logistic 4 3Electric bicycle 4 3 Nature 7 6Energy 14 10 Restaurants 11 7Entertainment 4 2 Travel 16 9Events 3 2 Urban development 3 2Farming 3 2 Water 3 2Total 97 82 Total 95 70Total Population 192 Total Sample 152

(iv) GreenOpportunities (GO) does your company iden-tify the Green Opportunities that the market offers

A brief presentation of the aims of the research was also sentto them in the context of the Spanish Corporate GovernanceCode [23] in what concerns corporate compliance Two hun-dred and twelve start-ups answered and asserted complyingwith the four attributes The survey was sent to them andone hundred fifty-two companies returned the survey withtheir answersThe team then emailed the compliance officersand several heads of departments between June and August2018 requesting they answer the survey The structure anddistribution of the population under study and the sample areexplained in Table 1

32 Surveys According to the literature review a survey wasdrafted to measure green entrepreneursrsquo attitudes towardsnature empirically Twenty statements were completed

Two focus groups held online meetings via Skype tovalidate the survey Skype was used to overcome the distancebetween participants Nine green entrepreneurs from differ-ent areas of Spain were involved Twenty original questionswere discussed during a period of two hours in eachmeetingThe survey was amended as a result Five of the originalitems were deleted and two added With this information anexperiential survey was conducted to validate the proposedsurvey Six interviews were made to contrast the clarity ofthe questionsThree questionswere furthermodified after thepretest The final survey is shown in Table 2

The items were analysed through the ten-point Likertscale to indicate the degree of importance of the factors (1= ldquofully disagreerdquo to 10 = ldquofully agreerdquo) (Allen and Seaman2007)

33 Model and Data Analysis Process SEM-PLS Path Mod-elling was used to ascertain the model and obtain results[62] SEM not only enables examining the relationshipsbetween observable indicators and constructs statistically[63] but also works with composite model latent variables[62]Themethods can be used for explanatory and predictiveresearch as well as complex modelsThe green start-up modelis composed of three endogenous constructs the Environ-mental Value Model (EVM) Green Attitude towards Nature(GAN) and Environmental Corporate Compliance (ECC)and four exogenous ones Balanced Green Investment (BGI)Innovation andCompetitiveness (IC) Environmental Impact(EI) and Green Opportunities (GO) (see Figure 2)

Seven hypotheses were analysed in the study

H1 Environmental Corporate Compliance (ECC)

positively influences Green Attitudes towards Nature(GAN) among entrepreneursH2 Environmental Corporate Compliance (ECC)

positively influences the Environmental Value Model(EVM)H3 the Environmental ValueModel (EVM)positively

influences Green Attitudes towards Nature (GAN)H4 the Environmental ValueModel (EVM)positively

influences Balanced Green Investment (BGI)H5 the Environmental ValueModel (EVM)positively

influences Innovation and Competitiveness (IC)H6 the Environmental ValueModel (EVM)positively

influences Environmental Impact (EI)H7 the Environmental ValueModel (EVM)positively

influences Green Opportunities (GO)

6 Complexity

Table 2 Latent variables and the elaborated questionnaire

Latent variables Questions

GANGreen Attitudes towardsNature

Do you think it is important for your company to be aware of being an appropriate steward ofnatural resources (GAN

1)

Do you think it is important for your company to reduce hazardous emissions or toxic materialsto improve health and safety at work (GAN

2)

ECC EnvironmentalCorporate Compliance

Do you think it is important to implement the prevention model to supervise the compliance ofgreen innovations and procedures and services with the law (ECC

1)

Do you think it is important to undertake protocols and training procedures for staff members toapply these surveillance and organisational measures (ECC

2)

Do you think it is important to establish a disciplinary system to prevent non-compliance withthe law (ECC

3)

The Environmental ValueModel (EVM)

Do you think it is important to build entrepreneurial models to create environmental valuesbased on sustainable principles (EVM

1)

Do you think it is important to describe analyse and communicate such values in your company(EVM

2)

Balanced GreenInvestment(BGI)

Do you think it is important to seek a balance between profit nature and people throughsustainable management to protect nature (BGI

1)

Do you think it is important to reach environmental economic and social standards to respectfuture generations in an innovation context (BGI

2)

Environmental Impact (EI)

Do you think it is important to produce green technologies to prevent environmental degradationon the planet (EI

1)

Do you think it is important to reduce the consumption of toxic substances and harmfulemissions to prevent damages to the health and safety of employees at work (EI

2)

Green Opportunities (GO)

Do you think it is important to develop green skills among employees to build opportunities toexpand environmental values in the company (GO

1)

Do you think it is important to implement a new generation of manufacturing processes to reducepollution in production (GO

2)

Do you think it is important to implement eco-friendly technologies to overcome the marketfailures (GO

3)

Innovation andCompetitiveness (IC)

Do you think it is important to propose new ideas and support creative processes (IC1)

Do you think it is important to become a pioneer in green innovation ideas and practices (IC2)

Do you think it is important to respond faster than your competitors to the needs of customers(IC3)

4 Results

41 Measurement Model Reliability and validity are the firsttwo conditions used to assess the model The process canbe structured in four steps (1) individual item reliability(2) construct reliability (3) convergent validity and (4)discriminant validity First individual reliability is measuredthrough the load (120582) of each item The minimum levelestablished for acceptance as part of the construct is typically120582 gt = 0707 [64] This condition was validated in the model(see Figure 3)

Composite reliability (CR) was applied to test the consis-tency of the constructs This evaluation measures the rigourwith which these elements measure the same latent variable[65]

Cronbachs alpha index also determines the consistencyof the model for every latent variable Values higher than 07are typically accepted [66] Table 3 shows that the reliabilityof each construct was accepted

AVE measures the convergent validity the acceptablelimit of which is 05 or higher This indicator providesinformation about the level of convergence of the constructswith their indicators [67] Table 3 also shows that the values

meet the criteria Rho A was also measured Results exceedthe value of 07 [68]

Table 4 explains the correlations between the constructson the left A construct should share more variance with itsindicators than with other latent variables in the model [69]However Henseler et al [70] did detect a lack of discriminantvalidity in this respect Ratio Heterotrait-monotrait (HTMT)provides a better approach to this indicator The resultsobtained in this sense are on the right The HTMT ratio foreach pair of factors is lt090

42 Structural Model Analyses The structural model ofassessment proposed is explained in Table 5 The generalcriterion used to evaluate the structural model is the coeffi-cient of determination (R-squared) R-squared analyses theproportion of variance (in percentage) in the exogenousvariable that can be conveyed by the endogenous variableThe R-squared value can be expressed from 0 to 1 Valuesclose to 1 define the predictive accuracy Chin [71] proposeda rule of thumb for acceptable R-squared with 067 033 and019 They are defined as substantial moderate and weakpredictive power respectively

Complexity 7

Table 3 Cronbach Alpha rho A Composite Reliability and AVE

Cronbachs Alpha rho A Composite Reliability Average VarianceExtracted (AVE)

ECC 0874 0895 0876 0706EI 0810 0812 0810 0681EVM 0853 0854 0854 0745GAN 0820 0826 0822 0698BGI 0817 0827 0820 0696GO 0849 0853 0848 0651IC 0864 0869 0865 0683

Table 4 Measurement model Discriminant validity

Fornell-Larcker Criterion Heterotrait-monotrait ratio (HTMT)ECC EI EVM GAN BGI GO IC ECC EI EVM GAN BGI GO IC

ECC 0840EI 0644 0825 0655EVM 0697 0730 0863 0694 0729GAN 0793 0557 0830 0836 0793 0561 0832BGI 0703 0406 0780 0745 0834 0712 0408 0783 0760GO 0456 0491 0773 0557 0571 0807 0452 0494 0770 0562 0576IC 0499 0268 0597 0652 0550 0552 0826 0496 0269 0599 0652 0557 0556

Environmental Corporate

Compliance(ECC)


(EVM)

Green Attitudes towards Nature

(GAN)

GAN1 GAN2

ECC2

EI2

EI1 Environmental

Impact

(EI)

Innovation and

Competitiveness

(IC)

Green

Opportunities

(GO)

Balanced Green

Investment

(BGI)

IC2

IC3

IC1

GO2

GO1

BGI2

BGI1

ECC1

ECC3

EVM2EVM1

GO3

Figure 2 Conceptual scheme of the structural equation model utilized BAN Green Attitudes towards Nature ECC EnvironmentalCorporate Compliance EVM Environmental Value Model BGI Balanced Green Investment EI environmental impact IC innovationand competitiveness GO Green Opportunities EP Educational Process

8 Complexity

Table 5 Comparison of hypothesis

Hypotheses Effect Path coefficient(120573)

ConfidentInterval(25)

ConfidentInterval (95)

t-statistic(120573STDEV)

p-Value Supported

H1 ECC997888rarrGAN 0417 0096 0665 5116 0009 Yes lowastlowast

H2 ECC 997888rarrEVM 0697 0551 0825 2905 0000 Yes lowast lowast lowast

H3 EVM 997888rarrGAN 0539 0288 0864 4760 0001 Yes lowastlowast

H4 EVM 997888rarrBGI 0780 0673 0878 6375 0000 Yes lowast lowast lowast

H5 EVM 997888rarr IC 0597 0462 0727 2737 0000 Yes lowastlowastH6 EVM 997888rarr EI 0730 0548 0873 8395 0000 Yes lowast lowast lowastH7 EVM 997888rarr GO 0773 0692 0850 4476 0000 Yes lowastlowastNote For n = 5000 subsamples for t-distribution (499) Students in single queue lowast p lt 005 (t (005499) = 164791345) lowastlowast p lt 001 (t(001499) = 2333843952)lowast lowast lowast p lt 0001 (t(0001499) = 3106644601)

(ECC)0487

(EVM)

0779(GAN)

GAN1 GAN2

0888

07770780

ECC2

EI2

EI10532(EI)

0356(IC)

0598(GO)

0609(BGI)

0772

IC2

IC3

IC1

GO2

GO3

GO1

BGI2

BGI1

ECC1

ECC3

0697

EVM2EVM1

0815

0887

0852

0798

08180729

0867

0597

0730

0773

0417

09790740

07820844 0882

0539

0793 0876

Figure 3 Individual item reliability

Our model presents constructs with high predictivevalue (R-squared997888rarrGAN=0776) and moderate predictivevalue (R-squared997888rarrBGI=0606 R-squared997888rarrGO=0595 R-squared997888rarrEI=0529 R-squared997888rarrIC=0352) and weak R-squared997888rarrEVM=0268) Therefore the evidence shows thatthis model is applicable for green start-ups when developingGreen Attitudes towards Nature due to its strong predictivepower and explanatory capacity

Table 5 also shows the hypothesis testing using 5000bootstrapped resamples After analysing the path coefficientsthere are grounds to assert that all hypothesised relationshipsare significant at 999 confidence levels except two of themthe first one is ECC997888rarrGAN 120573=0417 Statistical T=5116 andthe second one is EVM997888rarrGAN120573=0539 Statistical T=4760These are supported with a 99 confidence level

As part of the assessment of the structured model SRMSalso needs to bemeasured to analyse the good fit of themodelIn the research SRMR is 0075 which is less than 008 as Huand Bentler [72] had expressly indicated

Blindfolding is also assessed within the model It mea-sures the predictive capacity of the model through the Stone-Geisser test (Q2) [73 74] The result revealed that the modelis predictive (Q2 = 0474) since Q2 gt 0

43 Results for Unobserved Heterogeneity The unobservedheterogeneity can to be analysed with different PLS seg-mentation methods [75 76] We select FIMIX-PLS and PLSprediction-oriented segmentation (PLS-POS) methodologyfor two reasons [76] First according to the evaluation ofthese methods Sarstedt [75] concludes that FIMIX-PLS is

Complexity 9

Table 6 Indices FIT Criteria for model choice

K=2 K=3 K=4 K=5AIC (criterio de informacion de Akaike) 1948641 1907489 1884002 1864526AIC3 (modificado de AIC con Factor 3) 1975641 1948489 1939002 1933526AIC4 (modificado de AIC con Factor 4) 2002641 1989489 1994002 2002526BIC (criterio de informacion Bayesiano) 2030285 2031468 2050315 2073173CAIC (AIC consistente) 2057285 2072468 2105315 2142173MDL5 (longitud de descripcion mınima con Factor 5) 2572865 2855384 3155569 3459765LnL (LogLikelihood) -947320 -912744 -887001 -863263EN (estadıstico de entropıa (normalizado)) 0672 0724 0722 0767

viewed as the proper commonly used approach to captureheterogeneity in PLS Path Modelling Second PLS-POSinforms about nonobserved heterogeneity in the structuralmodel as well as the constructsmeasures with both formativeand reflective models [76]

In order to achieve a major capacity of prediction PLS-POS provides ongoing improvements of the objective tar-geted Due to the hill-climbing approach iterations of thealgorithmmight generate an intermediate solution not goodenough to be validated For this reason it is importantto run the application of PLS-POS with different startingsegmentations [76] In our case we have applied the PLS-POSalgorithm with 5 4 3 and 2 partitions

Regarding Becker et al [76] the bias from using eitherof the two methods (FIMIX-PLS or PLS-POS) is much lowerthan that obtained from analysing the overall sample withoutuncovering heterogeneity [76]

In addition FIMIX-PLS is understood as better methodsfor reducing biases in parameters estimates and avoidinginferential Becker et al [76] find an exception in lowstructural model heterogeneity and high formative measure-mentmodel heterogeneity Regarding this condition FIMIX-PLS generate more biased results than those resulting fromignoring heterogeneity [76]

In our case we do not find unobserved heterogeneitywithPLS-POS algorithm because the results indicate one grouptoo small for the study in all iterations

As a result we have used de FIMIX-PLS This methodol-ogy considers the possibility of acceptance in any segmentobserved The observations are adapted depending on thenumber of segments Through the linear regression func-tions a group of potential segments is given Every case isattached to the segment with the highest probability FIMIXmethodology was applied to structure the sample into severalsegments Selecting the convenient number of segments wasthe first problem found It is usual to repeat the FIMIX-PLSmethod with successive numbers of latent classes [77] Inthis case taking into account the sample size n = 152 thecalculation was made for k=2 k=3 k=4 and k=5 Differentinformation criteria offered by the FIT indices were used tocompare the results obtained The controlled AIC (CAIC)Akaike (AIC) the standardized entropy statistic (EN) andthe Bayesian information criterion (BIC) were compared

The study is implemented in four stages in the first placeFIMIX provides the number of best segments Hence theconstruct that confirms these segments is found As a result

the segments and the model are estimated Table 6 shows theresults provided for the FIT indices

Firstly in order to find the number of samplings it canbe organised into the FIMIX test applied The algorithm forthe size of the sample so as to be able to use PLS-SEMwith 10repetitions was constructed As a result the new compositionwas carried out using the expected maximization algorithm(EM) The EM algorithm switches between maximizationstep (M) and performing an expectation step (E) [78] StepE assesses the accepted estimation of the variables Step Mmeasures the parameters by making as big as possible thelogarithmic registration likelihood obtained in step E StepsE and M are used continuously until the results are balancedThe equilibrium is attained when no essential progress in thevalues is attained

The results after running FIMIX with different numbersof partitions are shown in Table 6 As the number of segmentswas unknown beforehand the different segment numberswere contrasted in terms of appropriateness and statisticalanalysis [79 80]

Trial and error information can be conveyed within theEN and information criteria due to their sensitiveness to thefeatures of themodel and their dataThe data-based approachgave a rough guide of the number of segments [78]

As a result the criteria applied were assessed The abilityto accomplish aim of the information criteria in FIMIX-PLS was studied for a broad range of data groups [81]Their outputs indicated that researchers should take intoaccount AIC 3 and CAIC While these two criteria expressthe equivalent number of segments the results possiblyshow the convenient number of segments Table 6 showsthat in our study these outputs do not indicate the samenumber of segments so AIC was utilized with factor 4(AIC 4 [82]) This index usually behaves appropriately Thesame number of segments was shown in the case study(see Table 6) which was k=3 Then it is understood to bea strong miscalculation even though MDL5 expressed theminimum number of segments k+1 In this case it wouldimply 3 [78] The regulated entropy statistic (EN) was one ofthe measurements of entropy which was also esteemed [83]EN applied the likelihood that an observation is addressedto a segment to express if the separation is trustworthy ornot The greater the chance of being included to a segment isfor an assessment the higher segment relationship isThe ENindex varies between 0 and 1 The greatest values express thebetter quality segmentation Other researches in which EN

10 Complexity

Table 7 Relative segment sizes

K Segment 1 Segment 2 Segment 3 Segment 4 Segment 52 0537 04633 0384 0310 03074 0342 0325 0258 00745 0462 0222 0138 0131 0047

Table 8 Path coefficients for the general model and for the three segments

PathCoefficients Global model k=1 (384)

n=54k=2 (31)

n=51k=3 (307)

n=47MGAk1 vs k2

MGAk1 vs k3

MGAk2 vs k3

ECC997888rarrEVM 0620lowast lowast lowast 0274lowast 0742lowast lowast lowast 0932lowast lowast lowast 0468 ns 0659 ns 0190 nsECC997888rarrGAN 0340lowast lowast lowast 0363lowastlowast -0033 ns 0015 ns 0397lowastlowast 0348lowast 0049 nsEVM997888rarrEI 0606lowast lowast lowast 0456lowast lowast lowast 0548lowast lowast lowast 0794lowast lowast lowast 0091 ns 0338 ns 0246 nsEVM997888rarrGAN 0549lowast lowast lowast 0311lowast 0986lowast lowast lowast 0978lowast lowast lowast 0675 ns 0667 ns 0008 nsEVM997888rarrGBI 0692lowast lowast lowast 0567lowast lowast lowast 0956lowast lowast lowast 0541lowast lowast lowast 0389 ns 0026 ns 0415lowast lowast lowastEVM997888rarrGO 0649lowast lowast lowast 0550lowast lowast lowast 0728lowast lowast lowast 0570lowast lowast lowast 0178 ns 0021 ns 0157lowastEVM997888rarrIC 0537lowast lowast lowast 0095 ns 0943lowast lowast lowast 0550lowast lowast lowast 0848 ns 0455 ns 0393lowast lowast lowastNote ns not supported lowast plt005 lowastlowast plt001 lowast lowast lowast plt0001

values are above 050 which offer easy understanding of thedata into the chosen number of segments are also appointed[84 85] Table 6 shows that all the segmentations conveyedEN valuesgt050 even though the greatest value is achievedfor k = 5 with EN=0767 for k=5 and EN=0724 for k=3

Accordingly the number of best segments was k=3FIMIX-PLS indicates then the number of segments due tothe lowest size of the segmentation which in this case is307 Table 7 shows that for the k=3 solution and a samplen=152 segment partitioning is 384 (54) 31 (51) and307 (47) [72 86] In spite of the percentages the segmentsizes are significant so they are enough to use PLS Due tothe covariance the sample size can be substantially lowestin PLS than in SEM [77] It means that it might be morevariables than observations which imply data missing canbe obtained [87 88] Similarly in similar cases other authorshave indicated the reduced size of the sample [89] whoseminimummight attain the number of 20 in PLS [90]

The process of the FIMIX-PLS strategy is completed withthese analyses On the other hand other researches suggesttesting to see if the numerical variation between the pathcoefficients of the segment is also significantly distinctive byusing multigroup analysis (see Table 8) Several approachesfor multigroup analysis were found in document researchwhich are discussed in more detail by Sarstedt et al [81] andHair et al [66] The use of the permutation approach wassuggested by Hair et al [78] which was also used in theSmartPLS 3 software

However before making the interpretation of the multi-group analysis outcomes researchers must ensure that themeasurement models are invariable in the groups Oncethe measurement invariance (MICOM) was checked [79] amultigroup analysis (MGA) was carried out to find if therewere any significant discrepancies between the segmentsTheresults are shown in the three right hand side columns ofTable 9

As proven by the results obtained from nonparamet-ric testing the multigroup PLS-MGA analysis confirmedthe parametric tests and also found significant discrepancybetween segments 2 and 3

There is divergence between the first and second seg-ments but only k=2 and K=3 in EVMGBI EVMGO andEVMIC show a significant difference

Table 9 shows the validity of the segment measurementmodel and its explanatory capacity using R-squared andclassified by segment The values of k=2 for CR and AVE areshown to be below the limits

431 Assessment of the Predictive Validity PLS can beused for both explanatory and predictive research In otherwords the model has the ability to predict the currentand future observations Predictive validity shows that themeasurements for several constructs can predict a dependentlatent variable as in our case Green Attitudes towardsNature (GAN) The prediction outside the sample knownas predictive validity was assessed using cross-validationwith maintained samples This research used the approachrecommended by Shmueli et al [91]

By using other authorsrsquo research [92ndash94] the PLS predictalgorithm in the updated SmartPLS software version 327was used The results for k-fold cross prediction errors andthe essence of prediction errors It was expresses throughthe mean absolute error (MAE) and root mean square error(RMSE) Then the expected achievement of the PLS modelfor latent variables and indicators is assessed The followingcriterion expressed by the SmartPLS team was applied toappraise the expected achievement of the model [91ndash94]

(1)The Q2 value in PLS predict the miscalculations of thePLS model with the mean future projection are compared

The PLS-SEMrsquos miscalculation data can be lower than theprediction error of simply using mean values then the Q2value is positive Therefore the PLS-SEM model provided

Complexity 11

Table 9 Reliability measurements for the general model and for the three segments

Global model k=1 (384) k=2 (31) K=3 (307)CR AVE R-squared CR AVE R-squared CR AVE R-squared CR AVE R-squared

ECC 0922 0797 - 0947 0857 - 0791 0561 - 0912 0775 -EI 0913 0840 0368 0935 0877 0208 0857 0750 0300 0886 0795 0631EVM 0920 0794 0384 0861 0675 0075 0918 0789 0551 0959 0885 0869GAN 0917 0847 0648 0908 0831 0290 0861 0756 0924 0940 0886 0985GBI 0916 0845 0479 0851 0741 0322 0893 0807 0914 0945 0895 0293GO 0909 0768 0421 0878 0707 0302 0861 0676 0529 0929 0814 0325IC 0917 0786 0288 0904 0759 0009 0872 0694 0899 0938 0835 0303

Table 10 Summary of dependent variable prediction

Construct GAN RMSE MAE Q2

Complete sample 0591 0425 0429Segment 1 0486 0376 0120Segment 2 0619 0468 0515Segment 3 0723 0490 0761

convenient predictive performance which is the case in thetwo subsamples of segments 2 and 3 (see Table 10) in thedependent construct Green Attitude towards Nature (GAN)(Table 10) Then the prediction results were achieved

(2) The linear regression model (LM) approach a regres-sion of the exogenous indicators in every endogenous indi-cator was performed Then better prediction errors can beachieved when this comparison is considered This can beseen when the MAE and RMSE values are smaller than thoseof the LM model If this occurs predictions can be madeThis methodology is only used for indicators As shown inTable 10 the MAE and RMSE values were mainly negative Itexpressed excellent predictive power

It is also contrasted the predictions the real compositescores within the sample and outside the sample with [91]With this intention the research by Danks Ray and Shmueli[95] was applied

By using this methodology the measure was appliedfor the Green Attitudes towards Nature (GAN) constructRMSE for the complete sample (see Table 10) was 0591 andhad a higher value in segment 3 (0723 difference=0132)and lower values in segment 1 (0486 difference=0105) andsegment 2 (0619 difference=0028) The complex values arenormalized in which the value of mean is 0 and variance 1RMSE expressed the standard deviation measure Since thedifference in RMSE is not considerable excess capacity is nota problem for this study

In relation to Q2 the following metrics were found forthe GAN construct RMSE for the complete sample (seeTable 10) was 0429 and had a higher value in segment 3(0761 difference=0332) and lower values in segment 1 (0120difference=0309) and segment 2 (0515 difference=0086)

The density diagrams of the residues within the sampleand outside the sample are provided in Figure 4

Due to the result of the different analyses this researchfound enough evidence to accept the predictive validity of the

0

5

10

15

20

25

minus25

0minus2

25

minus20

0minus1

75

minus15

0minus1

25

minus10

0minus0

75

minus05

0minus0

25

000

025

050

075

100

125

150

175

200

225

250

275

300

GAN

Figure 4 Residue density within the sample and outside the sample

model expressed As a result themodel conveys appropriatelythe intention to apply in further samples They are quitedistinctive from the data used to check the theoretical model[96]

44 Considerations for the Management of Internet SearchEngines (IPMA) According to research that studied dataheterogeneity [77 96] the IPMA-PLS technique was usedto find more precise recommendations for marketing ofInternet search engines IPMA is a framework study thatuses matrices that enable combining the average value scorefor ldquoperformancerdquo with the estimation ldquoimportancerdquo in PLS-SEMrsquos total effects [77 97 98]The outcomes are shown in animportance-performance chart of four fields [77 99]

According to Groszlig [97] the analysis of the four quadrantsis shown in the chart (Figure 5) They are expressed conse-quently in the following points

(i) Quadrant I conveys acceptance attributes that aremuch more valued for performance and importance

(ii) Quadrant II explains acceptance attributes of highimportance but small performance It must be devel-oped

(iii) Quadrant III considers acceptance attributes thathave reduced importance and performance

(iv) Quadrant IV expresses acceptance attributes withgreat performance index but small equitable impor-tance

12 Complexity

000 010 020 030 040 050 060 070 080 090 100

Importance-Performance Map



EVMECC

EVMECC

EVMECC

0

20

40

60

80

100

GA

N

010 020 030 040 050 060 070 080 090 100000Total-Effects

Total-Effects

Total-Effects

0

20

40

60

80

100

GA

N

000 010 020 030 040 050 060 070 080 090 1000

20

40

60

80

100

GA

N

Figure 5 Importance-performance maps for k=1 k=2 and k=3

The results are unequal for each segment (Figure 5) Forusers belonging to k=1 all constructs are low and provideperformance lt50 showing that ECC and EVM obtained ascore of 045 and 031

Finally the results for k=3 show a different situationThese entrepreneurs valued importance gt90 showing thatECC and EVM is the most valued and obtained a score of093 and 098

5 Discussion

51 Theoretical Implications Spanish green start-ups haveincreasingly developed better green attitudes in recentdecades The new environmental regulatory system has beenthe driving force behind several changes in the way greenstart-ups must comply with the law On the one hand theSpanish regulatory compliance has established a new legal

framework for companies On the other hand it proposedthe implementation of an organisational model to preventcompanies committing unexpected unlawful behaviour

The FIMIX-PLS analysis has been split into two groupsCoincidentally the size of the segments and the coefficient ofdetermination R-squared as the FIT indices are divided intotwo samples groups as well

Nonparametric unobserved segmentation in complexsystems has provided us enough information to compare thestatistical results To present the complete analyses withinFIMIX-PLS method multigroup and permutation approachhave been taken into account Results show that segmentshave presented differences using the multigroup analysis(MGA) namely between the first and second segmentsEVM997888rarrGBI EVM997888rarrGO and EVM997888rarrIC

To revise the validity of the segmentsmeasured SmartPLSsoftware version 327 was used This statistical package wasused to set up the predictive performance of the model Theprediction errors of the PLS model were compared with thesimple mean predictions In the two subsamples of segments2 and 3 in the dependent construct GAN the prediction errorof the PLS-SEM results was less than the prediction errorof simply using the mean values It means that the modelprovides acceptable prediction results Similarly the linearregression model (LM) approach was studied in order toget better prediction errors It is achieved when the value ofRMSE andMAE are lower than those of the LMmodel As theresults have shown the indicators show the valued of RMSEand MAE were mostly negative It can be deduced that thosevalues provide a strong predictive power

Such findings are manifested through the significanceof the path coefficients particularly among the influence ofEnvironmental Corporate Compliance in the EnvironmentalValue Model (H

2 ECC997888rarrEVM 120573 = 0697 T-statistic =

2905) and in the Green Attitudes towards Nature (H1

ECC997888rarrGAN 120573 = 0417 T-statistic = 5116) As a resultregulatory compliance not only prevents companies fromcommitting unlawful environmental actions [39] but alsofrom running the risk of any negative implications for nature[4]

The T-statistic in both path coefficients shows little differ-ence Nevertheless it is safe to say that the new regulation hasa higher effect on improving attitudes towards nature than theEnvironmental Value Model

At the empirical level the new regulation has a significanteffect on Spanish green start-ups as a way of improvingattitudes towards nature and its Environmental Value ModelIn other words the new Criminal Code raises awareness onmonitoring a voluntary environmental surveillance processbased on the rules of the organisation [36] Simultaneouslystart-ups are willing to update the environmental standardsthat allow them to comply with the law by developing apreventive system to deter unlawful decisions taken by itsmanagers [34]

Start-ups should therefore implement effective monitor-ing systems including organisational monitoring to avoidunexpected criminal sanctions [38] and keep the compliancemodel up to date when relevant infractions are revealed [40]

Complexity 13

The findings also show that the hypothesised rela-tionships between Environmental Value Model and thefour attributes (Balanced Green Investment EnvironmentalImpact of green entrepreneurs Innovation and Competi-tiveness and Green Opportunities) are indeed significantBased on the results obtained each attribute creates areal environmental value for start-ups The environmentalvalue makes them more innovative and competitive (H

5

EVM997888rarrIC 120573 = 0597 le 0001) and green-oriented towardsmarket opportunities (H

7 EVM 997888rarr GO 120573 = 0773 p value

le 0001) by balancing economy and ecology (H4 EVM 997888rarr

BGI 120573 = 0780 p value le 0001) All these factors contributetowards generating a positive impact in nature (H

6 EVM997888rarr

EI 120573 = 0730 p value le 0001)

52 Practical Implications Practical implications can bedrawn from the findings not only for green start-ups but alsofor public and legal authoritiesThe first implication is relatedto the need for empirical studies to test the recent approvalof the Spanish regulatory compliance as well as nurture theirlegal and public decisions The second implication is directlyrelated to the findings Results have offered an unexpectedpicture of what is commonly understood about coercive regu-lations Results show that the new regulatory compliance sys-temhas emerged as the critical factor to foster green start-upsrespect for nature The new regulation positively influencesstart-ups improvement towards nature (H1 ECC997888rarrGAN=0417 T-statistic=5116) and has a positive effect on theEnvironmental Value Model (H2 ECC997888rarrEVM = 0697 T-statistic=2905) The new environmental law therefore playsa key role to explain how these entrepreneurs respect naturewhether by setting up a department to supervise the com-pliance of green innovations [34] or by undertaking trainingprotocols to implement surveillance norms in the companies[36] The research also contributes as the third implica-tion towards protecting companies from unlawful actionsand corruption by developing organisational indicators Inother words corruption which is supposed to be negativelyassociated with private investment and growth has becomean outstanding setback and international phenomenon [98]This explains the current trend to avert bribery extortionor fraud by developing international efforts to address thisdilemma from the legal and organisational perspective

The fourth implication alludes to the model designed Asthe results have shown H4 H5 H6 and H7 are significantIn other words the four attributes appropriately reflect theEnvironmental Value Model On top of that a balancedeconomy and ecology are considered themain attributes (H4EVM997888rarrBGI =0780 p value 0001) Finally green start-upsare willing to concentrate efforts on environmental andman-agerial solutions by discovering new Green Opportunities todefend nature This trend devised in green entrepreneursmight positively influence the future of start-ups as con-firmed by the predictive value of the model (Q2 = 0474)

6 Conclusion

Two relevant contributions have been made to the literaturereview The first one has been the methodology it has been

applied This is so not only because it has been used non-parametric unobserved segmentation in complex systemsbut also because it gives you another approach about data canbe organised to provide statistical results

The second contribution is that the outcome of this studyprovides relevant theoretical and practical implications to beincorporated into the literature review Findings shed lighton not only the way green start-ups comply with the recentSpanish Criminal Code but also how they develop greenattitudes towards nature to avoid threats to nature

This research highlights three aspects regarding theeffects of the new regulatory compliance system in Spain (1)the implications of the new regulation on green start-ups byimposing coercive rules and designing voluntary surveillancemeasures (2) the significance of the Environmental ValueModel reflected in four green attributes and (3) the influenceof the Environmental Value Model and the new regulation ingreen start-ups attitudes towards nature

The findings provided a robust explanatory capacity ofthe completemodel (R-squaredGAN=0779) In otherwordsall the constructs explain 779 of the variance of start-upsGreen Attitudes towards Nature

Theoretical and practical implications can also be learntfrom this robust result Even though the literature about greencompanies is vast very little has been published about theempirical impact of environmental regulation on green start-ups

This study helps to shed light on this aspect Due tothe gradual increase in the number of start-ups in the lastdecades these findings can be used by public authoritiesto impose legal requirements and encourage companies todevelop surveillance measures through their complianceofficers

To be more precise we recommend the authorities focuson promoting organisational measures by rewarding com-panies that establish protocols or training procedures andset up surveillance standards within the company This mea-sure can also be undertaken by developing interconnectedmechanisms between public authorities and start-ups basedon cooperative rules as a priority to improve the employeesattitudes towards nature Three limitations of the study mustalso be appropriately conveyed

First A new list of Spanish green start-ups has been proposedin the study by selecting four attributes from the literaturereview

Second Not all start-ups were aware of the recent publicationof the Spanish Corporate Compliance Criminal Code letalone implemented it

ThirdThe researchwas based on collecting data onhowgreenstart-ups perceived the new regulation and their influenceon the model presented Unfortunately the research didnot offer further information about the design process oforganisational standards to prevent unlawful decisions Inother words a practical and theoretical approach to thatphenomenon might yield different results

Concerning future research lines it would be advisableto work on these limitations to achieve a more accurate

14 Complexity

approach to the current research Moreover the researchoutcomes obtained can be compared with nonenvironmentalstart-ups to see how they see nature as a critical element tobe respected by every company to preserve the environmentfor our future generations

The research concludes by stating that the three rela-tionships were successfully tested and contrasted by themodel The new Spanish regulatory compliance has providedunexpected results that could contribute to improve the legaldecisions taken by public authorities

Data Availability



There are no conflicts of interest

Authorsrsquo Contributions

Rafael Robina-Ramırez wrote the theoretical and empiri-cal part Antonio Fernandez-Portillo addressed the FIMIXresults Juan Carlos Dıaz-Casero has revised the paper

Acknowledgments

INTERRA (Research Institute for Sustainable TerritorialDevelopment) has collaborated in the research We have notreceived any funds

References

[1] European Commission EU SBA Fact Sheet - EuropeanCommission - EuropaEURef Ares (2018) 2717562 - 250520182018 Retrieve httpseceuropaeudocsroomdocuments29489attachments27translationsenrenditionspdf

[2] G H Brundtland ldquoBrundtland reportrdquo Our Common FutureComissao Mundial 1987

[3] R Barkemeyer ldquoBeyond compliance - below expectations CSRin the context of international developmentrdquo Business Ethics AEuropean Review vol 18 no 3 pp 273ndash289 2009

[4] B Cohen and M I Winn ldquoMarket imperfections opportunityand sustainable entrepreneurshiprdquo Journal of Business Ventur-ing vol 22 no 1 pp 29ndash49 2007

[5] J Randjelovic A ROrsquoRourke andR J Orsato ldquoThe emergenceof green venture capitalrdquo Business Strategy and the Environmentvol 12 no 4 pp 240ndash253 2003

[6] L U Hendrickson and D B Tuttle ldquoDynamic management ofthe environmental enterprise a qualitative analysisrdquo Journal ofOrganizational ChangeManagement vol 10 no 4 pp 363ndash3821997

[7] J D Ivanko ECOpreneuring Putting Purpose and the PlanetBefore Profits New Society Publishers Gabriola Island Canada2008

[8] C Leadbeater Social Enterprise and Social Innovation Strategiesfor the Next Ten Years A Social Enterprise Think Piece for the

Office of the Third Sector Cabinet Office Office of the ThirdSector London UK 2007

[9] S Schaltegger ldquoA framework for ecopreneurship Leadingbioneers and environmental managers to ecopreneurshiprdquoGreener Management International no 38 pp 45ndash58 2002

[10] T J Dean and J S McMullen ldquoToward a theory of sustain-able entrepreneurship Reducing environmental degradationthrough entrepreneurial actionrdquo Journal of Business Venturingvol 22 no 1 pp 50ndash76 2007

[11] S Schaltegger ldquoA framework and typology of ecopreneurshipleading bioneers and environmental managers to ecopreneur-shiprdquo in Making Ecopreneurs M Schaper Ed pp 95ndash114Routledge London UK 2016

[12] Y Li Z Wei and Y Liu ldquoStrategic orientations knowledgeacquisition and firm performance the perspective of thevendor in cross-border outsourcingrdquo Journal of ManagementStudies vol 47 no 8 pp 1457ndash1482 2010

[13] E E Walley and D W Taylor ldquoOpportunists championsMavericks a typology of green entrepreneursrdquo Greener Man-agement International no 38 pp 31ndash43 2002

[14] A R Anderson ldquoCultivating the garden of eden environmentalentrepreneuringrdquo Journal of Organizational Change Manage-ment vol 11 no 2 pp 135ndash144 1998

[15] R Isaak ldquoGlobalisation and green entrepreneurshiprdquo GreenerManagement International vol 18 pp 80ndash90 1997

[16] K Hockerts and R Wustenhagen ldquoGreening Goliaths versusemerging Davids - Theorizing about the role of incumbentsand new entrants in sustainable entrepreneurshiprdquo Journal ofBusiness Venturing vol 25 no 5 pp 481ndash492 2010

[17] A Triguero L Moreno-Mondejar andM A Davia ldquoDrivers ofdifferent types of eco-innovation in European SMEsrdquo EcologicalEconomics vol 92 pp 25ndash33 2013

[18] S-P Chuang and C-L Yang ldquoKey success factors when imple-menting a green-manufacturing systemrdquo Production Planningand Control vol 25 no 11 pp 923ndash937 2014

[19] J G Covin and G T Lumpkin ldquoEntrepreneurial orienta-tion theory and research reflections on a needed constructrdquoEntrepreneurshipTheory and Practice vol 35 no 5 pp 855ndash8722011

[20] D J Teece ldquoDynamic capabilities and entrepreneurial man-agement in large organizations toward a theory of the(entrepreneurial) firmrdquo European Economic Review vol 86 pp202ndash216 2016

[21] K Woldesenbet M Ram and T Jones ldquoSupplying large firmsThe role of entrepreneurial and dynamic capabilities in smallbusinessesrdquo International Small Business Journal vol 30 no 5pp 493ndash512 2012

[22] D J Teece ldquoA dynamic capabilities-based entrepreneurialtheory of the multinational enterpriserdquo Journal of InternationalBusiness Studies vol 45 no 1 pp 8ndash37 2014

[23] Spanish Corporate Governance Code 12015 of 31 MarchRetrieved httpswwwboeesboedias20150331pdfsBOE-A-2015-3439pdf

[24] F Partnoy Infectious Greed How Deceit and Risk Corrupted theFinancial Markets Profile Books London UK 2003

[25] I Melay and S Kraus ldquoGreen entrepreneurship definitionsof related conceptsrdquo International Journal Strategegic Manage-ment vol 12 pp 1ndash12 2012

[26] A V Banerjee E Duflo and K Munshi ldquoThe (MIS) allocationof capitalrdquo Journal of the European Economic Association vol 1no 2-3 pp 484ndash494 2003

Complexity 15

[27] W L Koe and I A Majid ldquoSocio-cultural factors and inten-tion towards sustainable entrepreneurshiprdquo Eurasian Journal ofBusiness and Economics vol 7 no 13 pp 145ndash156 2014

[28] D Y Choi and E R Gray ldquoThe venture development processesof ldquosustainablerdquo entrepreneursrdquo Management Research Newsvol 31 no 8 pp 558ndash569 2008

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[30] T Gliedt and P Parker ldquoGreen community entrepreneurshipcreative destruction in the social economyrdquo International Jour-nal of Social Economics vol 34 no 8 pp 538ndash553 2007

[31] I-M Garcıa-Sanchez B Cuadrado-Ballesteros and J-V Frias-Aceituno ldquoImpact of the institutional macro context on thevoluntary disclosure of CSR informationrdquo Long Range Planningvol 49 no 1 pp 15ndash35 2016

[32] J M Tamarit Sumalla ldquoLa responsabilidad penal de laspersonas jurıdicasrdquo in La reforma penal de 2010 analisis ycomentarios Quintero Olivares Dir Cizur Menor p 58 2010

[33] P W Moroz and K Hindle ldquoEntrepreneurship as a processtoward harmonizing multiple perspectivesrdquo EntrepreneurshipTheory and Practice vol 36 no 4 pp 781ndash818 2012

[34] J A Ingrisano and S A Mathews ldquoPractical guide to avoidingfailure to supervise liabilityrdquo Preventive Law Reporter vol 14no 12 1995

[35] K B Huff ldquoThe role of corporate compliance programs indetermining corporate criminal liability a suggested approachrdquoColumbia Law Review vol 96 no 5 pp 1252ndash1298 1996

[36] S Sadiq G Governatori and K Namiri ldquoModeling controlobjectives for business process compliancerdquo in Proceedings ofthe International Conference on Business Process Managementpp 149ndash164 Berlin Germany 2007

[37] J G York and S Venkataraman ldquoThe entrepreneur-environment nexus Uncertainty innovation and allocationrdquoJournal of Business Venturing vol 25 no 5 pp 449ndash463 2010

[38] A Amicelle ldquoTowards a new political anatomy of financialsurveillancerdquo Security Dialogue vol 42 no 2 pp 161ndash178 2011

[39] E De Porres Ortiz de Urbina Responsabilidad Penal de laspersonas jurıdicas El Derecho Madrid Spain 2015

[40] D R Campbell M Campbell and G W Adams ldquoAddingsignificant value with internal controlsrdquo The CPA Journal vol76 no 6 p 20 2006

[41] I MacNeil and X Li ldquoComply or explain market disciplineand non-compliance with the combined coderdquo Corporate Gov-ernance An International Review vol 14 no 5 pp 486ndash4962006

[42] R Bampton and C J Cowton ldquoTaking stock of accountingethics scholarship a review of the journal literaturerdquo Journal ofBusiness Ethics vol 114 no 3 pp 549ndash563 2013

[43] R Calderon I Ferrero and D M Redin ldquoEthical codes andcorporate responsibility of the most admired companies of theworld toward a third generation ethicsrdquo Business and Politicsvol 14 no 4 pp 1ndash24 2012

[44] M P Vandenbergh ldquoBeyond elegance a testable typology ofsocial norms in corporate environmental compliancerdquo StanfordEnvironmental Law Journal vol 22 no 55 2003

[45] B V Todeschini M N Cortimiglia D Callegaro-de-Menezesand A Ghezzi ldquoInnovative and sustainable business modelsin the fashion industry entrepreneurial drivers opportunitiesand challengesrdquo Business Horizons vol 60 no 6 pp 759ndash7702017

[46] K Fletcher ldquoSlow fashion an invitation for systems changerdquoTheJournal of Design Creative Process and the Fashion Industry vol2 no 2 pp 259ndash265 2010

[47] OCDE ldquoSustainable development programmes and initiatives(2009-2010)rdquo 2009 Retrieve httpswwwoecdorggreengrowth47445613pdf

[48] A T Bon and E M Mustafa ldquoImpact of total quality manage-ment on innovation in service organizations literature reviewand new conceptual frameworkrdquo Procedia Engineering vol 53pp 516ndash529 2013

[49] M Mazzucato and C C R Penna Beyond Market FailuresThe Market Creating and Shaping Roles of State InvestmentBanks Levy Economics Institute New York NY USA 2015(Working Paper October 2015) Available at httpwwwlevyinstituteorgpubswp 831pdf

[50] J Kirkwood and S Walton ldquoHow green is green Ecopreneursbalancing environmental concerns and business goalsrdquo Aus-tralasian Journal of Environmental Management vol 21 no 1pp 37ndash51 2014

[51] M Shrivastava and J P Tamvada ldquoWhich green matters forwhom Greening and firm performance across age and sizedistribution of firmsrdquo Small Business Economics pp 1ndash18 2017

[52] D S Dhaliwal S Radhakrishnan A Tsang and Y G YangldquoNonfinancial disclosure and analyst forecast accuracy inter-national evidence on corporate social responsibility disclosurerdquoThe Accounting Review vol 87 no 3 pp 723ndash759 2012

[53] P M Clarkson Y Li G D Richardson and F P VasvarildquoRevisiting the relation between environmental performanceand environmental disclosure an empirical analysisrdquo Account-ing Organizations and Society vol 33 no 4-5 pp 303ndash3272008

[54] D F Pacheco T J Dean and D S Payne ldquoEscaping the greenprison Entrepreneurship and the creation of opportunities forsustainable developmentrdquo Journal of Business Venturing vol 25no 5 pp 464ndash480 2010

[55] A Perez-Luno J Wiklund and R V Cabrera ldquoThe dualnature of innovative activity How entrepreneurial orientationinfluences innovation generation and adoptionrdquo Journal ofBusiness Venturing vol 26 no 5 pp 555ndash571 2011

[56] J W Carland J A Carland and J W Pearce ldquoRisk takingpropensity among entrepreneurs small business owners andmanagersrdquo Journal of Business and Entrepreneurship vol 7 no1 pp 15ndash23 1995

[57] G Shirokova K Bogatyreva T Beliaeva and S PufferldquoEntrepreneurial orientation and firm performance in differ-ent environmental settings contingency and configurationalapproachesrdquo Journal of Small Business and Enterprise Develop-ment vol 23 no 3 pp 703ndash727 2016

[58] M Lenox and J G York ldquoEnvironmental entrepreneurshiprdquoin Oxford Handbook of Business and the Environment A JHoffman and T Bansal Eds Oxford University Press OxfordUK 2011

[59] Pricewater House Coopers (PwC) Kuala Lumpur Green taxincentives for Malaysia Oct 86 2010

[60] D J Teece ldquoDynamic capabilities routines versus entrepreneu-rial actionrdquo Journal of Management Studies vol 49 no 8 pp1395ndash1401 2012

[61] J G York I OrsquoNeil and S D Sarasvathy ldquoExploring environ-mental entrepreneurship identity coupling venture goals andstakeholder incentivesrdquo Journal of Management Studies vol 53no 5 pp 695ndash737 2016

16 Complexity

[62] M Sarstedt C M Ringle and J F Hair ldquoPartial least squaresstructural equationmodelingrdquo inHandbook ofMarket Researchpp 1ndash40 Springer International Publishing 2017

[63] C B Astrachan V K Patel andGWanzenried ldquoA comparativestudy of CB-SEM and PLS-SEM for theory development infamily firm researchrdquo Journal of Family Business Strategy vol5 no 1 pp 116ndash128 2014

[64] E Carmines and R Zeller Reliability and Validity Assessmentvol 17 SAGE Publications Inc Thousand Oaks Calif USA1979

[65] O Gotz K Liehr-Gobbers and M Krafft ldquoEvaluation ofstructural equation models using the partial least squares (PLS)approachrdquo in Handbook of Partial Least Squares pp 691ndash711Springer Berlin Germany 2010

[66] J Hair W Black B Babin R Anderson and R TathamMultivariate data analysis Prentice Hall Upper Saddle RiverNJ USA 5th edition 2005

[67] C Fornell and D F Larcker ldquoStructural equation models withunobservable variables and measurement error algebra andstatisticsrdquo Journal of Marketing Research vol 18 no 3 pp 382ndash388 2018

[68] T K Dijkstra and J Henseler ldquoConsistent partial least squarespath modelingrdquo MIS Quarterly Management Information Sys-tems vol 39 no 2 pp 297ndash316 2015

[69] J Henseler CM Ringle and R R Sinkovics ldquoThe use of partialleast squares pathmodeling in international marketingrdquo inNewChallenges to International Marketing pp 277ndash319 EmeraldGroup Publishing Limited 2009

[70] J Henseler C M Ringle and M Sarstedt ldquoA new criterionfor assessing discriminant validity in variance-based structuralequation modelingrdquo Journal of the Academy of MarketingScience vol 43 no 1 pp 115ndash135 2015

[71] W W Chin ldquoThe partial least squares approach to structuralequationmodelingrdquoModernMethods for Business Research vol295 no 2 pp 295ndash336 1998

[72] L T Hu and P M Bentler ldquoFit indices in covariance structuremodeling sensitivity to under-parameterized model misspeci-ficationrdquo Psychological Methods vol 3 no 4 pp 424ndash453 1998

[73] M Stone ldquoCross-validatory choice and assessment of statisticalpredictionsrdquo Journal of the Royal Statistical Society Series B(Methodological) vol 36 pp 111ndash147 1974

[74] S Geisser ldquoA predictive approach to the random effect modelrdquoBiometrika vol 61 pp 101ndash107 1974

[75] J-M Becker A Rai CMRingle and FVolckner ldquoDiscoveringunobserved heterogeneity in structural equation models toavert validity threatsrdquoMIS Quarterly Management InformationSystems vol 37 no 3 pp 665ndash694 2013


[77] P Palos-Sanchez FMartin-Velicia and J R Saura ldquoComplexityin the acceptance of sustainable search engines on the internetan analysis of unobserved heterogeneity with FIMIX-PLSrdquoComplexity vol 2018 Article ID 6561417 19 pages 2018

[78] J F Hair Jr M Sarstedt L M Matthews and C MRingle ldquoIdentifying and treating unobserved heterogeneitywith FIMIX-PLS part I ndash methodrdquo European Business Reviewvol 28 no 1 pp 63ndash76 2016


[80] M Sarstedt C M Ringle D Smith R Reams and J F HairldquoPartial least squares structural equation modeling (PLS-SEM)a useful tool for family business researchersrdquo Journal of FamilyBusiness Strategy vol 5 no 1 pp 105ndash115 2014

[81] M Sarstedt J Becker C M Ringle andM Schwaiger ldquoUncov-ering and treating unobserved heterogeneity with FIMIX-PLS which model selection criterion provides an appropriatenumber of segmentsrdquo Schmalenbach Business Review vol 63no 1 pp 34ndash62 2011

[82] H Bozdogan ldquoMixture-model cluster analysis using modelselection criteria and a new informational measure of com-plexityrdquo in Proceedings of the First USJapan Conference on theFrontiers of Statistical Modeling An Informational ApproachH Bozdogan Ed pp 69ndash113 Kluwer Academic PublishersBoston London 1994

[83] V RamaswamyW S Desarbo D J Reibstein andW T Robin-son ldquoAn empirical pooling approach for estimating marketingmix elasticities with PIMS datardquo Marketing Science vol 12 no1 pp 103ndash124 1993

[84] C M Ringle S Wende and A Will ldquoCustomer segmentationwith FIMIX-PLSrdquo in Proceedings of the PLS-05 InternationalSymposium T Aluja J Casanovas and V Esposito Eds pp507ndash514 PAD TestampGo Paris France 2005

[85] C M Ringle S Wende and A Will ldquoFinite mixture partialleast squares analysis methodology and numerical examplesrdquoin Handbook of Partial Least Squares V Esposito Vinzi W WChin J Henseler and H Wang Eds vol 2 of Springer hand-books of computational statistics series pp 195ndash218 SpringerLondon UK 2010

[86] L Hu and P M Bentler ldquoCutoff criteria for fit indexes incovariance structure analysis conventional criteria versus newalternativesrdquo Structural Equation Modeling A MultidisciplinaryJournal vol 6 no 1 pp 1ndash55 1999

[87] J Mondejar-Jimenez M Segarra-Ona A Peiro-Signes A MPaya-Martınez and F J Saez-Martınez ldquoSegmentation of theSpanish automotive industry with respect to the environmentalorientation of firms towards an ad-hoc vertical policy topromote eco-innovationrdquo Journal of Cleaner Production vol 86pp 238ndash244 2015

[88] M Tenenhaus V E Vinzi Y Chatelin and C Lauro ldquoPLS pathmodelingrdquoComputational StatisticsampDataAnalysis vol 48 no1 pp 159ndash205 2005

[89] H O Wold ldquoIntroduction to the second generation of multi-variate analysisrdquo inTheoretical Empiricism A General Rationalefor Scientific Model-Building H O Wold Ed Paragon HouseNew York NY USA 1989

[90] WChing andPNewsted ldquoChapter 12 structural equationmod-eling analysis with small samples using partial least squaresrdquo inStatisticas Strategies for Smart Sample Researchs E R H HoyleEd Sage Publications Thousand Oaks CA USA 1999

[91] G Shmueli andO R Koppius ldquoPredictive analytics in informa-tion systems researchrdquo MIS Quarterly Management Informa-tion Systems vol 35 no 3 pp 553ndash572 2011

[92] C M Felipe J L Roldan and A L Leal-Rodrıguez ldquoImpact oforganizational culture values on organizational agilityrdquo Sustain-ability vol 9 no 12 p 2354 2017

[93] M Sarstedt C M Ringle G Schmueli J H Cheah and HTing ldquoPredictive model assessment in PLS-SEM guidelines forusing PLSpredictrdquoWorking Paper 2018

[94] C M Ringle S Wende and J M Becker ldquoSmartPLS 3rdquo Boen-ningstedt SmartPLS GmbH 2015 httpwwwsmartplscom

Complexity 17

[95] N Danks S Ray and G Shmueli ldquoEvaluating the predictiveperformance of constructs in PLS path modelingrdquo WorkingPaper 2018

[96] A G Woodside ldquoMoving beyond multiple regression analysisto algorithms Calling for adoption of a paradigm shift fromsymmetric to asymmetric thinking in data analysis and craftingtheoryrdquo Journal of Business Research vol 66 no 4 pp 463ndash4722013

[97] MGroszlig ldquoHeterogeneity in consumersrsquomobile shopping accep-tance A finite mixture partial least squares modelling approachfor exploring and characterising different shopper segmentsrdquoJournal of Retailing and Consumer Services vol 40 pp 8ndash182018

[98] A Argandona ldquoThe united nations convention against cor-ruption and its impact on international companiesrdquo Journal ofBusiness Ethics vol 74 no 4 pp 481ndash496 2007

[99] JHenseler GHubona andPA Ray ldquoUsing PLS pathmodelingin new technology research updated guidelinesrdquo IndustrialManagement amp Data Systems vol 116 no 1 pp 2ndash20 2016

Research ArticleA CEEMDAN and XGBOOST-Based Approach toForecast Crude Oil Prices

Yingrui Zhou1 Taiyong Li 12 Jiayi Shi1 and Zijie Qian1

1School of Economic Information Engineering Southwestern University of Finance and Economics Chengdu 611130 China2Sichuan Province Key Laboratory of Financial Intelligence and Financial EngineeringSouthwestern University of Finance and Economics Chengdu 611130 China

Correspondence should be addressed to Taiyong Li litaiyonggmailcom

Received 13 October 2018 Accepted 13 January 2019 Published 3 February 2019


Copyright copy 2019 Yingrui Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Crude oil is one of the most important types of energy for the global economy and hence it is very attractive to understand themovement of crude oil prices However the sequences of crude oil prices usually show some characteristics of nonstationarityand nonlinearity making it very challenging for accurate forecasting crude oil prices To cope with this issue in this paper wepropose a novel approach that integrates complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)and extreme gradient boosting (XGBOOST) so-called CEEMDAN-XGBOOST for forecasting crude oil prices Firstly we useCEEMDAN to decompose the nonstationary and nonlinear sequences of crude oil prices into several intrinsic mode functions(IMFs) and one residue Secondly XGBOOST is used to predict each IMF and the residue individually Finally the correspondingprediction results of each IMF and the residue are aggregated as the final forecasting results To demonstrate the performance of theproposed approach we conduct extensive experiments on the West Texas Intermediate (WTI) crude oil prices The experimentalresults show that the proposed CEEMDAN-XGBOOST outperforms some state-of-the-art models in terms of several evaluationmetrics

1 Introduction

As one of the most important types of energy that power theglobal economy crude oil has great impacts on every countryevery enterprise and even every person Therefore it is acrucial task for the governors investors and researchers toforecast the crude oil prices accurately However the existingresearch has shown that crude oil prices are affected by manyfactors such as supply and demand interest rate exchangerate speculation activities international and political eventsclimate and so on [1 2] Therefore the movement of crudeoil prices is irregular For example starting from about 11USDbarrel inDecember 1998 theWTI crude oil prices grad-ually reached the peak of 14531 USDbarrel in July 2008 andthen the prices drastically declined to 3028USDbarrel in thenext five months because of the subprime mortgage crisisAfter that the prices climbed to more than 113 USDbarrelin April 2011 and once again they sharply dropped to about26 USDbarrel in February 2016The movement of the crude

oil prices in the last decades has shown that the forecastingtask is very challenging due to the characteristics of highnonlinearity and nonstationarity of crude oil prices

Many scholars have devoted efforts to trying to forecastcrude oil prices accurately The most widely used approachesto forecasting crude oil prices can be roughly divided intotwo groups statistical approaches and artificial intelligence(AI) approaches Recently Miao et al have explored thefactors of affecting crude oil prices based on the least absoluteshrinkage and selection operator (LASSO) model [1] Ye etal proposed an approach integrating ratchet effect for linearprediction of crude oil prices [3] Morana put forward asemiparametric generalized autoregressive conditional het-eroskedasticity (GARCH) model to predict crude oil pricesat different lag periods even without the conditional averageof historical crude oil prices [4] Naser found that using thedynamic model averaging (DMA) with empirical evidence isbetter than linear models such as autoregressive (AR) modeland its variants [5] Gong and Lin proposed several new


2 Complexity

heterogeneous autoregressive (HAR) models to forecast thegood and bad uncertainties in crude oil prices [6] Wen et alalso used HAR models with structural breaks to forecast thevolatility of crude oil futures [7]

Although the statistical approaches improve the accuracyof forecasting crude oil prices to some extent the assumptionof linearity of crude oil prices cannot be met accordingto some recent research and hence it limits the accuracyTherefore a variety of AI approaches have been proposedto capture the nonlinearity and nonstationarity of crude oilprices in the last decades [8ndash11] Chiroma et al reviewed theexisting research associated with forecasting crude oil pricesand found that AI methodologies are attracting unprece-dented interest from scholars in the domain of crude oil priceforecasting [8]Wang et al proposed anAI system frameworkthat integrated artificial neural networks (ANN) and rule-based expert system with text mining to forecast crude oilprices and it was shown that the proposed approach wassignificantly effective and practically feasible [9] Barunikand Malinska used neural networks to forecast the termstructure in crude oil futures prices [10] Most recentlyChen et al have studied forecasting crude oil prices usingdeep learning framework and have found that the randomwalk deep belief networks (RW-DBN) model outperformsthe long short term memory (LSTM) and the random walkLSTM (RW-LSTM) models in terms of forecasting accuracy[11] Other AI-methodologies such as genetic algorithm[12] compressive sensing [13] least square support vectorregression (LSSVR) [14] and cluster support vector machine(ClusterSVM) [15] were also applied to forecasting crudeoil prices Due to the extreme nonlinearity and nonstation-arity it is hard to achieve satisfactory results by forecastingthe original time series directly An ideal approach is todivide the tough task of forecasting original time series intoseveral subtasks and each of them forecasts a relativelysimpler subsequence And then the results of all subtasksare accumulated as the final result Based on this idea aldquodecomposition and ensemblerdquo framework was proposed andwidely applied to the analysis of time series such as energyforecasting [16 17] fault diagnosis [18ndash20] and biosignalanalysis [21ndash23] This framework consists of three stages Inthe first stage the original time series was decomposed intoseveral components Typical decomposition methodologiesinclude wavelet decomposition (WD) independent compo-nent analysis (ICA) [24] variational mode decomposition(VMD) [25] empirical mode decomposition (EMD) [226] and its extension (ensemble EMD (EEMD)) [27 28]and complementary EEMD (CEEMD) [29] In the secondstage some statistical or AI-based methodologies are appliedto forecast each decomposed component individually Intheory any regression methods can be used to forecast theresults of each component In the last stage the predictedresults from all components are aggregated as the finalresults Recently various researchers have devoted effortsto forecasting crude oil prices following the framework ofldquodecomposition and ensemblerdquo Fan et al put forward a novelapproach that integrates independent components analy-sis (ICA) and support vector regression (SVR) to forecastcrude oil prices and the experimental results verified the

effectiveness of the proposed approach [24] Yu et al usedEMD to decompose the sequences of the crude oil pricesinto several intrinsic mode functions (IMFs) at first andthen used a three-layer feed-forward neural network (FNN)model for predicting each IMF Finally the authors usedan adaptive linear neural network (ALNN) to combine allthe results of the IMFS as the final forecasting output [2]Yu et al also used EEMD and extended extreme learningmachine (EELM) to forecast crude oil prices following theframework of rdquodecomposition and ensemblerdquo The empiri-cal results demonstrated the effectiveness and efficiency ofthe proposed approach [28] Tang et al further proposedan improved approach integrating CEEMD and EELM forforecasting crude oil prices and the experimental resultsdemonstrated that the proposed approach outperformed allthe listed state-of-the-art benchmarks [29] Li et al usedEEMD to decompose raw crude oil prices into several com-ponents and then used kernel and nonkernel sparse Bayesianlearning (SBL) to forecast each component respectively [3031]

From the perspective of decomposition although EMDand EEMD are capable of improving the accuracy of fore-casting crude oil prices they still suffer ldquomode mixingrdquoand introducing new noise in the reconstructed signalsrespectively To overcome these shortcomings an extensionof EEMD so-called complete EEMD with adaptive noise(CEEMDAN) was proposed by Torres et al [32] Later theauthors put forward an improved version of CEEMDAN toobtain decomposed components with less noise and morephysical meaning [33] The CEEMDAN has succeeded inwind speed forecasting [34] electricity load forecasting [35]and fault diagnosis [36ndash38] Therefore CEEMDAN mayhave the potential to forecast crude oil prices As pointedout above any regression methods can be used to forecasteach decomposed component A recently proposed machinelearning algorithm extreme gradient boosting (XGBOOST)can be used for both classification and regression [39]The existing research has demonstrated the advantages ofXGBOOST in forecasting time series [40ndash42]

With the potential of CEEMDAN in decomposition andXGBOOST in regression in this paper we aim at proposinga novel approach that integrates CEEMDAN and XGBOOSTnamely CEEMDAN-XGBOOST to improve the accuracy offorecasting crude oil prices following the ldquodecompositionand ensemblerdquo framework Specifically we firstly decomposethe raw crude oil price series into several components withCEEMDAN And then for each component XGBOOSTis applied to building a specific model to forecast thecomponent Finally all the predicted results from everycomponent are aggregated as the final forecasting resultsThe main contributions of this paper are threefold (1) Wepropose a novel approach so-called CEEMDAN-XGBOOSTfor forecasting crude oil prices following the ldquodecompositionand ensemblerdquo framework (2) extensive experiments areconducted on the publicly-accessed West Texas Intermediate(WTI) to demonstrate the effectiveness of the proposedapproach in terms of several evaluation metrics (3) wefurther study the impacts of several parameter settings withthe proposed approach

Complexity 3

50

40

30

20

10

15

20

original sequencelocal meanupper envelopeslower envelopes

1 5 10Time

Am

plitu

de

(a) The line of the original sequence local mean upper envelopes and lowerenvelopes

10864

minus2minus4minus6minus8

minus10

2

2

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

TimeAm

plitu

de

(b) The first IMF decomposed by EMD from the above original sequence

Figure 1 An illustration of EMD

The remainder of this paper is organized as followsSection 2 describes CEEMDAN and XGBOOST Section 3formulates the proposed CEEMDAN-XGBOOST approachin detail Experimental results are reported and analyzedin Section 4 We also discussed the impacts of parametersettings in this section Finally Section 5 concludes this paper

2 Preliminaries

21 EMD EEMD and CEEMDAN EMD was proposed byHuang et al in 1998 and it has been developed and appliedin many disciplines of science and engineering [26] The keyfeature of EMD is to decompose a nonlinear nonstationarysequence into intrinsic mode functions (IMFs) in the spiritof the Fourier series In contrast to the Fourier series theyare not simply sine or cosine functions but rather functionsthat represent the characteristics of the local oscillationfrequency of the original data These IMFs need to satisfy twoconditions (1) the number of local extrema and the numberof zero crossing must be equal or differ at most by one and (2)the curve of the ldquolocal meanrdquo is defined as zero

At first EMDfinds out the upper and the lower envelopeswhich are computed by finding the local extrema of the orig-inal sequence Then the local maxima (minima) are linkedby two cubic spines to construct the upper (lower) envelopesrespectively The mean of these envelopes is considered asthe ldquolocal meanrdquo Meanwhile the curve of this ldquolocal meanrdquois defined as the first residue and the difference betweenoriginal sequence and the ldquolocal meanrdquo is defined as the firstIMF An illustration of EMD is shown in Figure 1

After the first IMF is decomposed by EMD there is still aresidue (the local mean ie the yellow dot line in Figure 1(a))between the IMF and the original data Obviously extremaand high-frequency oscillations also exist in the residueAnd EMD decomposes the residue into another IMF andone residue If the variance of the new residue is not smallenough to satisfy the Cauchy criterion EMD will repeat to

decompose new residue into another IMF and a new residueFinally EMD decomposes original sequence into severalIMFs and one residue The difference between the IMF andthe residues is defined as

119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 [119905] 119896 = 2 119870 (1)

where 119903119896[119905] is the k-th residue at the time t and K is the totalnumber of IMFs and residues

Subsequently Huang et al thought that EMD could notextract the local features from the mixed features of theoriginal sequence completely One of the reasons for thisis the frequent appearance of the mode mixing The modemixing can be defined as the situation that similar piecesof oscillations exist at the same corresponding position indifferent IMFs which causes that a single IMF has lostits physical meanings What is more if one IMF has thisproblem the following IMFs cannot avoid it either To solvethis problem Wu and Huang extended EMD to a newversion namely EEMD that adds white noise to the originaltime series and performs EMDmany times [27] Given a timeseries and corresponding noise the new time series can beexpressed as

119909119894 [119905] = 119909 [119905] + 119908119894 [119905] (2)

where 119909[119905] stands for the original data and 119908119894[119905] is the i-thwhite noise (i=12 N and N is the times of performingEMD)

Then EEMD decomposes every 119909119894[119905] into 119868119872119865119894119896[119905] Inorder to get the real k-th IMF 119868119872119865119896 EEMD calculates theaverage of the 119868119872119865119894119896[119905] In theory because the mean of thewhite noise is zero the effect of the white noise would beeliminated by computing the average of 119868119872119865119894119896[119905] as shownin

119868119872119865119896 = 1119873119873sum119894=1

119868119872119865119894119896 [119905] (3)

4 Complexity

However Torres et al found that due to the limitednumber of 119909119894[119905] in empirical research EEMD could not com-pletely eliminate the influence of white noise in the end Forthis situation Torres et al put forward a new decompositiontechnology CEEMDAN on the basis of EEMD [32]

CEEMDAN decomposes the original sequence into thefirst IMF and residue which is the same as EMD ThenCEEMDAN gets the second IMF and residue as shown in

1198681198721198652 = 1119873119873sum119894=1

1198641 (1199031 [119905] + 12057611198641 (119908119894 [119905])) (4)

1199032 [119905] = 1199031 [119905] minus 1198681198721198652 (5)

where 1198641() stands for the first IMF decomposed from thesequence and 120576119894 is used to set the signal-to-noise ratio (SNR)at each stage

In the same way the k-th IMF and residue can becalculated as

119868119872119865119896 = 1119873119873sum119894=1

1198641 (119903119896minus1 [119905] + 120576119896minus1119864119896minus1 (119908119894 [119905])) (6)

119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 (7)

Finally CEEMDAN gets several IMFs and computes theresidue as shown in

R [119905] = x [119905] minus 119870sum119896=1

119868119872119865119896 (8)

The sequences decomposed by EMD EEMD and CEEM-DAN satisfy (8) Although CEEMDAN can solve the prob-lems that EEMD leaves it still has two limitations (1) theresidual noise that themodels contain and (2) the existence ofspurious modes Aiming at dealing with these issues Torreset al proposed a new algorithm to improve CEEMDAN [33]

Compared with the original CEEMDAN the improvedversion obtains the residues by calculating the local meansFor example in order to get the first residue shown in (9)it would compute the local means of N realizations 119909119894[119905] =119909[119905] + 12057601198641(119908119894[119905])(i=1 2 N)

1199031 [119905] = 1119873119873sum119894=1

119872(119909119894 [119905]) (9)

whereM() is the local mean of the sequenceThen it can get the first IMF shown in

1198681198721198651 = 119909 [119905] minus 1199031 [119905] (10)

For the k-th residue and IMF they can be computed as(11) and (12) respectively

119903119896 [119905] = 1119873119873sum119894=1

119872(119903119896minus1 [119905] + 120576119896minus1119864k (119908119894 [119905])) (11)

119868119872119865119896 = 119903119896minus1 [119905] minus 119903119896 [119905] (12)

The authors have demonstrated that the improvedCEEMDAN outperformed the original CEEMDAN in signaldecomposition [33] In what follows we will refer to theimproved version of CEEMDAN as CEEMDAN unlessotherwise statedWithCEEMDAN the original sequence canbe decomposed into several IMFs and one residue that is thetough task of forecasting the raw time series can be dividedinto forecasting several simpler subtasks

22 XGBOOST Boosting is the ensemble method that cancombine several weak learners into a strong learner as

119910119894 = 0 (119909119894) = 119870sum119896=1

119891119896 (119909119894) (13)

where 119891119896() is a weak learner and K is the number of weaklearners

When it comes to the tree boosting its learners aredecision trees which can be used for both regression andclassification

To a certain degree XGBOOST is considered as treeboost and its core is the Newton boosting instead of GradientBoosting which finds the optimal parameters by minimizingthe loss function (0) as shown in

119871 (0) = 119899sum119894=1

119897 (119910119894 119910119894) + 119870sum119896=1

Ω(119891119896) (14)

Ω(119891119896) = 120574119879 + 12120572 1205962 (15)

where Ω(119891119896) is the complexity of the k-th tree model n isthe sample size T is the number of leaf nodes of the decisiontrees 120596 is the weight of the leaf nodes 120574 controls the extentof complexity penalty for tree structure on T and 120572 controlsthe degree of the regularization of 119891119896

Since it is difficult for the tree ensemble model to mini-mize loss function in (14) and (15) with traditional methodsin Euclidean space the model uses the additive manner [43]It adds 119891119905 that improves the model and forms the new lossfunction as

119871119905 = 119899sum119894=1

119897 (119910119894 119910119894(119905minus1) + 119891119905 (119909119894)) + Ω (119891119905) (16)

where 119910119894(119905) is the prediction of the i-th instance at the t-thiteration and 119891119905 is the weaker learner at the t-th iteration

Then Newton boosting performs a Taylor second-orderexpansion on the loss function 119897(119910119894 119910119894(119905minus1) + 119891119905(119909119894)) to obtain119892119894119891119905(119909119894) + (12)ℎ1198941198912119905 (119909119894) because the second-order approxi-mation helps to minimize the loss function conveniently andquickly [43]The equations of 119892119894 ℎ119894 and the new loss functionare defined respectively as

119892119894 = 120597119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (17)

Complexity 5

ℎ119894 = 1205972119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (18)

119871 119905 = 119899sum119894=1

[119897 (119910119894 119910119894(119905minus1)) + 119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)]+ Ω (119891119905)

(19)

Assume that the sample set 119868119895 in the leaf node j is definedas 119868119895=119894 | 119902(119909119894) = 119895 where q(119909119894) represents the tree structurefrom the root to the leaf node j in the decision tree (19) canbe transformed into the following formula as show in

(119905) = 119899sum119894=1

[119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)] + 120574119879 + 12120572119879sum119895=1

1205962119895= 119879sum119895=1

[[(sum119894isin119868119895

119892119894)120596119895 + 12 (sum119894isin119868119895

ℎ119894 + 119886)1205962119895]] + 120574119879(20)

The formula for estimating the weight of each leaf in thedecision tree is formulated as

120596lowast119895 = minus sum119894isin119868119895 119892119894sum119894isin119868119895 ℎ119894 + 119886 (21)

According to (21) as for the tree structure q the lossfunction at the leaf node j can be changed as

(119905) (119902) = minus12119879sum119895=1

(sum119894isin119868119895 119892119894)2sum119894isin119868119895 ℎ119894 + 119886 + 120574119879 (22)

Therefore the equation of the information gain afterbranching can be defined as

119866119886119894119899= 12 [[[

(sum119894isin119868119895119871 119892119894)2sum119894isin119868119895119871 ℎ119894 + 119886 + (sum119894isin119868119895119877 119892119894)2sum119894isin119868119895119877 ℎ119894 + 119886 minus (sum119894isin119868119895 119892119894)2sum119894isin119868119895 ℎ119894 + 119886]]]minus 120574(23)

where 119868119895119871 and 119868119895119877 are the sample sets of the left and right leafnode respectively after splitting the leaf node j

XGBOOST branches each leaf node and constructs basiclearners by the criterion of maximizing the information gain

With the help of Newton boosting the XGBOOSTcan deal with missing values by adaptively learning To acertain extent XGBOOST is based on the multiple additiveregression tree (MART) but it can get better tree structure bylearning with Newton boosting In addition XGBOOST canalso subsample among columns which reduces the relevanceof each weak learner [39]

3 The Proposed CEEMDAN-XGBOOSTApproach

From the existing literature we can see that CEEMDAN hasadvantages in time series decomposition while XGBOOST

does well in regression Therefore in this paper we inte-grated these two methods and proposed a novel approachso-called CEEMDAN-XGBOOST for forecasting crude oilprices The proposed CEEMDAN-XGBOOST includes threestages decomposition individual forecasting and ensembleIn the first stage CEEMDAN is used to decompose the rawseries of crude oil prices into k+1 components includingk IMFs and one residue Among the components someshow high-frequency characteristics while the others showlow-frequency ones of the raw series In the second stagefor each component a forecasting model is built usingXGBOOST and then the built model is applied to forecasteach component and then get an individual result Finally allthe results from the components are aggregated as the finalresult Although there exist a lot of methods to aggregatethe forecasting results from components in the proposedapproach we use the simplest way ie addition to sum-marize the results of all components The flowchart of theCEEMDAN-XGBOOST is shown in Figure 2

From Figure 2 it can be seen that the proposedCEEMDAN-XGBOOST based on the framework of ldquodecom-position and ensemblerdquo is also a typical strategy of ldquodivideand conquerrdquo that is the tough task of forecasting crude oilprices from the raw series is divided into several subtasks offorecasting from simpler components Since the raw seriesis extremely nonlinear and nonstationary while each decom-posed component has a relatively simple form for forecastingthe CEEMDAN-XGBOOST has the ability to achieve higheraccuracy of forecasting crude oil prices In short the advan-tages of the proposed CEEMDAN-XGBOOST are threefold(1) the challenging task of forecasting crude oil prices isdecomposed into several relatively simple subtasks (2) forforecasting each component XGBOOST can build modelswith different parameters according to the characteristics ofthe component and (3) a simple operation addition is usedto aggregate the results from subtasks as the final result

4 Experiments and Analysis

41 Data Description To demonstrate the performance ofthe proposed CEEMDAN-XGBOOST we use the crudeoil prices from the West Texas Intermediate (WTI) asexperimental data (the data can be downloaded fromhttpswwweiagovdnavpethistRWTCDhtm) We usethe daily closing prices covering the period from January 21986 to March 19 2018 with 8123 observations in total forempirical studies Among the observations the first 6498ones from January 2 1986 to September 21 2011 accountingfor 80 of the total observations are used as trainingsamples while the remaining 20 ones are for testing Theoriginal crude oil prices are shown in Figure 3

We perform multi-step-ahead forecasting in this paperFor a given time series 119909119905 (119905 = 1 2 119879) the m-step-aheadforecasting for 119909119905+119898 can be formulated as119909119905+119898 = 119891 (119909119905minus(119897minus1) 119909119905minus(119897minus2) 119909119905minus1 119909119905) (24)where 119909119905+119898 is the m-step-ahead predicted result at time t f isthe forecasting model 119909119894 is the true value at time i and l isthe lag order

6 Complexity

X[n]

CEEMDAN

IMF1 IMF2 IMF3 IMFk R[n]

XGBoost XGBoost XGBoost XGBoost

Decomposition

Individual forecasting

Ensemble

XGBoost

x[n]

R[n]IMF1IMF2

IMF3IMFk

k

i=1IMFi +

R[n]sum

Figure 2 The flowchart of the proposed CEEMDAN-XGBOOST

Jan 02 1986 Dec 22 1989 Dec 16 1993 Dec 26 1997 Jan 15 2002 Feb 07 2006 Feb 22 2010 Mar 03 2014 Mar 19 2018

Date

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Pric

es (U

SDb

arre

l)

WTI crude oil prices

Figure 3 The original crude oil prices of WTI

For SVR and FNN we normalize each decomposed com-ponent before building the model to forecast the componentindividually In detail the normalization process can bedefined as

1199091015840119905 = 119909119905 minus 120583120590 (25)

where 1199091015840119905 is the normalized series of crude oil prices series 119909119905is the data before normalization 120583 is the mean of 119909119905 and 120590 isthe standard deviation of 119909119905 Meanwhile since normalizationis not necessary for XGBOOST and ARIMA for models withthese two algorithms we build forecasting models from eachof the decomposed components directly

Complexity 7

42 Evaluation Criteria When we evaluate the accuracy ofthe models we focus on not only the numerical accuracybut also the accuracy of forecasting direction Therefore weselect the root-mean-square error (RMSE) and the meanabsolute error (MAE) to evaluate the numerical accuracy ofthe models Besides we use directional statistic (Dstat) as thecriterion for evaluating the accuracy of forecasting directionThe RMSE MAE and Dstat are defined as

RMSE = radic 1119873119873sum119894=1

(119910119905 minus 119910119905)2 (26)

MAE = 1119873 ( 119873sum119894=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816) (27)

Dstat = sum119873119894=2 119889119894119873 minus 1 119889119894 =

1 (119910119905 minus 119910119905minus1) (119910119905 minus 119910119905minus1) gt 00 (119910119905 minus 119910119905minus1) (119910119905 minus 119910119905minus1) lt 0

(28)

where 119910119905 is the actual crude oil prices at the time t119910119905 is theprediction and N is the size of the test set

In addition we take the Wilcoxon signed rank test(WSRT) for proving the significant differences between theforecasts of the selectedmodels [44]TheWSRT is a nonpara-metric statistical hypothesis test that can be used to evaluatewhether the population mean ranks of two predictions fromdifferent models on the same sample differ Meanwhile it isa paired difference test which can be used as an alternative tothe paired Student t-test The null hypothesis of the WSRTis whether the median of the loss differential series d(t) =g(119890119886(t)) minus g(119890119887(t)) is equal to zero or not where 119890119886(t) and 119890119887(t)are the error series of model a and model b respectively andg() is a loss function If the p value of pairs of models is below005 the test rejects the null hypothesis (there is a significantdifference between the forecasts of this pair of models) underthe confidence level of 95 In this way we can prove thatthere is a significant difference between the optimal modeland the others

However the criteria defined above are global If somesingular points exist the optimal model chosen by thesecriteria may not be the best one Thus we make the modelconfidence set (MCS) [31 45] in order to choose the optimalmodel convincingly

In order to calculate the p-value of the statistics accuratelythe MCS performs bootstrap on the prediction series whichcan soften the impact of the singular points For the j-thmodel suppose that the size of a bootstrapped sample isT and the t-th bootstrapped sample has the loss functionsdefined as

119871119895119905 = 1119879ℎ+119879sum119905=ℎ+1

lfloor119910119905 minus 119910119905rfloor (29)

Suppose that a set M0=mi i = 1 2 3 n thatcontains n models for any two models j and k the relativevalues of the loss between these two models can be defined as

119889119895119896119905 = 119871119895119905 minus 119871119896119905 (30)

According to the above definitions the set of superiormodels can be defined as

119872lowast equiv 119898119895 isin M0 119864 (119889119895119896119905) le 0 forall119898119896 isin M0 (31)

where E() represents the average valueThe MCS repeatedly performs the significant test in

M0 At each time the worst prediction model in the set iseliminated In the test the hypothesis is the null hypothesisof equal predictive ability (EPA) defined as

1198670 119864 (119889119895119896119905) = 0 forall119898119895 119898119896 isin 119872 sub M0 (32)

The MCS mainly depends on the equivalence test andelimination criteria The specific process is as follows

Step 1 AssumingM=1198720 at the level of significance 120572 use theequivalence test to test the null hypothesis1198670119872Step 2 If it accepts the null hypothesis and then it defines119872lowast1minus120572 = 119872 otherwise it eliminates the model which rejectsthe null hypothesis from M according to the eliminationcriteria The elimination will not stop until there are not anymodels that reject the null hypothesis in the setM In the endthe models in119872lowast1minus120572 are thought as surviving models

Meanwhile the MCS has two kinds of statistics that canbe defined as

119879119877 = max119895119896isin119872

1003816100381610038161003816100381611990511989511989610038161003816100381610038161003816 (33)

119879119878119876 = max119895119896isinM

1199052119895119896 (34)

119905119895119896 = 119889119895119896radicvar (119889119895119896) (35)

119889119895119896 = 1119879ℎ+119879sum119905=ℎ+1

119889119895119896119905 (36)

where T119877 and 119879119878119876 stand for the range statistics and thesemiquadratic statistic respectively and both statistics arebased on the t-statistics as shown in (35)-(36) These twostatistics (T119877 and 119879119878119876) are mainly to remove the model whosep-value is less than the significance level 120572 When the p-valueis greater than the significance level120572 themodels can surviveThe larger the p-value the more accurate the prediction ofthe modelWhen the p-value is equal to 1 it indicates that themodel is the optimal forecasting model

43 Parameter Settings To test the performance ofXGBOOST and CEEMDAN-XGBOOST we conducttwo groups of experiments single models that forecast crude

8 Complexity

Table 1 The ranges of the parameters for XGBOOST by grid search

Parameter Description rangeBooster Booster to use lsquogblinearrsquo lsquogbtreersquoN estimators Number of boosted trees 100200300400500Max depth Maximum tree depth for base learners 345678Min child weight Maximum delta step we allow each treersquos weight estimation to be 123456Gamma Minimum loss reduction required to make a further partition on a leaf node of the tree 001 005010203 Subsample Subsample ratio of the training instance 060708091Colsample Subsample ratio of columns when constructing each tree 060708091Reg alpha L1 regularization term on weights 00100501Reg lambda L2 regularization term on weights 00100501Learning rate Boosting learning rate 0010050070102

oil prices with original sequence and ensemble models thatforecast crude oil prices based on the ldquodecomposition andensemblerdquo framework

For single models we compare XGBOOST with onestatistical model ARIMA and two widely used AI-modelsSVR and FNN Since the existing research has shownthat EEMD significantly outperforms EMD in forecastingcrude oil prices [24 31] in the experiments we onlycompare CEEMDAN with EEMD Therefore we com-pare the proposed CEEMDAN-XGBOOST with EEMD-SVR EEMD-FNN EEMD-XGBOOST CEEMDAN-SVRand CEEMDAN-FNN

For ARIMA we use the Akaike information criterion(AIC) [46] to select the parameters (p-d-q) For SVR weuse RBF as kernel function and use grid search to opti-mize C and gamma in the ranges of 2012345678 and2minus9minus8minus7minus6minus5minus4minus3minus2minus10 respectively We use one hiddenlayer with 20 nodes for FNNWe use a grid search to optimizethe parameters for XGBOOST the search ranges of theoptimized parameters are shown in Table 1

We set 002 and 005 as the standard deviation of theadded white noise and set 250 and 500 as the numberof realizations of EEMD and CEEMDAN respectively Thedecomposition results of the original crude oil prices byEEMD and CEEMDAN are shown in Figures 4 and 5respectively

It can be seen from Figure 4 that among the componentsdecomposed byEEMD the first six IMFs show characteristicsof high frequency while the remaining six components showcharacteristics of low frequency However regarding thecomponents by CEEMDAN the first seven ones show clearhigh-frequency and the last four show low-frequency asshown in Figure 5

The experiments were conducted with Python 27 andMATLAB 86 on a 64-bit Windows 7 with 34 GHz I7 CPUand 32 GB memory Specifically we run FNN and MCSwith MATLAB and for the remaining work we use PythonRegarding XGBoost we used a widely used Python pack-age (httpsxgboostreadthedocsioenlatestpython) in theexperiments

Table 2The RMSE MAE and Dstat by single models with horizon= 1 3 and 6

Horizon Model RMSE MAE Dstat

1

XGBOOST 12640 09481 04827SVR 12899 09651 04826FNN 13439 09994 04837

ARIMA 12692 09520 04883

3

XGBOOST 20963 16159 04839SVR 22444 17258 05080FNN 21503 16512 04837

ARIMA 21056 16177 04901

6

XGBOOST 29269 22945 05158SVR 31048 24308 05183FNN 30803 24008 05028

ARIMA 29320 22912 05151

44 Experimental Results In this subsection we use a fixedvalue 6 as the lag order andwe forecast crude oil prices with 1-step-ahead 3-step-ahead and 6-step-ahead forecasting thatis to say the horizons for these three forecasting tasks are 1 3and 6 respectively

441 Experimental Results of Single Models For single mod-els we compare XGBOOST with state-of-the-art SVR FNNand ARIMA and the results are shown in Table 2

It can be seen from Table 2 that XGBOOST outperformsother models in terms of RMSE and MAE with horizons 1and 3 For horizon 6 XGBOOST achieves the best RMSE andthe second best MAE which is slightly worse than that byARIMA For horizon 1 FNNachieves theworst results amongthe four models however for horizons 3 and 6 SVR achievesthe worst results Regarding Dstat none of the models canalways outperform others and the best result of Dstat isachieved by SVR with horizon 6 It can be found that theRMSE and MAE values gradually increase with the increaseof horizon However the Dstat values do not show suchdiscipline All the values of Dstat are around 05 ie from

Complexity 9

minus505

IMF1

minus4minus2

024

IMF2

minus505

IMF3

minus505

IMF4

minus4minus2

024

IMF5

minus505

IMF6

minus200

20

IMF7

minus100

1020

IMF8

minus100

10

IMF9

minus20minus10

010

IMF1

0

minus200

20

IMF1

1


Resid

ue

Figure 4 The IMFs and residue of WTI crude oil prices by EEMD

04826 to 05183 which are very similar to the results ofrandom guesses showing that it is very difficult to accuratelyforecast the direction of the movement of crude oil priceswith raw crude oil prices directly

To further verify the advantages of XGBOOST over othermodels we report the results by WSRT and MCS as shownin Tables 3 and 4 respectively As for WSRT the p-valuebetween XGBOOST and other models except ARIMA isbelow 005 which means that there is a significant differenceamong the forecasting results of XGBOOST SVR and FNNin population mean ranks Besides the results of MCS showthat the p-value of119879119877 and119879119878119876 of XGBOOST is always equal to1000 and prove that XGBOOST is the optimal model among

Table 3Wilcoxon signed rank test between XGBOOST SVR FNNand ARIMA

XGBOOST SVR FNN ARIMAXGBOOST 1 40378e-06 22539e-35 57146e-01SVR 40378e-06 1 46786e-33 07006FNN 22539e-35 46786e-33 1 69095e-02ARIMA 57146e-01 07006 69095e-02 1

all the models in terms of global errors and most local errorsof different samples obtained by bootstrap methods in MCSAccording to MCS the p-values of 119879119877 and 119879119878119876 of SVR onthe horizon of 3 are greater than 02 so SVR becomes the

10 Complexity

minus505

IMF1

minus505

IMF2

minus505

IMF3

minus505

IMF4

minus505

IMF5

minus100

10

IMF6

minus100

10

IMF7

minus20

0

20

IMF8

minus200

20

IMF9

minus200

20

IMF1

0


Resid

ue

Figure 5 The IMFs and residue of WTI crude oil prices by CEEMDAN

Table 4 MCS between XGBOOST SVR FNN and ARIMA

HORIZON=1 HORIZON=3 HORIZON=6119879119877 119879119878119876 119879119877 119879119878119876 119879119877 119879119878119876XGBOOST 10000 10000 10000 10000 10000 10000SVR 00004 00004 04132 04132 00200 00200FNN 00002 00002 00248 00538 00016 00022ARIMA 00000 00000 00000 00000 00000 00000

survival and the second best model on this horizon Whenit comes to ARIMA ARIMA almost performs as good asXGBOOST in terms of evaluation criteria of global errorsbut it does not pass the MCS It indicates that ARIMA doesnot perform better than other models in most local errors ofdifferent samples

442 Experimental Results of Ensemble Models With EEMDor CEEMDAN the results of forecasting the crude oil pricesby XGBOOST SVR and FNN with horizons 1 3 and 6 areshown in Table 5

It can be seen from Table 5 that the RMSE and MAEvalues of the CEEMDAN-XGBOOST are the lowest ones

Complexity 11

Table 5 The RMSE MAE and Dstat by the models with EEMD orCEEMDAN


1

CEEMDAN-XGBOOST 04151 03023 08783EEMD-XGBOOST 09941 07685 07109CEEMDAN-SVR 08477 07594 09054

EEMD-SVR 11796 09879 08727CEEMDAN-FNN 12574 10118 07597

EEMD-FNN 26835 19932 07361

3



EEMD-FNN 12046 08637 06959

6



EEMD-FNN 27786 20495 06337

among those by all methods with each horizon For examplewith horizon 1 the values of RMSE and MAE are 04151 and03023which are far less than the second values of RMSE andMAE ie 08477 and 07594 respectively With the horizonincreases the corresponding values of each model increasein terms of RMSE and MAE However the CEEMDAN-XGBOOST still achieves the lowest values of RMSE andMAEwith each horizon Regarding the values of Dstat all thevalues are far greater than those by random guesses showingthat the ldquodecomposition and ensemblerdquo framework is effectivefor directional forecasting Specifically the values of Dstatare in the range between 06165 and 09054 The best Dstatvalues in all horizons are achieved by CEEMDAN-SVR orEEMD-SVR showing that among the forecasters SVR is thebest one for directional forecasting although correspondingvalues of RMSE and MAE are not the best As for thedecomposition methods when the forecasters are fixedCEEMDAN outperforms EEMD in 8 8 and 8 out of 9 casesin terms of RMSE MAE and Dstat respectively showingthe advantages of CEEMDAN over EEMD Regarding theforecasters when combined with CEEMDAN XGBOOST isalways superior to other forecasters in terms of RMSE andMAE However when combined with EEMD XGBOOSToutperforms SVR and FNN with horizon 1 and FNN withhorizon 6 in terms of RMSE and MAE With horizons 1and 6 FNN achieves the worst results of RMSE and MAEThe results also show that good values of RMSE usually areassociated with good values of MAE However good valuesof RMSE or MAE do not always mean good Dstat directly

For the ensemble models we also took aWilcoxon signedrank test and an MCS test based on the errors of pairs ofmodels We set 02 as the level of significance in MCS and005 as the level of significance in WSRT The results areshown in Tables 6 and 7

From these two tables it can be seen that regardingthe results of WSRT the p-value between CEEMDAN-XGBOOST and any other models except EEMD-FNN arebelow 005 demonstrating that there is a significant differ-ence on the population mean ranks between CEEMDAN-XGBOOST and any other models except EEMD-FNNWhatis more the MCS shows that the p-value of 119879119877 and 119879119878119876of CEEMDAN-XGBOOST is always equal to 1000 anddemonstrates that CEEMDAN-XGBOOST is the optimalmodel among all models in terms of global errors and localerrorsMeanwhile the p-values of119879119877 and119879119878119876 of EEMD-FNNare greater than othermodels except CEEMDAN-XGBOOSTand become the second best model with horizons 3 and 6in MCS Meanwhile with the horizon 6 the CEEMDAN-SVR is also the second best model Besides the p-values of119879119877 and 119879119878119876 of EEMD-SVR and CEEMDAN-SVR are up to02 and they become the surviving models with horizon 6 inMCS

From the results of single models and ensemble modelswe can draw the following conclusions (1) single modelsusually cannot achieve satisfactory results due to the non-linearity and nonstationarity of raw crude oil prices Asa single forecaster XGBOOST can achieve slightly betterresults than some state-of-the-art algorithms (2) ensemblemodels can significantly improve the forecasting accuracy interms of several evaluation criteria following the ldquodecom-position and ensemblerdquo framework (3) as a decompositionmethod CEEMDAN outperforms EEMD in most cases (4)the extensive experiments demonstrate that the proposedCEEMDAN-XGBOOST is promising for forecasting crudeoil prices

45 Discussion In this subsection we will study the impactsof several parameters related to the proposed CEEMDAN-XGBOOST

451The Impact of the Number of Realizations in CEEMDANIn (2) it is shown that there are N realizations 119909119894[119905] inCEEMDAN We explore how the number of realizations inCEEMDAN can influence on the results of forecasting crudeoil prices by CEEMDAN-XGBOOST with horizon 1 and lag6 And we set the number of realizations in CEEMDAN inthe range of 102550751002505007501000 The resultsare shown in Figure 6

It can be seen from Figure 6 that for RMSE and MAEthe bigger the number of realizations is the more accurateresults the CEEMDAN-XGBOOST can achieve When thenumber of realizations is less than or equal to 500 the valuesof both RMSE and MAE decrease with increasing of thenumber of realizations However when the number is greaterthan 500 these two values are increasing slightly RegardingDstat when the number increases from 10 to 25 the value ofDstat increases rapidly and then it increases slowly with thenumber increasing from 25 to 500 After that Dstat decreasesslightly It is shown that the value of Dstat reaches the topvalues with the number of realizations 500 Therefore 500is the best value for the number of realizations in terms ofRMSE MAE and Dstat

12 Complexity

Table 6 Wilcoxon signed rank test between XGBOOST SVR and FNN with EEMD or CEEMDAN

CEEMDAN-XGBOOST

EEMD-XGBOOST

CEEMDAN-SVR

EEMD-SVR

CEEMDAN-FNN

EEMD-FNN

CEEMDAN-XGBOOST 1 35544e-05 18847e-50 0 0028 16039e-187 00726EEMD-XGBOOST 35544e-05 1 45857e-07 0 3604 82912e-82 00556CEEMDAN-SVR 18847e-50 45857e-07 1 49296e-09 57753e-155 86135e-09EEMD-SVR 00028 0 3604 49296e-09 1 25385e-129 00007

CEEMDAN-FNN 16039e-187 82912e-82 57753e-155 25385e-129 1 81427e-196

EEMD-FNN 00726 00556 86135e-09 00007 81427e-196 1

Table 7 MCS between XGBOOST SVR and FNN with EEMD or CEEMDAN

HORIZON=1 HORIZON=3 HORIZON=6119879119877 119879119878119876 119879119877 119879119878119876 119879119877 119879119878119876CEEMDAN-XGBOOST 1 1 1 1 1 1EEMD-XGBOOST 0 0 0 0 0 00030CEEMDAN-SVR 0 00002 00124 00162 0 8268 08092EEMD-SVR 0 0 00008 0004 07872 07926CEEMDAN-FNN 0 0 00338 00532 02924 03866EEMD-FNN 0 00002 04040 04040 0 8268 08092

452 The Impact of the Lag Orders In this section weexplore how the number of lag orders impacts the predictionaccuracy of CEEMDAN-XGBOOST on the horizon of 1 Inthis experiment we set the number of lag orders from 1 to 10and the results are shown in Figure 7

According to the empirical results shown in Figure 7 itcan be seen that as the lag order increases from 1 to 2 thevalues of RMSE andMAEdecrease sharply while that of Dstatincreases drastically After that the values of RMSE of MAEremain almost unchanged (or increase very slightly) withthe increasing of lag orders However for Dstat the valueincreases sharply from 1 to 2 and then decreases from 2 to3 After the lag order increases from 3 to 5 the Dstat staysalmost stationary Overall when the value of lag order is upto 5 it reaches a good tradeoff among the values of RMSEMAE and Dstat

453 The Impact of the Noise Strength in CEEMDAN Apartfrom the number of realizations the noise strength inCEEMDAN which stands for the standard deviation of thewhite noise in CEEMDAN also affects the performance ofCEEMDAN-XGBOOST Thus we set the noise strength inthe set of 001 002 003 004 005 006 007 to explorehow the noise strength in CEEMDAN affects the predictionaccuracy of CEEMDAN-XGBOOST on a fixed horizon 1 anda fixed lag 6

As shown in Figure 8 when the noise strength in CEEM-DAN is equal to 005 the values of RMSE MAE and Dstatachieve the best results simultaneously When the noisestrength is less than or equal to 005 except 003 the valuesof RMSE MAE and Dstat become better and better with theincrease of the noise strength However when the strengthis greater than 005 the values of RMSE MAE and Dstat

become worse and worse The figure indicates that the noisestrength has a great impact on forecasting results and an idealrange for it is about 004-006

5 Conclusions

In this paper we propose a novelmodel namely CEEMDAN-XGBOOST to forecast crude oil prices At first CEEMDAN-XGBOOST decomposes the sequence of crude oil prices intoseveral IMFs and a residue with CEEMDAN Then it fore-casts the IMFs and the residue with XGBOOST individuallyFinally CEEMDAN-XGBOOST computes the sum of theprediction of the IMFs and the residue as the final forecastingresults The experimental results show that the proposedCEEMDAN-XGBOOST significantly outperforms the com-pared methods in terms of RMSE and MAE Although theperformance of the CEEMDAN-XGBOOST on forecastingthe direction of crude oil prices is not the best theMCS showsthat the CEEMDAN-XGBOOST is still the optimal modelMeanwhile it is proved that the number of the realizationslag and the noise strength for CEEMDAN are the vitalfactors which have great impacts on the performance of theCEEMDAN-XGBOOST

In the future we will study the performance of theCEEMDAN-XGBOOST on forecasting crude oil prices withdifferent periods We will also apply the proposed approachfor forecasting other energy time series such as wind speedelectricity load and carbon emissions prices

Data Availability

The data used to support the findings of this study areincluded within the article

Complexity 13

RMSEMAEDstat

Dsta

t

RMSE

MA

E

070

065

060

050

055

045

040

035

03010 25 50 500 750 1000 75 100 250

088

087

086

085

084

083

082

081

080

The number of realizations in CEEMDAN

Figure 6 The impact of the number of IMFs with CEEMDAN-XGBOOST

RMSEMAEDstat

Dsta

t

RMSE

MA

E

2 4 6 8 10lag order

088

086

084

082

080

078

076

074

12

10

08

06

04

Figure 7 The impact of the number of lag orders with CEEMDAN-XGBOOST

RMSEMAEDstat

Dsta

t

The white noise strength of CEEMDAN

088

086

084

082

080

078

076

074

10

08

09

06

07

04

05

03

001 002 003 004 005 006 007

RMSE

MA

E

Figure 8 The impact of the value of the white noise of CEEMDAN with CEEMDAN-XGBOOST

14 Complexity



Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (Grants no JBK1902029no JBK1802073 and no JBK170505) the Natural ScienceFoundation of China (Grant no 71473201) and the ScientificResearch Fund of Sichuan Provincial Education Department(Grant no 17ZB0433)

References

[1] H Miao S Ramchander T Wang and D Yang ldquoInfluentialfactors in crude oil price forecastingrdquo Energy Economics vol 68pp 77ndash88 2017

[2] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[3] M Ye J Zyren C J Blumberg and J Shore ldquoA short-run crudeoil price forecast model with ratchet effectrdquo Atlantic EconomicJournal vol 37 no 1 pp 37ndash50 2009

[4] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[5] H Naser ldquoEstimating and forecasting the real prices of crudeoil A data richmodel using a dynamic model averaging (DMA)approachrdquo Energy Economics vol 56 pp 75ndash87 2016

[6] XGong andB Lin ldquoForecasting the goodandbaduncertaintiesof crude oil prices using a HAR frameworkrdquo Energy Economicsvol 67 pp 315ndash327 2017

[7] F Wen X Gong and S Cai ldquoForecasting the volatility of crudeoil futures using HAR-type models with structural breaksrdquoEnergy Economics vol 59 pp 400ndash413 2016

[8] H Chiroma S Abdul-Kareem A Shukri Mohd Noor et alldquoA Review on Artificial Intelligence Methodologies for theForecasting of Crude Oil Pricerdquo Intelligent Automation and SoftComputing vol 22 no 3 pp 449ndash462 2016

[9] S Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo in Data Miningand Knowledge Management Y Shi W Xu and Z Chen Edsvol 3327 of Lecture Notes in Computer Science pp 233ndash242Springer Berlin Germany 2004

[10] J Barunık and B Malinska ldquoForecasting the term structure ofcrude oil futures prices with neural networksrdquo Applied Energyvol 164 pp 366ndash379 2016

[11] Y Chen K He and G K F Tso ldquoForecasting crude oil prices-a deep learning based modelrdquo Procedia Computer Science vol122 pp 300ndash307 2017

[12] R Tehrani and F Khodayar ldquoA hybrid optimized artificialintelligent model to forecast crude oil using genetic algorithmrdquoAfrican Journal of Business Management vol 5 no 34 pp13130ndash13135 2011

[13] L Yu Y Zhao and L Tang ldquoA compressed sensing basedAI learning paradigm for crude oil price forecastingrdquo EnergyEconomics vol 46 no C pp 236ndash245 2014

[14] L Yu H Xu and L Tang ldquoLSSVR ensemble learning withuncertain parameters for crude oil price forecastingrdquo AppliedSoft Computing vol 56 pp 692ndash701 2017

[15] Y-L Qi and W-J Zhang ldquoThe improved SVM method forforecasting the fluctuation of international crude oil pricerdquo inProceedings of the 2009 International Conference on ElectronicCommerce and Business Intelligence (ECBI rsquo09) pp 269ndash271IEEE China June 2009

[16] M S AL-Musaylh R C Deo Y Li and J F AdamowskildquoTwo-phase particle swarm optimized-support vector regres-sion hybrid model integrated with improved empirical modedecomposition with adaptive noise for multiple-horizon elec-tricity demand forecastingrdquo Applied Energy vol 217 pp 422ndash439 2018

[17] LHuang and JWang ldquoForecasting energy fluctuationmodel bywavelet decomposition and stochastic recurrent wavelet neuralnetworkrdquo Neurocomputing vol 309 pp 70ndash82 2018

[18] H Zhao M Sun W Deng and X Yang ldquoA new featureextraction method based on EEMD and multi-scale fuzzyentropy for motor bearingrdquo Entropy vol 19 no 1 Article ID14 2017

[19] W Deng S Zhang H Zhao and X Yang ldquoA Novel FaultDiagnosis Method Based on Integrating Empirical WaveletTransform and Fuzzy Entropy for Motor Bearingrdquo IEEE Accessvol 6 pp 35042ndash35056 2018

[20] Q Fu B Jing P He S Si and Y Wang ldquoFault FeatureSelection and Diagnosis of Rolling Bearings Based on EEMDand Optimized Elman AdaBoost Algorithmrdquo IEEE SensorsJournal vol 18 no 12 pp 5024ndash5034 2018

[21] T Li and M Zhou ldquoECG classification using wavelet packetentropy and random forestsrdquo Entropy vol 18 no 8 p 285 2016

[22] M Blanco-Velasco B Weng and K E Barner ldquoECG signaldenoising and baseline wander correction based on the empir-ical mode decompositionrdquo Computers in Biology and Medicinevol 38 no 1 pp 1ndash13 2008

[23] J Lee D D McManus S Merchant and K H ChonldquoAutomatic motion and noise artifact detection in holterECG data using empirical mode decomposition and statisticalapproachesrdquo IEEE Transactions on Biomedical Engineering vol59 no 6 pp 1499ndash1506 2012

[24] L Fan S Pan Z Li and H Li ldquoAn ICA-based supportvector regression scheme for forecasting crude oil pricesrdquoTechnological Forecasting amp Social Change vol 112 pp 245ndash2532016

[25] E Jianwei Y Bao and J Ye ldquoCrude oil price analysis andforecasting based on variationalmode decomposition and inde-pendent component analysisrdquo Physica A Statistical Mechanicsand Its Applications vol 484 pp 412ndash427 2017

[26] N E Huang Z Shen S R Long et al ldquoThe empiricalmode decomposition and the Hilbert spectrum for nonlinearand non-stationary time series analysisrdquo Proceedings of theRoyal Society of London Series A Mathematical Physical andEngineering Sciences vol 454 pp 903ndash995 1998

[27] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[28] L Yu W Dai L Tang and J Wu ldquoA hybrid grid-GA-basedLSSVR learning paradigm for crude oil price forecastingrdquoNeural Computing andApplications vol 27 no 8 pp 2193ndash22152016

[29] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technology ampDecision Making vol 14 no 1 pp 141ndash169 2015

Complexity 15

[30] T Li M Zhou C Guo et al ldquoForecasting crude oil price usingEEMD and RVM with adaptive PSO-based kernelsrdquo Energiesvol 9 no 12 p 1014 2016

[31] T Li Z Hu Y Jia J Wu and Y Zhou ldquoForecasting CrudeOil PricesUsing Ensemble EmpiricalModeDecomposition andSparse Bayesian Learningrdquo Energies vol 11 no 7 2018

[32] M E TorresMA Colominas G Schlotthauer andP FlandrinldquoA complete ensemble empirical mode decomposition withadaptive noiserdquo in Proceedings of the 36th IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo11) pp 4144ndash4147 IEEE Prague Czech Republic May 2011

[33] M A Colominas G Schlotthauer andM E Torres ldquoImprovedcomplete ensemble EMD a suitable tool for biomedical signalprocessingrdquo Biomedical Signal Processing and Control vol 14no 1 pp 19ndash29 2014

[34] T Peng J Zhou C Zhang and Y Zheng ldquoMulti-step aheadwind speed forecasting using a hybrid model based on two-stage decomposition technique and AdaBoost-extreme learn-ing machinerdquo Energy Conversion and Management vol 153 pp589ndash602 2017

[35] S Dai D Niu and Y Li ldquoDaily peak load forecasting basedon complete ensemble empirical mode decomposition withadaptive noise and support vector machine optimized bymodified grey Wolf optimization algorithmrdquo Energies vol 11no 1 2018

[36] Y Lv R Yuan TWangH Li andG Song ldquoHealthDegradationMonitoring and Early Fault Diagnosis of a Rolling BearingBased on CEEMDAN and Improved MMSErdquo Materials vol11 no 6 p 1009 2018

[37] R Abdelkader A Kaddour A Bendiabdellah and Z Der-ouiche ldquoRolling Bearing FaultDiagnosis Based on an ImprovedDenoising Method Using the Complete Ensemble EmpiricalMode Decomposition and the OptimizedThresholding Opera-tionrdquo IEEE Sensors Journal vol 18 no 17 pp 7166ndash7172 2018

[38] Y Lei Z Liu J Ouazri and J Lin ldquoA fault diagnosis methodof rolling element bearings based on CEEMDANrdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 231 no 10 pp 1804ndash18152017

[39] T Chen and C Guestrin ldquoXGBoost a scalable tree boostingsystemrdquo in Proceedings of the 22nd ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining KDD2016 pp 785ndash794 ACM New York NY USA August 2016

[40] B Zhai and J Chen ldquoDevelopment of a stacked ensemblemodelfor forecasting and analyzing daily average PM25 concentra-tions in Beijing Chinardquo Science of the Total Environment vol635 pp 644ndash658 2018

[41] S P Chatzis V Siakoulis A Petropoulos E Stavroulakis andN Vlachogiannakis ldquoForecasting stock market crisis eventsusing deep and statistical machine learning techniquesrdquo ExpertSystems with Applications vol 112 pp 353ndash371 2018

[42] J Ke H Zheng H Yang and X Chen ldquoShort-term forecastingof passenger demand under on-demand ride services A spatio-temporal deep learning approachrdquoTransportation Research PartC Emerging Technologies vol 85 pp 591ndash608 2017

[43] J Friedman T Hastie and R Tibshirani ldquoSpecial Invited PaperAdditive Logistic Regression A Statistical View of BoostingRejoinderdquo The Annals of Statistics vol 28 no 2 pp 400ndash4072000

[44] J D Gibbons and S Chakraborti ldquoNonparametric statisticalinferencerdquo in International Encyclopedia of Statistical ScienceM Lovric Ed pp 977ndash979 Springer Berlin Germany 2011

[45] P RHansen A Lunde and JMNason ldquoThemodel confidencesetrdquo Econometrica vol 79 no 2 pp 453ndash497 2011

[46] H Liu H Q Tian and Y F Li ldquoComparison of two newARIMA-ANN and ARIMA-Kalman hybrid methods for windspeed predictionrdquo Applied Energy vol 98 no 1 pp 415ndash4242012

Research ArticleDevelopment of Multidecomposition Hybrid Model forHydrological Time Series Analysis

Hafiza Mamona Nazir 1 Ijaz Hussain 1 Muhammad Faisal 23

Alaa Mohamd Shoukry45 Showkat Gani6 and Ishfaq Ahmad 78

1Department of Statistics Quaid-i-Azam University Islamabad Pakistan2Faculty of Health Studies University of Bradford Bradford BD7 1DP UK3Bradford Institute for Health Research Bradford Teaching Hospitals NHS Foundation Trust Bradford UK4Arriyadh Community College King Saud University Riyadh Saudi Arabia5KSAWorkers University El-Mansoura Egypt6College of Business Administration King Saud University Al-Muzahimiyah Saudi Arabia7Department of Mathematics College of Science King Khalid University Abha 61413 Saudi Arabia8Department of Mathematics and Statistics Faculty of Basic and Applied Sciences International Islamic University44000 Islamabad Pakistan

Correspondence should be addressed to Ijaz Hussain ijazqauedupk

Received 1 October 2018 Accepted 13 December 2018 Published 2 January 2019


Copyright copy 2019 Hafiza Mamona Nazir et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Accurateprediction of hydrological processes is key for optimal allocation of water resources In this study two novel hybridmodelsare developed to improve the prediction precision of hydrological time series data based on the principal of three stages as denoisingdecomposition and decomposed component prediction and summationTheproposed architecture is applied on daily rivers inflowtime series data of Indus Basin SystemThe performances of the proposed models are compared with traditional single-stagemodel(without denoised and decomposed) the hybrid two-stage model (with denoised) and existing three-stage hybrid model (withdenoised and decomposition) Three evaluation measures are used to assess the prediction accuracy of all models such as MeanRelative Error (MRE) Mean Absolute Error (MAE) andMean Square Error (MSE)The proposed three-stage hybrid models haveshown improvement in prediction accuracywith minimumMRE MAE andMSE for all case studies as compared to other existingone-stage and two-stage models In summary the accuracy of prediction is improved by reducing the complexity of hydrologicaltime series data by incorporating the denoising and decomposition

1 Introduction

Accurate prediction of hydrological processes is key foroptimal allocation of water resources It is challengingbecause of its nonstationary and multiscale stochastic char-acteristics of hydrological process which are affected notonly by climate change but also by other socioeconomicdevelopment projects The human activities also effected theclimate change through contributing in Earthrsquos atmosphereby burning of fossil fuels which release carbon dioxide inatmosphere Instead of these greenhouse and aerosols havemade effect on Earthrsquos atmosphere by altering in-out coming

solar radiations which is the part of Earthrsquos energy balanceThis makes the prediction of hydrological time series datachallenging To predict such hydrological processes twobroad types ofmodels are commonly used one is the process-based models which further included the lumped concep-tual models hydrological model and one-two-dimensionalhydrodynamic models [1] and the second is data drivenmodels which included autoregressive moving averages andartificial neural network (which are also known as black boxmodels) The process-based models considered the physi-cal mechanism of stochastic hydrological processes whichrequires a large amount of data for calibration and validation


2 Complexity

[2] Moreover physical-based models demand the scientificprinciples of energy and water movements spatiotemporallyZahidul [3] concluded that unavailability of sufficient amountof data and scientific knowledge of water movement can leadto poor understanding of hydrological system which makesthe hydrological modeling a challenging task In order toovercome these drawbacks hydrologists used data drivenmodels to efficiently model the hydrological process [4 5]The data driven models only take the advantage of inherentthe input-output relationship through data manipulationwithout considering the internal physical process The data-driven models are efficient over the process-driven models byappreciating the advantage of less demanding the quantitativedata simple formulation with better prediction performance[6] These data-driven models are further divided into twocategories simple traditional statistical techniques and morecomplex machine learning methods In the last few decadesmany traditional statistical time series models are devel-oped including Autoregressive (AR)Moving Averages (MA)Autoregressive Moving Averages (ARMA) and Autoregres-sive Integrated Moving Averages (ARIMA) [7] Applicationof ARIMA model to monitoring hydrological processes likeriver discharge is common and successfully applied [8] Butthe problem with all these traditional statistical techniquesrequired that the time series data to be stationary Howeverhydrological data was characterized as both nonstationaryand nonlinear due to its time varying nature Thereforethese techniques are not enough to capture the nonlinearcharacteristics of hydrological series [6] To rescue thedrawbacks of existing traditional models machine learning(ML) algorithms have been put forward and widely exploitedwhich provide powerful solution to the instability of hydro-logical time series data [4] ML models include ArtificialNeural Network (ANN) Support Vector Machine (SVM)and random forest and genetic algorithms [9ndash14] Riad etal [5] developed an ANN to model the nonlinear relationbetween rainfall and runoff and concluded that ANN modelis better to model the complex hydrological system over thetraditional statistical models However these ML methodshave their own drawbacks such as overfitting and beingsensitive to parameter selection In addition there are twomain drawbacks of using MLmodels first is that ML modelsignore the time varying characteristics of hydrological timeseries data and secondly the hydrological data containsnoises which deprive the researchers to accurately predictthe hydrological time series data in an effective way [15]These time varying and noise corrupted characteristics ofhydrological time series data require hybrid approaches tomodel the complex characteristics of hydrological time seriesdata [16]

To conquer the limitations of existing singlemodels somehybrid algorithms such as data preprocessing methods areutilized with data-driven models with the hope to enhancethe prediction performance of complex hydrological timeseries data by extracting time varying components withnoise reduction These preprocess based hybrid models havealready been applied in hydrology [2] The framework ofhybrid model usually comprised ldquodecompositionrdquo ldquopredic-tionrdquo and ldquoensemblerdquo [2 6 13] The most commonly used

data preprocessing method is wavelet analysis (WA) whichis used to decompose the nonlinear and nonstationaryhydrological data into multiscale components [13] Theseprocessed multiscale components are further used as inputsin black box models at prediction stage and finally predictedcomponents are ensemble to get final predictions Peng etal [6] proposed hybrid model by using empirical wavelettransform andANN for reliable stream flow forecastingTheydemonstrated their proposed hybrid model efficiency oversingle models Later Wu et al [11] exploited a two-stagehybrid model by incorporating Wavelet Multi-ResolutionAnalysis (WMRA) and other data preprocessing methods asMA singular spectrum analysis with ANN to enhance theestimate of daily flows They proposed five models includ-ing ANN-MA ANN-SSA1 ANN-SSA2 ANN-WMRA1 andANN-WMRA2 and suggested that decomposition with MAmodel performs better than WMRA An improvement inwavelet decomposition method has been made to get moreaccurate hybrid results comprising WA [17] However theproblemwhich reduces the performance ofWA ie selectionof mother wavelet basis function is still an open debate as theselection ofmother wavelet is subjectively determined amongmany wavelet basis functions The optimality of multiscalecharacteristics entirely depends on the choice of motherwavelet function as poorly selected mother wavelet functioncan lead to more uncertainty in time-scale components Toovercome this drawback Huang et al [18] proposed a purelydata-driven Empirical Mode Decomposition (EMD) tech-nique The objective of EMD is to decompose the nonlinearand nonstationary data adaptively into number of oscillatorycomponents called Intrinsic Mode Decomposition (IMF)A number of studies have been conducted combining theEMD with data driven models [15 18ndash21] Specifically inhydrology EMD is usedwithANN forwind speed and streamflow prediction [15 20] Agana and Homaifar [21] developedthe EMD-based predictive deep belief network for accuratelypredicting and forecasting the Standardized Stream flowIndex (SSI) Their study manifested that their proposedmodel is better than the existing standard methods with theimprovement in prediction accuracy of SSI However Kang etal [22] revealed that EMD suffers withmodemixing problemwhich ultimately affects the efficiency of decomposition Inorder to deal with this mode mixing problem Wu andHang [23] proposed an improved EMD by successivelyintroducing white Gauss noise in signals called EnsembleEmpirical Mode Decomposition (EEMD) that addresses theissue of frequent apparent of mode mixing in EMD LaterEEMD was effectively used as data decomposition methodto extract the multiscale characteristics [24ndash26] Di et al[2] proposed a four-stage hybrid model (based on EEMDfor decomposition) to improve the prediction accuracy byreducing the redundant noises and concluded that couplingthe appropriate data decomposition with EEMD methodwith data driven models could improve the prediction per-formance compared to existing EMD based hybrid modelJiang et al [26] proposed another two-stage hybrid approachcoupling EEMD with data-driven models to forecast highspeed rail passenger flow to estimate daily ridership Theysuggested that their proposed hybrid model is more suitable

Complexity 3

Decompose-

D-step

e original Time Series

WAEMD-based method

e denoised time series

CEEMDAN

Sum of predicted IMFs

step

IMF1

ARMA

RGMDH

RBFNN

IMF2 IMFl

ARMA

RGMDH

RBFNN

ARMA

RGMDH

RBFNN

P-step

Select the best

Predicted IMF

Component

Figure 1 The proposed WAEMD-CEEMDAN-MM structure to predict hydrological time series data

for short term prediction by accounting for the day to dayvariation over other hybrid and single models However dueto successive addition of independent white Gauss noise theperformance of EEMD is affected which reduces the accuracyof extracted IMFs through EEMD algorithm Dai et al [27]reported in their study that EEMD based hybrid modelsdid not perform appropriately due to independent noiseaddition

This study aimed to develop a robust hybrid modelto decompose the hydrological time varying characteristicsusing CEEMDAN [28] The CEEMDAN successively addswhite noise following the steps of EEMD with interferencein each decomposition level to overcome the drawback ofEEMD algorithm Dai et al [27] developed a model compris-ing CEEMDAN to daily peak load forecasting which showsrobust decomposed ability for reliable forecasting Thereforethe purpose of using CEEMDAN method for decompositionin this study is to find an effective way to decompose thenonlinear data which enhances the prediction accuracy of thecomplex hydrological time series data [27]

2 Proposed Methods

In this study two novel approaches are proposed to enhancethe prediction accuracy of the hydrological time series Bothmodels have the same layout except in stage of denoisingwhere two different approaches have been used to removenoises from hydrological time series data In both modelsat decomposition stage an improved version of EEMD ieCEEMDAN is used to find oscillations ie the high tolow frequencies in terms of IMF At prediction stage multi-models are used to accurately predict the extracted IMFs byconsidering the nature of IMFs instead of using only single

stochastic model The purpose of using multimodel is two-way one is for accurately predicting the IMFs by consideringthe nature of IMFs and the other is to assess the performanceof simple and complex models after reducing the complexityof hydrological time series data through decompositionPredicted IMFs are added to get the final prediction ofhydrological time series The proposed three stages involvedenoising (D-step) decomposition (Decompose-step) andcomponent prediction (P-step) which are briefly describedbelow

(1) D-step WA and EMD based denoising methods arepresented to remove the noises from hydrologicaltime series data

(2) Decomposed-step Using CEEMDAN two sepa-rately denoised series are decomposed into 119896 IMFsand one residual

(3) P-stepThe denoised-decomposed series into 119896 IMFsand one residual are predicted with linear stochasticand nonlinear machine learning models The modelwith the lowest error rate of prediction is selected bythree performance evaluation measures Finally thepredicted results are added to get the final prediction

For convenient two proposed methods as named asEMD (denoising) CEEMDAN (decomposing) MM (multi-models) ie EMD-CEEMDAN-MM and WA (denoising)CEEMDAN (denoising) and MM (multi-models) ie WA-CEEMDAN-MM The proposed architecture of WAEMD-CEEMDAN-MM is given in Figure 1

21 D-Step In hydrology time series data noises or stochasticvolatiles are inevitable component which ultimately reduced

4 Complexity

the performance of prediction To reduce the noise fromdata many algorithms have been proposed in literature suchas Fourier analysis spectral analysis WA and EMD [29]as besides decomposition these techniques have the abilityto remove the noises from data However the spectral andFourier analysis only considered the linear and stationarysignals whereasWA and EMD have the ability to address thenonlinear and nonstationary data with better performanceIn this study WA- and EMD-based threshold are adopted toreduce the stochastic volatiles from the hydrological data

(i) Wavelet analysis based denoising in order to removenoises discrete wavelet threshold method is recognized aspowerful mathematical functions with hard and soft thresh-oldWith the help of symlet 8 mother wavelet [30] hydrolog-ical time series data is decomposed into approximation anddetails coefficients with the following equations respectively[31]

119886119895119896 =2119873minus119895minus1

sum119896=0

2minus11989520 (2minus119895119905 minus 119896) (1)

and

119889119895119896 =119869

sum119895=1

2119873minus119895minus1

sum119896=0


After estimating the approximation and details coefficientsthreshold is calculated for each coefficient to remove noisesThe energy of data is distributed only on few wavelet coef-ficients with high magnitude whereas most of the waveletcoefficients are noisiest with low magnitude To calculate thenoise free coefficient hard and soft threshold rules are optedwhich are listed as follows respectively [31]

1198891015840119895119896 =

119889119895119896 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(3)

and

1198891015840119895119896 =

sgn (119889119895119896) (1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 minus 119879119895)

1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(4)

where119879119894 is the threshold calculated as 119879119895 = 119886radic2119864119895 ln(119873) 119895 =1 2 119869 where 119886 is constant which takes the values between04 and 14 with step of 01 and 119864119895 = 119898119890119889119894119886119899(|119889119895119896| 119896 =1 2 119873)06745 is median deviation of all details Thenthe decomposed data is reconstructed using the noise freedetails and approximations using the following equation

119909 (119905) =2119873minus119895minus1

sum119896=0

11988610158401198951198962minus11989520 (2minus119895119905 minus 119896)

+119869

sum119895=1


sum119896=0


(5)

where 1198861015840119895119896 is threshold approximation coefficient and 1198891015840119895119896 isthreshold detailed coefficient

(ii) Empirical mode decomposition based denoising anEMD is data-driven algorithm which has been recentlyproposed to decompose nonlinear and nonstationary datainto several oscillatory modes [18] Due to adaptive natureEMD directly decomposes data into number of IMFs bysatisfying two conditions as follows (a) From complete dataset the number of zero crossings and the number of extremesmust be equal or differ at most by one (b) the mean valueof the envelope which is smoothed through cubic splineinterpolation based on the local maxima and minima shouldbe zero at all points

The EMD structure is defined as follows

(1) Identify all local maxima and minima from timeseries 119909(119905) (119905 = 1 2 119873) and make upper envelopeof maxima 119890max(119905) and lower envelope minima 119890min(119905)through cubic spline interpolation

(2) Find the mean of upper and lower envelope 119898(119905) =(119890max(119905) + 119890min(119905))2 Find the difference between orig-inal series and extracted mean as

ℎ (119905) = 119909 (119905) minus 119898 (119905) (6)

(3) Check the properties defined in (a) and (b) of ℎ(119905) ifboth conditions are satisfied then mark this ℎ(119905) as 119894119905ℎIMF then the next step will be to replace the originalseries by 119903(119905) = 119909(119905) minus ℎ(119905) if ℎ(119905) is not IMF justreplace 119909(119905) with ℎ(119905)

(4) Repeat the process of (1-3) until the residue 119903(119905)becomes amonotone function fromwhich no furtherIMFs can be extracted

Finally original series can be written as the sum of allextracted IMFs and residue as

119909 (119905) =119898

sum119894

ℎ119894 (119905) + 119903 (119905) (7)

where 119898 is the number of IMFs as (119894 = 1 2 119898) andℎ119894(119905) is the 119894119905ℎ IMF and 119903(119905) is the trend of the signal Theway of denoised IMF is the same as mentioned in (3)-(5)except the last two IMFs which are used completely withoutdenoising due to low frequencies The subscript in EMD-based threshold case in (3)-(5) is replaced with 119894119905ℎ accordingto number of IMFs The denoised signal is reconstructed asfollows

119909 (119905) =119898minus2

sum119894=1

ℎ119894 (119905) +119898

sum119894=119898minus2

ℎ119894 (119905) + 119903 (119905) (8)

22 Decompose-Step Decomposition Step The EEMD meth-od the EEMD is a technique to stabilize the problem of modemixing which arises in EMD and decomposes the nonlinearsignals into number which contains the information of localtime varying characteristics The procedure of EEMD is asfollows

(a) Add a white Gaussian noise series to the original dataset

Complexity 5

(b) Decompose the signals with added white noise intoIMFs using conventional EMDmethod

(c) Repeat steps (a) and (b)119898119905ℎ time by adding differentwhite noises (119898 = 1 2 119897) in original signals

(d) Obtain the ensemble means of all IMFs119898119905ℎ ensembletime as the final results as 119868119872119865119896 = sum119897119898=1 119868119872119865119898119896 119897where 119896 = 1 2 119870 is 119896119905ℎ IMF

The CEEMDAN based decomposition although theEEMD can reduce the mode mixing problem to some extentdue to the successive addition of white Gauss noise in EEMDthe error cannot be completely eliminated from IMFs Toovercome this situation CEEMDAN function is introducedby Torres et al [28] We employed the CEEMDAN to decom-pose the hydrological time series data The CEEMDAN isbriefly described as follows

(1) In CEEMDAN extracted modes are defined as 119868119872119865119896in order to get complete decomposition we needto calculate the first residual by using the first1198681198721198651 which is calculated by EEMD as 1198681198721198651 =sum119897119898=1 1198681198721198651198981 119897

(2) Then replace 119909(119905) by 1199031(119905) where 1199031(119905) = 119909(119905) minus1198681198721198651 and add white Gaussian noises ie 119908119898(119905)119898119905ℎ time in 1199031(119905) and find the IMF by taking theaverage of first IMF to get the 1198681198721198652 Calculate 1199032(119905) =1199031(119905) minus 1198681198721198652 and repeat (2) until stoppage criteriaare meet However selection of number of ensemblemembers and amplitude of white noise is still anopen challenge but here in this paper the numberof ensemble members is fixed as 100 and standarddeviation of white noise is settled as 02

(3) The resulting 119896119905ℎ decomposedmodes iesum119870119896=1 119868119872119865119896and one residual 119877(119905) are used for further predictionof hydrological time series

More details of EMD EEMD and CEEMDAN are given in[20 28]

23 P-Step

Prediction of All IMFrsquos In prediction stage denoised IMFsare further used to predict the hydrological time series dataas inputs by using simple stochastic and complex machinelearning time series algorithms The reason of using two typesof model is that as first few IMFs contain high frequencieswhich are accurately predicted through complex ML modelswhenever last IMFs contain low frequencies which areaccurately predictable through simple stochastic modelsTheselected models are briefly described as follows

The IMF prediction with ARIMA model to predict theIMFs autoregressive moving average model is used as fol-lows

119868119872119865119905119894 = 1205721119868119872119865119905minus1119894 + sdot sdot sdot + 120572119901119868119872119865119905minus119901119894 + 120576119905119894 + 1205731120576119905minus1119894

+ sdot sdot sdot + 120573119902120576119905minus119902119894(9)

Table 1 Transfer functions of GMDH-NN algorithms

Transfer Functions

Sigmoid function 119911 = 1(1 + 119890minus119910)

Tangent function 119911 = tan (119910)Polynomial function 119911 = 119910Radial basis function 119911 = 119890minus1199102

Here 119868119872119865119905119894 is the 119894119905ℎ IMF and 120576119905119894 is the 119894119905ℎ residual ofCEEMDAN where 119901 is autoregressive lag and 119902 is movingaverage lag value Often the case time series is not stationary[7]made a proposal that differencing to an appropriate degreecan make the time series stationary if this is the case then themodel is said to be ARIMA (119901 119889 119902) where d is the differencevalue which is used to make the series stationary

The IMF prediction with group method of data handlingtype neural network ANN has been proved to be a pow-erful tool to model complex nonlinear system One of thesubmodels of NN which is constructed to improve explicitpolynomial model by self-organizing is Group Method ofData Handling-type Neural Network (GMDH-NN) [32]TheGMDH-NN has a successful application in a diverse rangeof area however in hydrological modeling it is still scarceThe algorithm of GMDH-NN worked by considering thepairwise relationship between all selected lagged variablesEach selected combination of pairs entered in a neuronand output is constructed for each neuron The structureof GMDH-NN is illustrated in Figure 2 with four variablestwo hidden and one output layer According to evaluationcriteria some neurons are selected as shown in Figure 2four neurons are selected and the output of these neuronsbecomes the input for next layer A prediction mean squarecriterion is used for neuron output selection The process iscontinued till the last layer In the final layer only single bestpredicted neuron is selected However the GMDH-NN onlyconsiders the two variable relations by ignoring the individualeffect of each variable The Architecture Group Method ofData Handling type Neural Network (RGMDH-NN) animproved form of GMDH-NN is used which simulates notonly the two-variable relation but also their individualsThe model for RGMDH-NN is described in the followingequation

119910119905 = 119886 +119903

sum119894=1

119887119894119909119894 +119903

sum119894=1

119903

sum119895=1

119888119894119895119909119894119909119895 (10)

The rest of the procedure of RGMDH-NN is the sameas GMDH-NN The selected neuron with minimum MeanSquare Error is transferred in the next layer by using thetransfer functions listed in Table 1 The coefficients of eachneuron are estimated with regularized least square estimationmethod as this method of estimation has the ability tosolve the multicollinearity problem which is usually theinherited part of time series data with multiple laggedvariables

Radial basis function neural network to predict thedenoised IMFs nonlinear neural network ie Radial Basis

6 Complexity

Input Layer Layer 1 Layer 2 Layer 3

Selected

Unselected

８1

８2

８3

８4

Figure 2 Architecture of GMDH-type neural network (NN) algorithms

Input X Output Y

x(1)

x(2)

x(n)

K(X )

K(X )

K(X )

Bias b

Figure 3 Topological structure of radial basis function

Function (RBFNN) is also adopted The reason for selectingRBFNN is that it has a nonlinear structure to find relationbetween lagged variables The RBFNN is a three-layer feed-forward neural network which consists of an input layer ahidden layer and an output layer The whole algorithm isillustrated in Figure 3 Unlike GMDH-NN RBFNN takesall inputs in each neuron with corresponding weights andthen hidden layer transfers the output by using radial basisfunction with weights to output The sigmoid basis functionis used to transfer the complex relation between laggedvariables as follows

0119894 (119909) = 11 + exp (119887119879119909 minus 1198870) (11)

3 Case Study and Experimental Design

Selection of Study Area In this study the Indus Basin System(IBS) known to be the largest river system in Pakistan isconsidered which plays a vital role in the power generationand irrigation system The major tributaries of this river areRiver Jhelum River Chenab and River Kabul These riversget their inflows mostly from rainfall snow and glacier meltAs in Pakistan glaciers covered 13680 km2 area in whichestimated 13 of the areas are covered by Upper Indus Basin(UIB) [33] About 50 melted water from these 13 areasadds the significant contribution of water in these majorriversThe Indus river and its tributaries cause flooding due toglacier and snow melting and rainfall [34] The major events

Complexity 7

AFGHANISTAN

INDIA

Arabian Sea

Sindh

Punjab

Kabul R

Figure 4 Rivers and irrigation network of Pakistan

of flood usually occur in summer due to heavy monsoonrainfall which starts from July and end in September Itwas reported [35] that due to excessive monsoon rainfallin 2010 floods have been generated in IBS which affected14 million people and around 20000000 inhabitants weredisplaced Moreover surface water system of Pakistan is alsobased on flows of IBS and its tributaries [36] Pappas [37]mentioned that around 65 of agriculture land is irrigatedwith the Indus water system Therefore for effective waterresources management and improving sustainable economicand social development and for proactive and integratedflood management there is a need to appropriately analyzeand predict the rivers inflow data of IBS and its tributaries

Data To thoroughly investigate the proposed models fourriversrsquo inflow data is used in this study which is comprisedof daily rivers inflow (1st-January to 19th-June) for the periodof 2015-2018 We consider the main river inflow of Indus atTarbela with its two principal one left and one right banktributaries [38] Jhelum inflow at Mangla Chenab at Maralaand Kabul at Nowshera respectively (see Figure 4) Data ismeasured in 1000 cusecs The rivers inflow data was acquiredfrom the site of Pakistan Water and Power DevelopmentAuthority (WAPDA)

Comparison of Proposed Study with Other Methods Theproposed models are compared with other predictionapproaches by considering with and without principals ofdenoising and decomposition For that purpose the follow-ing types of models are selected

(I) Without denoising and decomposing only singlestatistical model is selected ie ARIMA (for conve-nience we call one-stage model 1-S) as used in [8]

(II) Only denoised based models in this stage the noiseremoval capabilities of WA and EMD are assessedThe wavelet based models are WA-ARIMA WA-RBFNN and WA-RGMDH whereas the empiri-cal mode decomposition based models are EMD-ARIMA EMD-RBFNN and EMD-RGMDH The

different prediction models are chosen for the com-parison of traditional statistical models with artificialintelligence based models as RBFN and RGMDH (forconvenience we call two-stage model 2-S) The 2-Sselected models for comparison are used from [15 17]for the comparison with the proposed model

(III) With denoising and decomposition (existing meth-od) for that purpose three-stage EMD-EEMD-MMmodel is used from [2] for the comparison withproposed models Under this the multiple modelsare selected by keeping the prediction characteristicssimilar to proposed model for comparison purpose(for convenience we call three-stage model 3-S)

Evaluation Criteria The prediction accuracy of models isassessed using three evaluation measures such as MeanRelative Error (MRE) Mean Absolute Error (MAE) andMean Square Error (MSE)The following are their equationsrespectively

119872119877119864 = 1119899sum119899119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816

119891 (119905)(12)

119872119860119864 = 1119899119899

sum119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816 (13)

and

119872119878119864 = 1119899119899

sum119905=1

(119891 (119905) minus 119891 (119905))2 (14)

All proposed and selected models are evaluated using thesecriteria Moreover inGMDH-NNand RGMDH-NNmodelsneurons are selected according to MSE

4 Results

D-stage results the results of twonoise removal filters ieWAand EMD are described below

8 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w

Original InflowDenoised by EMDDenoised by Wavelet

2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow


Figure 5 The denoised series of the two hydrological time series of Indus and Jhelum rivers inflow The figure shows the denoised resultsobtained through the EMD-based threshold method (in red color) and the wavelet analysis-based threshold method (in blue color)

Wavelet based denoised after calculating the approxima-tions from (1) and details from (2) the hard and soft ruleof thresholding are used to remove noises from hydrologicaltime series coefficients Both hard and soft rules are calculatedfrom (3) and (4) respectively On behalf of lower MSE hardthreshold based denoised series are reconstructed through (5)for WA

EMD-based threshold to remove noises through EMDintrinsic mode functions are calculated from (7) and thenhard and soft thresholds are used to denoise the calculatedIMFs except the last two IMFs as due to smooth and lowfrequency characteristics there is no need to denoise the lasttwo IMFs Hard threshold based denoised IMFs are furtherused to reconstruct the noise free hydrological time seriesdata from (8)

The WA and EMD based denoised Indus and Jhelumrivers inflow are shown in Figure 5 The statistical measuresincluding mean (119909) standard deviation (120590) MRE MAEand MSE of original and denoised series for all case studiesof both noise removal methods are presented in Table 2The results show that the statistical measures are almost thesame for both denoising methods except MSE as for Indusand Jhelum inflow WA-based denoised series have lowerMSE than EMD however for Kabul and Chenab inflowEMD-based denoised series have lower MSE than WA-based

denoised series Overall it was concluded that both methodshave equal performance in denoising the hydrological timeseries data In decomposing stage both of WA and EMDbased denoised series are separately used as input to derivethe time varying characteristics in terms of high and lowfrequencies

Decompose-stage results to extract the local time varyingfeatures from denoised hydrological data the WAEMD-based denoised hydrological time series data are furtherdecomposed into nine IMFs and one residual The CEEM-DAN decomposition method is used to extract the IMFsfrom all four rivers EMD-based denoised-CEEMDAN-based decomposed results of Indus and Jhelum rivers infloware shown in Figure 6 whenever WA-CEEMDAN-basednoise free decomposed results of Indus and Jhelum riversinflow are shown in Figure 7 All four rivers are decom-posed into nine IMFs and one residual showing similarcharacteristics for both methods The extracted IMFs showthe characteristics of hydrological time series data wherethe starting IMFs represent the higher frequency whereaslast half IMFs show the low frequencies and residual areshown as trends as shown in Figures 6 and 7 The ampli-tude of white noise is set as 02 as in [2] and numbersof ensemble members are selected as maximum which is1000

Complexity 9

Table 2 Statistical measures of WA- and EMD-based denoised rivers inflow of four hydrological time series data sets

River Inflow Mode 119909 120590 MRE MAE MSEIndus Inflow Original series 802194 875044

EMD 805931 873925 39118 01275 367636Wavelet 802267 861632 38188 00987 229626

Jhelum Inflow Original series 302001 236743EMD 301412 231641 27118 01666 164864Wavelet 302023 227799 25579 01418 108837

Kabul Inflow Original series 291746 252352EMD 2523524 251181 25474 02036 125216Wavelet 2918118 2429148 27386 01615 122447

Chenab Inflow Original series 319557 294916EMD 320024 292734 2271784 01470 106797Wavelet 319585 282591 31958 017228 178353

minus20

020

minus40

24

6minus1

00

1020

minus30

minus10

10minus2

00

20

0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200

minus20

020

minus20

020

minus60

040

minus100

050

minus100

010

0

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

15

minus50

510

minus50

510

minus10

05

minus15

minus55

1030

50

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time Time Time Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Figure 6 The EMD-CEEMDAN decomposition of Indus (left) and Jhelum rivers inflow (right) The two series are decomposed into nineIMFs and one residue

P-step results for all extracted IMFs and residual threemethods of predictions are adopted to get the more preciseand near-to-reality results For that reason one traditionalstatistical method ie ARIMA (p d q) with two othernonlinear ML methods ie GMDH-NN and RBFNN areused to predict the IMFs and residuals of all four river inflowsThe rivers inflow data of all four rivers are split 70 fortraining set and 30 for testing set The parameters andstructure of models are estimated using 886 observations ofrivers inflow The validity of proposed and selected modelsis tested using 30 data of rivers inflow After successfulestimation of multimodels on each IMF and residual the bestmethod with minimum MRE MAE and MSE is selectedfor each IMF prediction The testing results of proposed

models with comparison to all othermodels for all four riversrsquoinflow ie Indus inflow Jhelum inflow Chenab inflow andKabul inflow are presented in Table 3 The proposed EMD-CEEMDAN-MM and WA-CEEMDAN-MM model predic-tion results fully demonstrate the effectiveness for all 4 caseswith minimum MRE MAE and MSE compared to all 1-S[8] 2-S [15 17] and 3-S [2] evaluation models Howeveroverall the proposed WA-CEEMDAN-MM model attainsthe lowest MSE as compared to other EMD-CEEMDAN-MM proposed models The worst predicted model is 1-S ieARIMA without denoising and without decomposing thehydrological time series datawith highestMSEThepredictedgraphs of proposed model ie EMD-CEEMDAN-MM withcomparison to 2-Smodels ie with EMD based denoised for

10 Complexity

minus20

020

minus40

24

6minus1

00

1020

minus20

020

minus20

020

0 200 400 600 800 1200

minus20

010

minus20

020

minus60

minus20

2060

minus100

050

minus100

010

0

0 200 400 600 800 1200

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

0 200 400 600 800 1200

minus50

5minus5

05

minus10

05

10minus1

5minus5

510

3050

0 200 400 600 800 1200

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time

Figure 7TheWA-CEEMDANdecomposition of Indus (left) and Jhelum rivers inflow (right)The two series are decomposed into nine IMFsand one residue

Indus and Jhelum river inflow are shown in Figure 8 andWA-CEEMDAN-MM with comparison to 2-S models ie withWA based denoised are shown in Figure 9

To improve the prediction accuracy of complex hydro-logical time series data from simple time series models onecan take the advantage from three principals of ldquodenoisingrdquoldquodecompositionrdquo and ldquoensembling the predicted resultsrdquoThe2-S model with simple ARIMA and GMDH can performwell as compared to 2-S models with complex models and1-S models by optimal decomposition methods Moreoverwith addition to extracting time varying frequencies fromdenoised series one can get the more precise results over 2-Smodels However from Table 3 it can be concluded that theproposedWA-CEEMDAN-MMand EMD-CEEMDAN-MMmodels perform more efficiently to predict the hydrologicaltime series data by decreasing the complexity of hydrologicaltime series data and enhancing the prediction performanceover 1-S 2-S and 3-S existing models

The following conclusions are drawn based on the testingerror presented in Table 3

Overall comparison the overall performances of proposedmodelsWA-CEEMDAN-MM andWA-CEEMDAN-MM arebetter than all other evaluation models selected from thestudy [2 8 15 17] with the lowest MAE MRE and MSEvalues for all case studies However among two proposedmodels WA-CEEMDAN-MM performs well by attaining onaverage 849 2419 and 543 lowest MAE MRE andMSE values respectively for all four riversrsquo inflow predictionas compared to EMD-CEEMDAN-MM as listed in Table 3It is shown that both proposed models perform well with

comparison to 1-S 2-S and existing 3-S Moreover it is alsonoticed that most of IMFs are precisely predicted with simpletraditional statistical ARIMAmodel except the first two IMFsas the first IMFs presented high frequencies showing morevolatile time varying characteristics with the rest of IMFsHowever overall WA-CEEMDAN-MM is more accurate inpredicting the rivers inflow

Comparison of proposed models with other only denoisedseries models removing the noise through WA- and EMD-based threshold filters before statistical analysis improved theprediction accuracy of complex hydrological time series Itcan be observed from Table 3 that the MAE MRE and MSEvalues of four cases perform well for 2-S model as comparedto 1-S model in both WA and EMD based denoised inputsHowever like overall performance of WA-CEEMDAN-MMWA-based denoisedmodels performwell compared to EMD-based denoised Moreover with denoised series the severalstatistical (simple) and machine learning (complex) methodsare adopted to further explore the performances betweensimple and complex methods to predict inflows This canbe seen from Table 3 where WA-RBFN and EMD-RBFNperform the worst compared to WA-ARIMA WA-RGMDHEMD-ARIMA and EMD-RGMDH This means that withdenoising the hydrological series one can move towardssimple models as compared to complex models like radialbasis function neural network WA-RGMDH and EMD-RGMDH attain the highest accuracy among all 2-S models

Comparison of proposed models with other denoised anddecomposed models in addition to denoising the decompo-sition of hydrological time series strategy effectively enhances

Complexity 11

Table 3 Evaluation index of testing prediction error of proposed models (EMD-CEEMDAN-MM and WA-CEEMDAN-MM) with allselected models for all four case studies

River Inflow Model Name Models MRE MAE MSEIndus Inflow 1-S ARIMA 42347 00685 647141

2-S WA-ARMA 32862 00430 534782WA-RGMDH 32548 00393 467382WA-RBFN 201949 02598 2301772

EMD-ARMA 49898 00960 761440EMD-RGMDH 49653 00915 760884EMD-RBFN 343741 07762 3931601

3-S EMD-EEMD-MM 52710 01721 440115WA-CEEMDAN-MM 15410 00349 55734EMD-CEEMDAN-MM 18009 00462 66983

Jhelum Inflow 1-S ARMA 35224 01201 4755292-S WA-ARMA 26129 00748 371441

WA-RGMDH 26208 00773 377954WA-RBFN 98608 07714 1807443



Kabul Inflow 1-S ARMA 24910 00883 2501362-S WA-ARMA 19999 00592 206874

WA-RGMDH 20794 00729 210612WA-RBFN 16565 00997 133554



Chenab Inflow 1-S ARMA 54157 04646 1081852-S WA-ARMA 39652 01087 842359

WA-RGMDH 36147 00943 816493WA-RBFN 41424 02757 476184

EMD-ARMA 47971 02721 1007013EMD-RGMDHA 44812 01865 956680EMD-RBFN 108228 21666 2845627


the prediction accuracy by reducing the complexity of hydro-logical data in multiple dimensions It is shown from Table 3that the 3-S (existing) performs better as on average for allfour rivers MAE MRE and MSE values are 1376 -655and 5479 respectively lower than 1-S model and 63406476 and 9678 lower than 2-S model (EMD-RBFNN)Further research work can be done to explore the ways toreduce the mathematical complexity of separate denoisingand decomposition like only single filter which not only

denoises but also decomposes the hydrological time seriesdata with the same filter to effectively predict or simulate thedata

5 Conclusion

The accurate prediction of hydrological time series data isessential for water supply and water resources purposesConsidering the instability and complexity of hydrological

12 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w

Original InflowPredicted by EMD-CEEMDAN-MMPredicted by EMD-ARIMApredicted by EMD-RBFNNpredicted by EMD-RGMDH

2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

owOriginal InflowPredicted by EMD-CEEMDAN-MMPredicted by EMD-ARIMApredicted by EMD-RBFNNpredicted by EMD-RGMDH

Original InflowPredicted by EMD-CEEMDAN-MM

predicted by EMD-RBFNNpredicted by EMD-RGMDH

Figure 8 Prediction results of Indus and Jhelum rivers inflow using proposed EMD-CEEMDAN-MMwith comparison to other EMD baseddenoised and predicted models

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w

Original InflowPredicted by WA-CEEMDAN-MMPredicted by WA-ARIMApredicted by WA-RBFNNpredicted by WA-RGMDH

2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow


Figure 9 Prediction results of Indus and Jhelum river inflow using proposed WA-CEEMDAN-MMwith comparison toWA based denoisedpredicted models

time series some data preprocessing methods are adoptedwith the aim to enhance the prediction of such stochasticdata by decomposing the complexity of hydrological timeseries data in an effective way This research proposed twonewmethodswith three stages as ldquodenoisedrdquo decomposition

and prediction and summation named as WA-CEEMDAN-MM and EMD-CEEMDAN-MM for efficiently predictingthe hydrological time series For the verification of proposedmethods four cases of rivers inflow data from Indus BasinSystem are utilizedTheoverall results show that the proposed

Complexity 13

hybrid prediction model improves the prediction perfor-mance significantly and outperforms some other popularprediction methods Our two proposed three-stage hybridmodels show improvement in prediction accuracy with min-imumMREMAE andMSE for all four rivers as compared toother existing one-stage [8] and two-stage [15 17] and three-stage [2] models In summary the accuracy of prediction isimproved by reducing the complexity of hydrological timeseries data by incorporating the denoising and decompo-sition In addition these new prediction models are alsocapable of solving other nonlinear prediction problems

Data Availability



The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through research group no RG-1437-027

References

[1] D P Solomatine and A Ostfeld ldquoData-driven modelling somepast experiences and new approachesrdquo Journal of Hydroinfor-matics vol 10 no 1 pp 3ndash22 2008

[2] C Di X Yang and X Wang ldquoA four-stage hybrid model forhydrological time series forecastingrdquo PLoS ONE vol 9 no 8Article ID e104663 2014

[3] Z Islam Literature Review on Physically Based HydrologicalModeling [Ph D thesis] pp 1ndash45 2011

[4] A R Ghumman Y M Ghazaw A R Sohail and K WatanabeldquoRunoff forecasting by artificial neural network and conven-tional modelrdquoAlexandria Engineering Journal vol 50 no 4 pp345ndash350 2011

[5] S Riad J Mania L Bouchaou and Y Najjar ldquoRainfall-runoffmodel usingan artificial neural network approachrdquoMathemati-cal and Computer Modelling vol 40 no 7-8 pp 839ndash846 2004

[6] T Peng J Zhou C Zhang andW Fu ldquoStreamflow ForecastingUsing Empirical Wavelet Transform and Artificial Neural Net-worksrdquoWater vol 9 no 6 p 406 2017

[7] G E P Box and G M Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1970

[8] B N S Ghimire ldquoApplication of ARIMA Model for RiverDischarges Analysisrdquo Journal of Nepal Physical Society vol 4no 1 pp 27ndash32 2017

[9] O Kisi ldquoStreamflow forecasting using different artificial neuralnetwork algorithmsrdquo Journal of Hydrologic Engineering vol 12no 5 pp 532ndash539 2007

[10] C A G Santos and G B L D Silva ldquoDaily streamflowforecasting using a wavelet transform and artificial neuralnetwork hybrid modelsrdquo Hydrological Sciences Journal vol 59no 2 pp 312ndash324 2014

[11] CWu K Chau andY Li ldquoMethods to improve neural networkperformance in daily flows predictionrdquo Journal of Hydrologyvol 372 no 1-4 pp 80ndash93 2009

[12] T Partal ldquoWavelet regression and wavelet neural networkmodels for forecasting monthly streamflowrdquo Journal of Waterand Climate Change vol 8 no 1 pp 48ndash61 2017

[13] Z M Yaseen M Fu C Wang W H Mohtar R C Deoand A El-shafie ldquoApplication of the Hybrid Artificial NeuralNetwork Coupled with Rolling Mechanism and Grey ModelAlgorithms for Streamflow Forecasting Over Multiple TimeHorizonsrdquo Water Resources Management vol 32 no 5 pp1883ndash1899 2018

[14] M Rezaie-Balf and O Kisi ldquoNew formulation for forecastingstreamflow evolutionary polynomial regression vs extremelearning machinerdquo Hydrology Research vol 49 no 3 pp 939ndash953 2018

[15] H Liu C Chen H-Q Tian and Y-F Li ldquoA hybrid model forwind speed prediction using empirical mode decompositionand artificial neural networksrdquo Journal of Renewable Energy vol48 pp 545ndash556 2012

[16] Z Qu K Zhang J Wang W Zhang and W Leng ldquoA Hybridmodel based on ensemble empirical mode decomposition andfruit fly optimization algorithm for wind speed forecastingrdquoAdvances in Meteorology 2016

[17] Y Sang ldquoA Practical Guide to DiscreteWavelet Decompositionof Hydrologic Time Seriesrdquo Water Resources Management vol26 no 11 pp 3345ndash3365 2012

[18] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London A Mathematical Physical and EngineeringSciences vol 454 Article ID 1971 pp 903ndash995 1998

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society of London A MathematicalPhysical and Engineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] Z Wang J Qiu and F Li ldquoHybrid Models Combining EMDEEMD and ARIMA for Long-Term Streamflow ForecastingrdquoWater vol 10 no 7 p 853 2018

[21] N A Agana and A Homaifar ldquoEMD-based predictive deepbelief network for time series prediction an application todrought forecastingrdquo Hydrology vol 5 no 1 p 18 2018

[22] A Kang Q Tan X Yuan X Lei and Y Yuan ldquoShort-termwind speed prediction using EEMD-LSSVM modelrdquo Advancesin Meteorology 2017

[23] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis (AADA) vol 1 no 1 pp 1ndash41 2009

[24] W-C Wang K-W Chau D-M Xu and X-Y Chen ldquoImprov-ing Forecasting Accuracy of Annual Runoff Time Series UsingARIMA Based on EEMD Decompositionrdquo Water ResourcesManagement vol 29 no 8 pp 2655ndash2675 2015

[25] H Su H Li Z Chen and Z Wen ldquoAn approach usingensemble empirical mode decomposition to remove noise fromprototypical observations on dam safetyrdquo SpringerPlus vol 5no 1 2016

[26] X-S Jiang L Zhang and M X Chen ldquoShort-term forecastingof high-speed rail demand A hybrid approach combiningensemble empirical mode decomposition and gray support

14 Complexity

vector machine with real-world applications in Chinardquo Trans-portation Research Part C Emerging Technologies vol 44 pp110ndash127 2014

[27] S Dai D Niu and Y Li ldquoDaily Peak Load Forecasting Basedon Complete Ensemble Empirical Mode Decomposition withAdaptive Noise and Support Vector Machine Optimized byModified Grey Wolf Optimization Algorithmrdquo Energies vol 11no 1 p 163 2018

[28] M E TorresMA Colominas G Schlotthauer andP FlandrinldquoA complete ensemble empirical mode decomposition withadaptive noiserdquo in Proceedings of the 36th IEEE InternationalConference onAcoustics Speech and Signal Processing pp 4144ndash4147 Prague Czech Republic May 2011

[29] A Jayawardena and A Gurung ldquoNoise reduction and pre-diction of hydrometeorological time series dynamical systemsapproach vs stochastic approachrdquo Journal of Hydrology vol228 no 3-4 pp 242ndash264 2000

[30] M Yang Y Sang C Liu and Z Wang ldquoDiscussion on theChoice of Decomposition Level forWavelet BasedHydrologicalTime Series ModelingrdquoWater vol 8 no 5 p 197 2016

[31] J Kim C Chun and B H Cho ldquoComparative analysis ofthe DWT-based denoising technique selection in noise-ridingDCV of the Li-Ion battery packrdquo in Proceedings of the 2015 9thInternational Conference on Power Electronics and ECCE Asia(ICPE 2015-ECCE Asia) pp 2893ndash2897 Seoul South KoreaJune 2015

[32] H Ahmadi MMottaghitalab and N Nariman-Zadeh ldquoGroupMethod of Data Handling-Type Neural Network Prediction ofBroiler Performance Based on Dietary Metabolizable EnergyMethionine and Lysinerdquo The Journal of Applied PoultryResearch vol 16 no 4 pp 494ndash501 2007

[33] A S Shakir and S Ehsan ldquoClimate Change Impact on RiverFlows in Chitral Watershedrdquo Pakistan Journal of Engineeringand Applied Sciences 2016

[34] B KhanM J Iqbal andM A Yosufzai ldquoFlood risk assessmentof River Indus of Pakistanrdquo Arabian Journal of Geosciences vol4 no 1-2 pp 115ndash122 2011

[35] K Gaurav R Sinha and P K Panda ldquoThe Indus flood of 2010in Pakistan a perspective analysis using remote sensing datardquoNatural Hazards vol 59 no 3 pp 1815ndash1826 2011

[36] A Sarwar and A S Qureshi ldquoWater management in the indusbasin in Pakistan challenges and opportunitiesrdquo MountainResearch and Development vol 31 no 3 pp 252ndash260 2011

[37] G Pappas ldquoPakistan and water new pressures on globalsecurity and human healthrdquo American Journal of Public Healthvol 101 no 5 pp 786ndash788 2011

[38] J LWescoatA Siddiqi andAMuhammad ldquoSocio-Hydrologyof Channel Flows in Complex River Basins Rivers Canals andDistributaries in Punjab Pakistanrdquo Water Resources Researchvol 54 no 1 pp 464ndash479 2018

Research ArticleEnd-Point Static Control of Basic OxygenFurnace (BOF) Steelmaking Based on WaveletTransform Weighted Twin Support Vector Regression

Chuang Gao12 Minggang Shen 1 Xiaoping Liu3 LidongWang2 andMaoxiang Chu2

1School of Materials and Metallurgy University of Science and Technology Liaoning Anshan Liaoning China2School of Electronic and Information Engineering University of Science and Technology Liaoning Anshan Liaoning China3School of Information and Electrical Engineering Shandong Jianzhu University Jinan Shandong China

Correspondence should be addressed to Minggang Shen lnassmg163com

Received 6 July 2018 Revised 22 October 2018 Accepted 10 December 2018 Published 1 January 2019

Academic Editor Michele Scarpiniti

Copyright copy 2019 Chuang Gao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A static control model is proposed based on wavelet transform weighted twin support vector regression (WTWTSVR) Firstly newweighted matrix and coefficient vector are added into the objective functions of twin support vector regression (TSVR) to improvethe performance of the algorithm The performance test confirms the effectiveness of WTWTSVR Secondly the static controlmodel is established based on WTWTSVR and 220 samples in real plant which consists of prediction models control modelsregulating units controller and BOF Finally the results of proposed prediction models show that the prediction error bound with0005 in carbon content and 10∘C in temperature can achieve a hit rate of 92 and 96 respectively In addition the double hitrate of 90 is the best result by comparing with four existingmethodsThe results of the proposed static controlmodel indicate thatthe control error bound with 800 Nm3 in the oxygen blowing volume and 55 tons in the weight of auxiliary materials can achieve ahit rate of 90 and 88 respectively Therefore the proposed model can provide a significant reference for real BOF applicationsand also it can be extended to the prediction and control of other industry applications

1 Introduction

With the development of end-point control technology forbasic oxygen furnace (BOF) the static control model can beestablished to overcome the randomness and inconsistencyof the artificial experience control models According to theinitial conditions of hot metal the relative calculations canbe carried out to guide production The end-point hit ratewould be improved through this approach Unfortunatelythe control parameters would not be adjusted during thesmelting process which restricts the further improvement ofthe end-point hit rate To solve this problem the sublancebased dynamic control model could be adopted by usingthe sublance technology By combining the static model withthe dynamic model the end-point hit rate of BOF could beguaranteed The establishment of the static control model isthe foundation of the dynamic model The accuracy of thestatic model will directly affect the hit rates of the dynamic

control model thus it plays an important role in the parame-ter optimization of BOF control process Therefore the staticcontrol model is still a practical and reliable technologyto guide the production and improve the technology andmanagement level of steelmaking plants

In recent years some significant developments of BOFprediction and control modelling have been achieved Blancoet al [1] designed a mixed controller for carbon and siliconin a steel converter in 1993 In 2002 three back propaga-tion models are adopted to predict the end-blow oxygenvolume and the weight of coolant additions [2] In 2006 adynamic model is constructed to predict the carbon contentand temperature for the end-blow stage of BOF [3] Basedon multivariate data analysis the slopping prediction wasproposed by Bramming et al [4] In 2014 the multi-levelrecursive regression model was established for the predictionof end-point phosphorus content during BOF steelmakingprocess [5] An antijamming endpoint prediction model


2 Complexity

of extreme learning machine (ELM) was proposed withevolving membrane algorithm [6] By applying input vari-ables selection technique the input weighted support vectormachinemodellingwas proposed [7] and then the predictionmodel was established on the basis of improving a case-based reasoning method [8] The neural network predictionmodellings [9ndash12] were carried out to achieve aimed end-point conditions in liquid steel A fuzzy logic control schemewas given for the basic oxygen furnace in [13] Most ofthese achievements are based on the statistical and intelligentmethods

As an intelligent method Jayadeva et al [14] proposeda twin support vector machine (TSVM) algorithm in 2007The advantage of this method is that the computational com-plexity of modelling can be reduced by solving two quadraticprogramming problems instead of one in traditional methodIt is also widely applied to the classification applications In2010 Peng [15] proposed a twin support vector regression(TSVR) which can be used to establish the prediction modelfor industrial data After that some improved TSVRmethods[16ndash22] were proposed By introducing a K-nearest neighbor(KNN) weighted matrix into the optimization problem inTSVR the modified algorithms [16 19] were proposed toimprove the performance of TSVR A V-TSVR [17] andasymmetric V-TSVR [20] were proposed to enhance the gen-eralization ability by tuning new model parameters To solvethe ill-conditioned problem in the dual objective functionsof the traditional TSVR an implicit Lagrangian formulationfor TSVR [18] was proposed to ensure that the matrices inthe formulation are always positive semidefinite matricesParastalooi et al [21] added a new term into the objectivefunction to obtain structural information of the input dataBy comparing with the neural network technology thedisadvantage of neural network is that the optimizationprocess may fall into the local optimum The optimization ofTSVR is a pair of quadratic programming problems (QPPs)which means there must be a global optimal solution foreach QPP All above modified TSVR algorithms are focusedon the improvements of the algorithm accuracy and thecomputation speed Currently the TSVR algorithm has neverbeen adopted in the BOF applications Motivated by this theTSVR algorithm can be used to establish a BOF model

Wavelet transform can fully highlight the characteristicsof some aspects of the problem which has attracted moreand more attention and been applied to many engineeringfields In this paper the wavelet transform technique isused to denoise the output samples during the learningprocess and it is a new application of the combinationof wavelet transform and support vector machine methodThen a novel static control model is proposed based onwavelet transform weighted twin support vector regression(WTWTSVR) which is an extended model of our previouswork In [23] we proposed an end-point prediction modelwith WTWTSVR for the carbon content and temperature ofBOF and the accuracy of the prediction model is expectedHowever the prediction model cannot be used to guidereal BOF production directly Hence a static control modelshould be established based on the prediction model tocalculate the oxygen volume and the weight of auxiliary raw

materials and the accuracy of the calculations affects thequality of the steel Therefore the proposed control modelcan provide a guiding significance for real BOF productionIt is also helpful to other metallurgical prediction and controlapplications To improve the performance of the controlmodel an improvement of the traditional TSVR algorithmis carried out A new weighted matrix and a coefficientvector are added into the objective function of TSVR Alsothe parameter 120576 in TSVR is not adjustable anymore whichmeans it is a parameter to be optimized Finally the staticcontrol model is established based on the real datasetscollected from the plant The performance of the proposedmethod is verified by comparing with other four existingregression methods The contributions of this work includethe following (1) It is the first attempt to establish the staticcontrol model of BOF by using the proposed WTWTSVRalgorithm (2)Newweightedmatrix and coefficient vector aredetermined by the wavelet transform theory which gives anew idea for the optimization problem inTSVR areas (3)Theproposed algorithm is an extension of the TSVR algorithmwhich is more flexible and accurate to establish a predictionand control model (4)The proposed control model providesa new approach for the applications of BOF control Theapplication range of the proposed method could be extendedto other metallurgical industries such as the prediction andcontrol in the blast furnace process and continuous castingprocess

Remark 1 Themaindifference betweenprimalKNNWTSVR[16] and the proposed method is that the proposed algo-rithm utilizes the wavelet weighted matrix instead of KNNweighted matrix for the squared Euclidean distances fromthe estimated function to the training points Also a waveletweighted vector is introduced into the objective functions forthe slack vectors 1205761 and 1205762 are taken as the optimized param-eters in the proposed algorithm to enhance the generalizationability Another difference is that the optimization problemsof the proposed algorithm are solved in the Lagrangian dualspace and that of KNNWTSVR are solved in the primal spacevia unconstrained convex minimization

Remark 2 By comparing with available weighted techniquelike K-nearest neighbor the advantages of the wavelet trans-form weighting scheme are embodied in the following twoaspects

(1) The Adaptability of the Samples The proposed method issuitable for dealing with timespatial sequence samples (suchas the samples adopted in this paper) due to the character ofwavelet transform Thewavelet transform inherits and devel-ops the idea of short-time Fourier transform The weights ofsample points used in the proposed algorithm are determinedby calculating the difference between the sample valuesand the wavelet-regression values for Gaussian functionwhich can mitigate the noise especially the influence ofoutliers While KNN algorithm determines the weight of thesample points by calculating the number of adjacent points(determined by Euclidian distance) which is more suitablefor the samples of multi-points clustering type distribution

Complexity 3

(2)TheComputational Complexity KNNalgorithm requires alarge amount of computations because the distance betweeneach sample to all known samples must be computed toobtain its K nearest neighbors By comparing with KNNweighting scheme the wavelet transform weighting schemehas less computational complexity because it is dealingwith one dimensional output samples and the computationcomplexity is proportional to the number of samples l KNNscheme is dealing with the input samples the computationcomplexity of KNN scheme is proportional to 1198972 With theincreasing of dimensions and number of the samples it willhave a large amount of computations

Therefore the wavelet transform weighting scheme ismore competitive than KNN weighting scheme for the timesequence samples due to its low computational complexity

2 Background

21 Description of BOF Steelmaking BOF is used to producethe steel with wide range of carbon alloy and special alloysteels Normally the capacity of BOF is between 100 tonsand 400 tons When the molten iron is delivered to theconverter through the rail the desulphurization process isfirstly required In BOF the hot metal and scrap lime andother fluxes are poured into the converter The oxidationreaction is carried out with carbon silicon phosphorusmanganese and some iron by blowing in a certain volumeof oxygen The ultimate goal of steelmaking is to produce thesteel with specific chemical composition at suitable tappingtemperature The control of BOF is difficult because thewhole smelting process is only half an hour and there isno opportunity for sampling and analysis in the smeltingprocess The proportion of the iron and scrap is about 31 inBOF The crane loads the waste into the container and thenpours the molten iron into the converter The water cooledoxygen lance enters the converter the high purity oxygenis blown into BOF at 16000 cubic feet per minute and theoxygen is reacted with carbon and other elements to reducethe impurity in the molten metal and converts it into a cleanhigh quality liquid steel The molten steel is poured into theladle and sent to the metallurgical equipment of the ladle [13]

Through the oxidation reaction of oxygen blown in BOFthe molten pig iron and the scrap can be converted into steelIt is a widely used steelmaking method with its higher pro-ductivity and low production cost [3] However the physicaland chemical process of BOF is very complicated Also thereare various types of steel produced in the same BOF whichmeans that the grade of steel is changed frequently Thereforethe BOF modelling is a challenge task The main objective ofthe modelling is to obtain the prescribed end-point carboncontent and temperature

22 Nonlinear Twin Support Vector Regression Support vec-tor regression (SVR) was proposed for the applications ofregression problems For the nonlinear case it is based onthe structural riskminimization andVapnik 120576-insensitive lossfunction In order to improve the training speed of SVR Pengproposed a TSVR algorithm in 2010 The difference betweenTSVR and SVR is that SVR solves one large QPP problem and

TSVR solves two smallQPPproblems to improve the learningefficiency Assume that a sample is an 119899-dimensional vectorand the number of the samples is 119897 which can be expressedas (x1 1199101) (x119897 119910119897) Let A = [x1 x119897]119879 isin 119877119897times119899 be theinput data set of training samples y = [1199101 y119897]119879 isin 119877119897 bethe corresponding output and e = [1 1]119879 be the onesvector with appropriate dimensions Assume119870( ) denotes anonlinear kernel function Let119870(AA119879) be the kernel matrixwith order 119897 and its (119894 119895)-th element (119894 119895 = 1 2 119897) bedefined by

[119870 (AA119879)]119894119895= 119870 (x119894 x119895) = (Φ (x119894) sdot Φ (x119895)) isin 119877 (1)

Here the kernel function 119870(x x119894) represents the inner prod-uct of the nonlinear mapping functions Φ(x119894) and Φ(x119895) inthe high dimensional feature space Because there are variouskernel functions the performance comparisons of kernelfunctions will be discussed later In this paper the radial basiskernel function (RBF) is chosen as follows

119870 (x x119894) = exp(minus10038161003816100381610038161003816100381610038161003816x minus x11989410038161003816100381610038161003816100381610038161003816221205902 ) (2)

where 120590 is the width of the kernel function Let 119870(x119879A119879) =(119870(x x1) 119870(x x2) 119870(x x119897)) be a row vector in 119877119897 Thentwo 120576-insensitive bound functions 1198911(x) = 119870(x119879A119879)1205961 + 1198871and1198912(x) = 119870(x119879A119879)1205962+1198872 can be obtained where12059611205962 isin119877119897 are the normal vectors and 1198871 1198872 isin 119877 are the bias valuesTherefore the final regression function 119891(x) is determined bythe mean of 1198911(x) and 1198912(x) that is

119891 (x) = 12 (1198911 (x) + 1198912 (x))= 12119870 (x119879A119879) (1205961 + 1205962) + 12 (1198871 + 1198872)

(3)

Nonlinear TSVR can be obtained by solving two QPPs asfollows

min1205961 1198871120585

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus 1205761e minus (119870(AA119879)1205961 + 1198871e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198881e119879120585st y minus (119870 (AA119879)1205961 + 1198871e) ge 1205761e minus 120585 120585 ge 0119890 (4)

and

min1205962 1198872120574

12 1003816100381610038161003816100381610038161003816100381610038161003816y + 1205762e minus (119870 (AA119879)1205962 + 1198872e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198882e119879120574st 119870(AA119879)1205962 + 1198872e minus y ge 1205762e minus 120574 120574 ge 0e (5)

By introducing the Lagrangian function and Karush-Kuhn-Tucker conditions the dual formulations of (4) and (5)can be derived as follows

max120572

minus 12120572119879G (G119879G)minus1G119879120572 + g119879G (G119879G)minus1G119879120572minus g119879120572

st 0e le 120572 le 1198881e(6)

4 Complexity

and

max120573

minus 12120573119879G (G119879G)minus1G119879120573 minus h119879G (G119879G)minus1G119879120573+ h119879120573

st 0e le 120573 le 1198882e(7)

whereG = [119870(AA119879) e] g = y minus 1205761e and h = y + 1205762eTo solve the above QPPs (6) and (7) the vectors1205961 1198871 and1205962 1198872 can be obtained

[12059611198871 ] = (G119879G)minus1G119879 (g minus 120572) (8)

and

[12059621198872 ] = (G119879G)minus1G119879 (h + 120573) (9)

By substituting the above results into (3) the final regres-sion function can be obtained

23 Nonlinear Wavelet Transform Based Weighted TwinSupport Vector Regression

231 Model Description of Nonlinear WTWTSVR In 2017Xu el al [20] proposed the asymmetric V-TSVR algorithmbased on pinball loss functions This new algorithm canenhance the generalization ability by tuning new modelparametersTheQPPs of nonlinear asymmetric V-TSVR wereproposed as follows

min1205961 11988711205761120585

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus (119870 (AA119879)1205961 + 1198871e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198881V11205761+ 1119897 1198881e119879120585

st y minus (119870 (AA119879)1205961 + 1198871e)ge minus1205761e minus 2 (1 minus 119901) 120585 120585 ge 0e 1205761 ge 0

(10)

and

min1205962 11988721205762 120574

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus (119870 (AA119879)1205962 + 1198872e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198882V21205762+ 1119897 1198882e119879120585lowast

st 119870(AA119879)1205962 + 1198872e minus y ge minus1205762e minus 2119901120585lowast120585lowast ge 0e 1205762 ge 0

(11)

where 1198881 1198882 V1 V2 ge 0 are regulating parameters and theparameter 119901 isin (0 1) is used to apply a slightly differentpenalty for the outliers The contributions of this method arethat 1205761 and 1205762 are introduced into the objective functions ofQPPs and the parameters V1 and V2 are used to regulate thewidth of 1205761 and 1205762 tubes Also the parameter 119901 is designed togive an unbalance weight for the slack vectors 120585 and 120585lowast Basedon the concept of asymmetric V-TSVR a nonlinear waveletstransform based weighted TSVR is firstly proposed in thispaper By comparing with the traditional TSVR algorithmthe proposed algorithm introduces awavelet transform basedweighted matrix D1 and a coefficient vector D2 into theobjective function of TSVR Also the parameters 1205761 and 1205762are not adjustable by user anymore which means they areboth introduced into the objective functions Simultaneouslythe regularization is also considered to obtain the optimalsolutions Therefore the QPPs of nonlinear WTWTSVR areproposed as follows

min1205961 11988711205851205761

12 (y minus (119870(AA119879)1205961 + 1198871e))119879D1 (y minus (119870(AA119879)1205961 + 1198871e)) + 11988812 (12059611987911205961 + 11988721 ) + 1198882 (D1198792 120585 + V11205761)st y minus (119870 (AA119879)1205961 + 1198871e) ge minus1205761e minus 120585 120585 ge 0e 1205761 ge 0

(12)

and

min1205962 1198872120585

lowast 1205762

12 (y minus (119870 (AA119879)1205962 + 1198872e))119879D1 (y minus (119870 (AA119879)1205962 + 1198872e)) + 11988832 (12059611987921205962 + 11988721) + 1198884 (D1198792 120585lowast + V21205762)st 119870(AA119879)1205962 + 1198872e minus y ge minus1205762e minus 120585lowast 120585lowast ge 0e 1205762 ge 0

(13)

where 1198881 1198882 1198883 1198884 V1 V2 ge 0 are regulating parametersD1 is adiagonal matrix with the order of 119897 times 119897 and D2 is a coefficient

vector with the length of 119897 times 1 The determination of D1 andD2 will be discussed later

Complexity 5

The first term in the objective function of (12) or (13) isused to minimize the sums of squared Euclidean distancesfrom the estimated function 1198911(x) = 119870(x119879A119879)1205961 + 1198871 or1198912(x) = 119870(x119879A119879)1205962 + 1198872 to the training points The matrixD1 gives different weights for each Euclidean distance Thesecond term is the regularization term to avoid the over-fitting problem The third term minimizes the slack vector120585 or 120585lowast and the width of 1205761-tube or 1205762-tube The coefficientvector D2 is a penalty vector for the slack vector To solve theproblem in (12) the Lagrangian function can be introducedthat is

119871 (1205961 1198871 120585 1205761120572120573 120574)= 12 (y minus (119870 (AA119879)1205961 + 1198871e))119879sdotD1 (y minus (119870 (AA119879)1205961 + 1198871e))+ 11988812 (12059611987911205961 + 11988721 ) + 1198882 (D1198792 120585 + V11205761)minus 120572119879 (y minus (119870(AA119879)1205961 + 1198871e) + 1205761e + 120585) minus 120573119879120585minus 1205741205761

(14)

where 120572 120573 and 120574 are the positive Lagrangian multipliers Bydifferentiating 119871 with respect to the variables 1205961 1198871 120585 1205761 wehave

1205971198711205971205961 = minus119870(AA119879)119879D1 (y minus (119870(AA119879)1205961 + 1198871e))+ 119870(AA119879)119879 120572 + 11988811205961 = 0

(15)

1205971198711205971198871 = minuse119879D1 (y minus (119870(AA119879)1205961 + 1198871e)) + e119879120572

+ 11988811198871 = 0 (16)

120597119871120597120585 = 1198882D2 minus 120572 minus 120573 = 0 (17)

1205971198711205971205761 = 1198882V1 minus e119879120572 minus 120574 = 0 (18)

The KKT conditions are given by

y minus (119870 (AA119879)1205961 + 1198871e) ge minus1205761e minus 120585120585 ge 0e

120572119879 (y minus (119870 (AA119879)1205961 + 1198871e) + 1205761e + 120585) = 0120572 ge 0e

120573119879120585 = 0120573 ge 0e

1205741205761 = 0 120574 ge 0

(19)

Combining (15) and (16) we obtain

minus [[119870(AA119879)119879

e119879]]D1 (y minus (119870 (AA119879)1205961 + 1198871e))

+ [[119870(AA119879)119879

e119879]]120572 + 1198881 [

12059611198871 ] = 0(20)

Let H = [119870(AA119879)e] and u1 = [1205961198791 1198871]119879 and (20) can berewritten as

minusH119879D1y + (H119879D1H + 1198881I) u1 +H119879120572 = 0 (21)

where I is an identity matrix with appropriate dimensionFrom (21) we get

u1 = (H119879D1H + 1198881I)minus1H119879 (D1y minus 120572) (22)

From (15) (16) and (19) the following constraints can beobtained

e119879120572 le 1198882V1 0e le 120572 le 1198882D2 (23)

Substituting (22) into Lagrangian function 119871 we can getthe dual formulation of (12)

min120572

12120572119879H (H119879D1H + 1198881I)minus1H119879120572 + y119879120572

minus y119879D1H (H119879D1H + 1198881I)minus1H119879120572st 0e le 120572 le 1198882D2 e119879120572 le 1198882V1

(24)

Similarly the dual formulation of (13) can be derived asfollows

min120578

12120578119879H (H119879D1H + 1198883I)minus1H119879120578+ y119879D1H (H119879D1H + 1198883I)minus1H119879120578 minus y119879120578

119904119905 0e le 120578 le 1198884D2 e119879120578 le 1198884V2(25)

To solve the above QPPs the vectors 1205961 1198871 and 1205962 1198872 canbe expressed by

[12059611198871 ] = (H119879D1H + 1198881I)minus1H119879 (D1y minus 120572) (26)

and

[12059621198872 ] = (H119879D1H + 1198883I)minus1H119879 (D1y + 120578) (27)


232 Determination of Wavelets Transform Based WeightedMatrixD1 and aCoefficientVectorD2 Thewavelet transformcan be used to denoise the time series signal Based onthe work of Ingrid Daubechies the Daubechies wavelets

6 Complexity

are a family of orthogonal wavelets for a discrete wavelettransform There is a scaling function (called father wavelet)for each wavelet type which generates an orthogonal mul-tiresolution analysis Daubechies orthogonal wavelets 1198891198871 minus11988911988710 are commonly used

The wavelet transform process includes three stagesdecomposition signal processing and reconstruction Ineach step of decomposition the signal can be decomposedinto two sets of signals high frequency signal and lowfrequency signal Suppose that there is a time series signal SAfter the first step of decomposition it generates two signalsone is high frequency part Sh1 and another is low frequencypart Sl1 Then in the second step of decomposition the lowfrequency signal Sl1 can be decomposed further into Sh2and Sl2 After 119896 steps of decomposition 119896 + 1 groups ofdecomposed sequence (Sh1 Sh2 Sh119896 Sl119896) are obtainedwhere Sl119896 represents the contour of the original signal Sand Sh1 Sh2 Sh119896 represent the subtle fluctuations Thedecomposition process of signal S is shown in Figure 1 anddefined as follows

Sl119896 (119899) = 119897119896minus1sum119898=1

120601 (119898 minus 2119899) Sl119896minus1 (119898) (28)

Sh119896 (119899) = 119897119896minus1sum119898=1

120593 (119898 minus 2119899) Sl119896minus1 (119898) (29)

where Sl119896minus1 with the length 119897119896minus1 is the signal to be decom-posed Sl119896 and Sh119896 are the results in the 119896-th step 120601(119898 minus 2119899)and 120593(119898 minus 2119899) are called scaling sequence (low pass filter)and wavelets sequence respectively [24] In this paper 1198891198872wavelet with the length of 4 is adopted After wavelet trans-forming appropriate signal processing can be carried outIn the stage of reconstruction the processed high frequencysignal Shlowast119896 and low frequency signal Sllowast119896 are reconstructed togenerate the target signal Slowast Let P and Q be the matrix withthe order of 119897119896times119897119896minus1 P119899119898 = 120601(119898minus2119899) andQ119899119898 = 120593(119898minus2119899)The reconstruction process is shown in Figure 2 Therefore(28) and (29) can be rewritten as follows

Sl119896 (119899) = P sdot Sl119896minus1 (119898) (30)

Sh119896 (119899) = Q sdot Sl119896minus1 (119898) (31)

The signal Sllowast119896minus1 can be generated by reconstructing Sllowast119896and Shlowast119896

Sllowast119896minus1 (119898) = Pminus1 sdot Sllowast119896 (119899) +Qminus1 sdot Shlowast119896 (119899) (32)

If the output vector S = [1198781 1198782 119878119897]119879 of a predictionmodel is a time sequence then the wavelets transform canbe used to denoise the output of the training samples Afterthe decomposition signal processing and reconstructionprocess a denoising sequence Slowast = [119878lowast1 119878lowast2 119878lowast119897 ]119879 can beobtained The absolute difference vector 119903119894 = |119878119894 minus 119878lowast119894 | 119894 =1 2 119897 and r = [1199031 1199032 119903119897]119879 between 119878 and 119878lowast denote thedistance from the training samples to the denoising samplesIn the WTWTSVR a small 119903119894 reflects a large weight on thedistance between the estimated value and training value of the

S

Sh1

Sl1Sh2

Sl2

Slk

Shk

Figure 1 The process of decomposition

SlowastShlowastk-1

Sllowastk-1

Sllowast1

Sllowastk

Shlowastk

Shlowast

1

Figure 2 The process of reconstruction

119894-th sample which can be defined as the following Gaussianfunction

119889119894 = exp(minus 11990321198942120590lowast2) 119894 = 1 2 119897 (33)

where 119889119894 denotes the weight coefficient and 120590lowast is the widthof the Gaussian function Therefore the wavelets transformbased weighted matrix D1 and the coefficient vector D2can be determined by D1 = diag(1198891 1198892 119889119897) and D2 =[1198891 1198892 119889119897]119879233 Computational Complexity Analysis The computationcomplexity of the proposed algorithm is mainly determinedby the computations of a pair of QPPs and a pair of inversematrices If the number of the training samples is l then thetraining complexity of dual QPPs is about 119874(21198973) while thetraining complexity of the traditional SVR is about 119874(81198973)which implies that the training speed of SVR is about fourtimes the proposed algorithm Also a pair of inverse matriceswith the size (119897 + 1) times (119897 + 1) in QPPs have the samecomputational cost 119874(1198973) During the training process itis a good way to cache the pair of inverse matrices withsome memory cost in order to avoid repeated computationsIn addition the proposed algorithm contains the wavelettransform weighted matrix and db2 wavelet with length of4 is used in this paper Then the complexity of wavelettransform is less than 8l By comparingwith the computationsof QPPs and inverse matrix the complexity of computing thewavelet matrix can be ignored Therefore the computationcomplexity of the proposed algorithm is about 119874(31198973)3 Establishment of Static Control Model

The end-point carbon content (denoted as C) and the tem-perature (denoted as T) are main factors to test the qualityof steelmaking The ultimate goal of steelmaking is to control

Complexity 7

WTWTSVRVM

Control ModelController BOF

R2 +

WTWTSVRCT

Prediction Model

R1 +

+

-

+

-

V

W

x1

x7

CP

TP

Tr

CrCgTg

Figure 3 Structure of proposed BOF control model

C and T to a satisfied region BOF steelmaking is a complexphysicochemical process so its mathematical model is verydifficult to establish Therefore the intelligent method such asTSVR can be used to approximate the BOF model From thecollected samples it is easy to see that the samples are listedas a time sequence which means they can be seen as a timesequence signal Hence the proposedWTWTSVR algorithmis appropriate to establish a model for BOF steelmaking

In this section a novel static control model for BOF isestablished based on WTWTSVR algorithm According tothe initial conditions of the hot metal and the desired end-point carbon content (denoted as 119862119892) and the temperature(denoted as119879119892) the relative oxygen blowing volume (denotedas V) and the weight of auxiliary raw material (denotedas W) can be calculated by the proposed model Figure 3shows the structure of the proposed BOF control modelwhich is composed of a prediction model for carbon contentand temperature (CT prediction model) a control modelfor oxygen blowing volume and the weight of auxiliary rawmaterials (VW control model) two parameter regulatingunits (R1 and R2) and a controller and a basic oxygenfurnace in any plant Firstly the WTWTSVR CT predictionmodel should be established by using the historic BOFsamples which consists of two individual models (C modeland T model) C model represents the prediction model ofthe end-point carbon content and T model represents theprediction of the end-point temperature The inputs of twomodels both include the initial conditions of the hot metaland the outputs of them are C and T respectively Theparameters of the prediction models can be regulated by R1to obtain the optimized models Secondly the WTWTSVRVW control model should be established based on theproposed prediction models and the oxygen blowing volumecan be calculated by V model and the weight of auxiliaryraw materials can be determined by W model The inputsof the control models both include the initial conditions ofthe hot metal the desired end-point carbon content 119862119892) and

the end-point temperature 119879119892The outputs of them areV andW respectively The parameters of the control models can beregulated by R2 After the regulation of R1 and R2 the staticcontrol model can be established For any future hot metalthe static control model can be used to calculate V and Wby the collected initial conditions and 119862119892 and 119879119892 Then thecalculated values of V and W will be sent to the controllerFinally the relative BOF system is controlled by the controllerto reach the satisfactory end-point region

31 Establishment of WTWTSVR CT Prediction Model Torealize the static BOF control an accurate prediction modelof BOF should be designed firstly The end-point predictionmodel is the foundation of the control model By collectingthe previous BOF samples the abnormal samples must bediscarded which may contain the wrong information ofBOF Through the mechanism analysis of BOF the influencefactors on the end-point information are mainly determinedby the initial conditions of hot metal which means theinfluence factors are taken as the independent input variablesof the predictionmodelsThe relative input variables are listedin Table 1 Note that the input variable x9 denotes the sumof the weight of all types of auxiliary materials which areincluding the light burned dolomite the dolomite stone thelime ore and scrap etc It has a guiding significance for realproduction and the proportion of the components can bedetermined by the experience of the userThe output variableof the model is the end-point carbon content C or the end-point temperature T

According to the prepared samples of BOF andWTWTSVR algorithm the regression function 119891119862119879(119909)can be given by

119891119862119879 (x) = 12119870 (x119879A119879) (1205961 + 1205962)119879 + 12 (1198871 + 1198872) (34)

where119891119862119879(x) denotes the estimation function of C model orT model and x = [1199091 1199092 1199099]119879

8 Complexity

Table 1 Independent input variables of prediction models

Name of input variable Symbol Units Name of input variable Symbol UnitsInitial carbon content 1199091 Sulphur content 1199096 Initial temperature 1199092 ∘C Phosphorus content 1199097 Charged hot metal 1199093 tons Oxygen blowing volume 1199098 (V) Nm3

Silicon content 1199094 Total weight of auxiliary raw materials 1199099 (W) tonsManganese content 1199095

Out of the historical data collected from one BOF of 260tons in some steel plant in China 220 samples have beenselected and prepared In order to establish the WTWTSVRprediction models the first 170 samples are taken as thetraining data and 50 other samples are taken as the test data toverify the accuracy of the proposed models In the regulatingunit R1 the appropriate model parameters (1198881 1198882 1198883 1198884 V1 V2and 1205901 120590lowast1 ) are regulatedmanually and then theC model andT model can be established In summary the process of themodelling can be described as follows

Step 1 Initialize the parameters of the WTWTSVR predic-tion model and normalize the prepared 170 training samplesfrom its original range to the range [-1 1] by mapminmaxfunction in Matlab

Step 2 Denoise the end-point carbon content C =[1198621 1198622 119862119897]119879 or temperature T = [1198791 1198792 119879119897]119879 in thetraining samples by using the wavelet transform describedin the previous section Then the denoised samples Clowast =[119862lowast1 119862lowast2 119862lowast119897 ]119879 and Tlowast = [119879lowast1 119879lowast2 119879lowast119897 ]119879 can beobtained

Step 3 By selecting the appropriate parameter 120590lowast1 in (33)determine the wavelets transform based weighted matrix D1and the coefficient vector D2

Step 4 Select appropriate values of 1198881 1198882 1198883 1198884 V1 V2 and 1205901 inthe regulating unit 1198771Step 5 Solve the optimization problems in (24) and (25) by119902119906119886119889119901119903119900119892 function in Matlab and return the optimal vector120572 or 120578

Step 6 Calculate the vectors1205961 1198871 and1205962 1198872 by (26) and (27)Step 7 Substitute the parameters in Step 5 into (3) to obtainthe function 119891119862119879(x)Step 8 Substitute the training samples into (34) to calculatethe relative criteria of the model which will be described indetails later

Step 9 If the relative criteria are satisfactory then theC model or T model is established Otherwise return toSteps from 4 to 8

32 Establishment of WTWTSVR VW Control Model Oncethe WTWTSVR CT prediction models are established theWTWTSVRVW control models are ready to build In order

to control the end-point carbon content and temperature tothe satisfactory region for any future hot metal the relativeoxygen blowing volume V and the total weight of auxiliaryraw materialsW should be calculated based on VW controlmodels respectively Through the mechanism analysis theinfluence factors on V and W are mainly determined by theinitial conditions of hot metal and the desired conditions ofthe steel are also considered as the the influence factorsThenthe independent input variables of the control models arelisted in Table 2 where the input variables x1-x7 are the sameas the CT prediction model and other two input variablesare the desired carbon content 119862119892 and desired temperature119879119892 respectively The output variable of the control model isVorW defined above

Similar to (34) the regression function 119891119881119882(x) can bewritten as

119891119881119882 (x) = 12119870 (x119879A119879) (1205963 + 1205964)119879 + 12 (1198873 + 1198874) (35)

where 119891119881119882(x) denotes the estimation function of V modelorW model and x = [1199091 1199092 1199099]119879

In order to establish the WTWTSVR control models thetraining samples and the test samples are the same as thatof the prediction models The regulating unit 1198772 is used toregulate the appropriate model parameters (1198885 1198886 1198887 1198888 V3 V4and 1205902 120590lowast2 ) manually then the V model and W model canbe established In summary the process of the modelling canbe described as follows

Steps 1ndash7 Similar to those in the section of establishing theprediction models except for the input variables listed inTable 2 and the output variable being the oxygen blowingvolume V = [1198811 1198812 119881119897]119879 or the total weight of auxiliarymaterials W = [11988211198822 119882119897]119879 Also select appropriatevalues of 1198885 1198886 1198887 1198888 V3 V4 and 1205902 in the regulating unit 1198772then the function 119891119881119882(x) is obtainedStep 8 Substitute the training samples into (35) to calculatethe relative predicted values 119881 and of V and W respec-tively

Step 9 and are combined with the input variables 1199091minus1199097in Table 1 to obtain 50 new input samples Then substitutingthem into the C model and T model 50 predicted values119862 and can be obtained Finally calculate the differencesbetween the desired values 119862119892 119879119892 and the predicted values119862 Also other relative criteria of the model should bedetermined

Complexity 9

Table 2 Independent input variables of control models

Name of input variable Symbol Units Name of input variable Symbol UnitsInitial carbon content 1199091 Sulphur content 1199096 Initial temperature 1199092 ∘C Phosphorus content 1199097 Charged hot metal 1199093 tons Desired carbon content 1199098 (Cg) Nm3

Silicon content 1199094 Desired temperature 1199099 (Tg) tonsManganese content 1199095

Step 10 If the obtained criteria are satisfactory then theV model or W model is established Otherwise return toSteps from 4 to 9

4 Results and Discussion

In order to verify the performances ofWTWTSVR algorithmand proposed models the artificial functions and practicaldatasets are adopted respectively All experiments are carriedout in Matlab R2011b on Windows 7 running on a PC withIntel (R) Core (TM) i7-4510U CPU 260GHz with 8GB ofRAM

The evaluation criteria are specified to evaluate theperformance of the proposed method Assume that the totalnumber of testing samples is 119899 119910119894 is the actual value at thesample point 119909119894 119910119894 is the estimated value of 119910119894 and 119910119894 =(sum119899119894=1 119910119894)119899 is the mean value of 1199101 1199102 119910119899 Therefore thefollowing criteria can be defined

119877119872119878119864 = radic 1119899119899sum119894=1

(119910119894 minus 119910119894)2 (36)

119872119860119864 = 1119899119899sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (37)

119878119878119864119878119878119879 = sum119899119894=1 (119910119894 minus 119910119894)2sum119899119894=1 (119910119894 minus 119910119894)2 (38)


119867119877 = Number of 1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 lt 119890119903119903119900119903 119887119900119906119899119889119899times 100 (40)

In the above equations RMSE denotes the root meansquared error MAE denotes the mean absolute error andSSE SST and SSR represent the sum of the squared devia-tion between any two of 119910119894 119910119894 119910119894 respectively Normally asmaller value of SSESST reflects that the model has a betterperformance The decrease in SSESST reflects the increasein SSRSST However if the value of SSESST is extremelysmall it will cause the overfitting problem of the regressor Animportant criteria for evaluating the performance of the BOFmodel is hit rate (119867119877) calculated by (40) which is defined as

the ratio of the number of the satisfactory samples over thetotal number of samples For any sample in the dataset if theabsolute error between the estimated value and actual valueis smaller than a certain error bound then the results of thesample hit the end point A large number of hit rate indicate abetter performance of the BOFmodel Generally a hit rate of90 for C or T is a satisfactory value in the real steel plants

41 Performance Test of WTWTSVR In this section theartificial function named Sinc function is used to testthe regression performance of the proposed WTWTSVRmethod which can be defined as 119910 = sin 119909119909 119909 isin [minus4120587 4120587]

In order to evaluate the proposed method effectivelythe training samples are polluted by four types of noiseswhich include the Gaussian noises with zero means andthe uniformly distributed noises For the train samples(119909119894 119910119894) 119894 = 1 2 119897 we have119910119894 = sin 119909119894119909119894 + 120579119894

119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 012) (41)

119910119894 = sin 119909119894119909119894 + 120579119894119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 022) (42)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 01] (43)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 02] (44)

where 119880[119898 119899] and 119873(119901 1199022) denote the uniformly randomvariable in [119898 119899] and the Gaussian variable with the mean119901 and variance 1199022 respectively For all regressions of theabove four Sinc functions 252 training samples and 500 testsamples are selected Note that the test samples are uniformlysampled from the Sinc functions without any noise For allalgorithms in this paper the following sets of parameters areexplored the penalty coefficient c is searched from the set1198941000 119894100 11989410 | 119894 = 1 2 10 The tube regulationcoefficient v is selected from the set 1 2 10Thewaveletcoefficient 120590 and kernel parameter 120590lowast are both searched overthe range 1198941000 119894100 11989410 119894 | 119894 = 1 2 10 In order toreduce the number of combinations in the parameter searchthe following relations are chosen c1= c3 c2= c4 and v1= v2

10 Complexity

minus15 minus10 minus5 0 5 10 15minus06minus04minus02

002040608

112

Outlier

OutlierOutlier

Outlier

Training samplesSinc(x)WTWTSVRWTWTSVR with outliersUpminusdown bound

Figure 4 Performance of WTWTSVR on Sinc function with and without outliers

Firstly the choice of kernel function should be consid-ered RBF kernel is an effective and frequently used kernelfunction in TSVR research papers Also the performanceof RBF has been evaluated by comparing with other threeexisting kernel functions (listed in Table 3) and the resultsare shown in Table 4 It is easy to see that the RBF kernelachieves the optimal results Then the average results of theproposed methods and other four existing methods (TSVR[15] V-TSVR [17] KNNWTSVR [19] and Asy V-TSVR [20])with 10 independent runs are shown in Table 5 whereType A B C and D denote four different types of noises(41)-(44) Obviously the results of Table 5 show that theproposed method achieves the optimal result of SSE Alsothe proposedmethod achieves the smallest SSESST results infour regressions of Sinc functions which are 00029 0012400231 and 00903 Especially the obvious performances areenhanced in Type A and B The SSRSST of the proposedmethod takes the first position in Type B the second positionin Type A and C and the third position in Type D Fromthe aspect of training time it can be seen that the proposedmethod achieves the optimal result in Type B In TypeA C and D the training time of the proposed method isfaster than that of TSVR and KNNWTSVR and close tov-Type and Asy v-Type It verifies that the computationalcomplexity of wavelet transform weighting scheme is lowerthan KNN weighting scheme For the effect of outliers wehave verified the performance of the proposed method onthe Sinc function with Type A noise Figure 4 shows that theprediction performance of the proposed method is satisfiedagainst the outliers as shown in the blue line and the resultsof SSE SSESST and SSRSST achieve 01911 00036 and10372 respectively By comparing with the results in Table 5it can be concluded that the overall performance of theproposed method with outliers is still better than TSVR v-TSVR and KNNWTSVRTherefore it can be concluded thatthe overall regression performance of the proposed methodis optimal

42 Verification of the Proposed BOF Models In order toverify the effectiveness of the proposed model 240 heatssamples for low carbon steel of 260119905119900119899119904 BOF were collected

Table 3 Type and functions of four existing kernel functions

Kernel Type Function

Radial Basis FunctionKernel (RBF) 119870(x x119894) = exp(minus10038161003816100381610038161003816100381610038161003816x minus x119894

10038161003816100381610038161003816100381610038161003816221205902 )Rational QuadraticKernel (RQ) 119870(x x119894) = 1 minus 10038161003816100381610038161003816100381610038161003816x minus x119894

100381610038161003816100381610038161003816100381610038162(10038161003816100381610038161003816100381610038161003816x minus x119894100381610038161003816100381610038161003816100381610038162 + 120590)

Multiquadric Kernel 119870(x x119894) = (10038161003816100381610038161003816100381610038161003816x minus x119894100381610038161003816100381610038161003816100381610038162 + 1205902)05

Log Kernel 119870(x x119894) = minus log (1 + 10038161003816100381610038161003816100381610038161003816x minus x11989410038161003816100381610038161003816100381610038161003816120590)

from some steel plant in China Firstly the preprocessing ofthe samples should be carried out The abnormal sampleswith the unexpected information must be removed whichexceeds the actual range of the signals It may be caused bythe wrong sampling operation Then 220 qualified samplesare obtained The next step is to analyze the informationof the qualified samples The end-point carbon content andtemperature are mainly determined by the initial conditionsof the iron melt the total blowing oxygen volume and theweight of material additions Therefore the unrelated infor-mation should be deleted such as the heat number the dateof the steelmaking and so on According to the informationof Tables 1 and 2 the datasets for the relative prediction andcontrol models can be well prepared After that the proposedmodels are ready to be established More details of the modelhave beendescribed in Section 3Theproposed controlmodelconsists of two prediction models (C model and T model)and two control models (V model and W model) For eachindividual model there are 8 parameters that need to beregulated to achieve the satisfactory results The regulationprocesses are related to the units R1 and R2 in Figure 3First of all the prediction models need to be establishedIn order to meet the requirements of the real productionthe prediction errors bound with 0005 for C model and10∘C for T model are selected Similarly the control errorsbound with 800 Nm3 for V model and 55 tons forW modelare selected The accuracy of each model can be reflected bythe hit rate with its relative error bound and a hit rate of90 within each individual error bound is a satisfied result

Complexity 11

Table 4 Comparisons of WTWTSVR on Sinc functions with different kernel functions

Noise Kernel Type SSE SSESST SSRSST

Type A

RBF 01577plusmn00099 00029plusmn00018 09972plusmn00264RQ 01982plusmn00621 00037plusmn00012 10157plusmn00297

Multiquadric 02884plusmn00762 00054plusmn00014 09911plusmn00345Log 04706plusmn01346 00088plusmn00025 10115plusmn00306

Type B



Type C



Type D



Table 5 Comparisons of four regression methods on Sinc functions with different noises

Noise Regressor SSE SSESST SSRSST Time s

Type A

WTWTSVR 01577plusmn00099 00029plusmn00018 09972plusmn00264 17997TSVR 02316plusmn01288 00043plusmn00024 10050plusmn00308 23815v-TSVR 04143plusmn01261 00077plusmn00023 09501plusmn00270 17397

Asy V-TSVR 01742plusmn01001 00032plusmn00019 10021plusmn00279 17866KNNWTSVR 02044plusmn00383 00038plusmn00007 10177plusmn00202 35104

Type B

WTWTSVR 06659plusmn02456 00124plusmn00046 10161plusmn00564 18053TSVR 08652plusmn03006 00161plusmn00056 10185plusmn00615 23296V-TSVR 08816plusmn03937 00164plusmn00073 09631plusmn00548 18582


Type C



Type D



Also an end-point double hit rate (119863119867119877) of the predictionmodel is another important criterion for the real productionwhich means the hit rates of end-point carbon content andtemperature are both hit for the same sample Hence 170samples are used to train the prediction and control modelsand 50 samples are adopted to test the accuracy of themodelsFor each runof the relativemodelling the results are returnedto the user with the information of the evaluation criteria Itshows the accuracy and fitness of the models with the specific

parameters Smaller values of RMSEMAE and SSESST andlarger values of SSRSST and hit rate are preferred whichmeans the model has a better generalization and higheraccuracy The regulation units R1 and R2 are used to balancethe above criteria by selecting the appropriate values of theparameters Note that the principle of the parameter selectionis satisfactory if the following results are obtained 119878119878119877119878119878119879 gt05 and119867119877 gt 85 with the smallest SSESST Especially thedouble hit rate DHR should be greater than 80 or higher For

12 Complexity

1670 1675 1680 1685 1690 1695 17001670

1675

1680

1685

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1700WTWTSVR

plusmn

125 13 135 14 145 15 155 16125

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NＧ

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plusmn800NＧ3

104

x 104

Figure 5 Performance of errors between predicted values and actual values with proposed static control model

the collected samples the parameters of WTWTSVR modelsare specified and shown in Table 6

By using these parameters the proposed BOF controlmodel has been established In order to evaluate the perfor-mance of the proposed method more simulations of otherexisting regression methods with the samples are carried outwhich are TSVR V-TSVR Asy V-TSVR and KNNWTSVRrespectively The comparison results of the carbon contentprediction are listed in Table 7 From the results it can be seenthat the results of RMSEMAE and SSESST in the proposedmethod are 00023 00026 and 11977 respectively Theyare all smaller than those of other three existing methodsand the SSRSST of 06930 is the highest result The errorperformance and distribution of end-point carbon contentbetween the predicted values and actual values are shown inFigures 5 and 6 It is clear that the proposed C model canachieve the hit rate of 92 which is better than other fourmethods From above analysis it illustrates that the proposedC model has the best fitting behaviour for carbon contentprediction

Similarly the performance comparisons of the tempera-ture prediction are also listed in Table 7 From the results ofTable 7 the best results of RMSE and SSESST and secondbest result of MAE of the proposed method are obtainedThe results of SSRSST is in the third position Figures 5and 6 show that the proposed T model can achieve thehit rate of 96 which is the optimal results by comparingwith other methods In addition the double hit rate is also

the key criterion in the real BOF applications the proposedmethod can achieve a double hit rate of 90 which is thebest result although the temperature hit rate of V-TSVR andKNNWTSVR method can also achieve 96 However theirdouble hit rates only achieve 80 and 84 respectively Inthe real applications a double hit rate of 90 is satisfactoryTherefore the proposed model is more efficient to providea reference for the real applications Also it meets therequirement of establishing the static control model

Based on the prediction models the control models(V model and W model) can be established by using therelative parameters in Table 6 The performance of theproposed V model is shown in Figure 7 By comparing theproposedmethod with the existing methods the comparisonresults of the oxygen blowing volume calculation are listedin Table 8 which shows that the results of RMSE MAEand SSESST in the proposed method are 3713953 4117855and 12713 respectively They are all smaller than those ofother four existing methods and the 119878119878119877119878119878119879 of 10868 is inthe fourth position Figures 5 and 6 show that the predictedvalues of the proposedmodel agreewell with the actual valuesof oxygen volume and the proposed V model has a best hitrate of 90 It verifies that the proposed V model has thebest fitting behaviour for the calculation of oxygen blowingvolume

Similarly the performance of the proposed W model isshown in Figure 8 and the performance comparisons of theweight of the auxiliary materials are also listed in Table 8

Complexity 13

WTWTSVR

Hit rate is 92

plusmn0005

WTWTSVR

Hit rate is 96

plusmn

minus1500 minus1000 minus500 0 500 1000 15000

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minus0005 0 0005 001minus001Predictive error of carbon content ()

minus10 minus5 0 5 10 15minus15

minus5 0 5 10minus10Predictive error of auxiliary raw materials (tons)

plusmn800NＧ3

Predictive error of oxygen volume (NＧ3)

Predictive error of temperature (∘C)

10∘C

Figure 6 Performance of error distributions with proposed static control model

Table 6 Specified parameters of prediction and control models

Prediction Model c1 = c3 c2 = c4 V1 = V2 1205901 120590lowast1C model 0005 002 3 0005 1T model 0002 005 5 004 03Control Model c5 = c7 c6= c8 V3 = V4 1205902 120590lowast2V model 0002 001 2 02 3W model 0001 002 1 05 5

Table 7 Performance comparisons of CT prediction models with four methods

Model Criteria WTWTSVR TSVR v-TSVR Asy v-TSVR KNNWTSVR

C model (plusmn0005 )RMSE 00023 00026 00026 00026 00025MAE 00026 00031 00031 00031 00030

SSESST 11977 16071 15675 15214 14680SSRSST 06930 04616 03514 03835 03500HR 92 82 84 84 88Time s 04523 02915 03706 03351 05352

T model (plusmn10 ∘C)RMSE 40210 42070 43630 41013 40272MAE 47380 50363 51461 47970 47165

SSESST 17297 18935 20365 17996 17351SSRSST 07968 06653 07578 09141 08338HR 96 94 92 96 96Time s 00977 01181 00861 00538 02393DHR 90 78 78 80 84

14 Complexity

Table 8 Performance comparisons of VW control models with four methods


V model (plusmn800 Nm3)RMSE 3713953 3830249 3832387 3998635 3997568MAE 4117855 4163151 4230260 4270896 4270415


W model (plusmn55 tons)RMSE 24158 28824 28431 36781 29077MAE 27057 31254 31632 39979 32588


Hit ratesHR (C) 86 82 88 90 86HR (T) 92 94 94 80 94DHR 82 78 82 82 82

0 5 10 15 20 25 30 35 40 45 50Heat

Actual valuePredicted value

Ｒ 104

125

13

135

14

145

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Oxy

gen

volu

me (

NＧ

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Figure 7 Performance of proposed V model

The proposedmodel achieves the best results of RMSEMAEand SSESST Figures 5 and 6 show that the proposed methodachieves the best hit rate of 88 In addition the training timeof the proposed models is faster than that of KNNWTSVRmethod and slower than that of other three methods Itillustrates that the weighting scheme takes more time toobtain higher accuracy and the performance of proposedweighting scheme is better than that of KNN weightingscheme for time sequence samples For 50 test samplesthere are 50 calculated values of V and W by V model andW modelThen they are taken as the input variables into theproposed C model and T model to verify the end-point hitrate From the results in Table 8 the proposed models canachieve a hit rate of 86 in C 92 in T and 82 in 119863119867119877is in the optimal result and the same as that of V-TSVR AsyV-TSVR and KNNWTSVR In the real productions 119863119867119877 ispaid more attention rather than the individual HR in C or T


35

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70W

eigh

t of a

uxili

ary

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Figure 8 Performance of proposedW model

Therefore the proposed control models are verified to be ableto guide the real production

Based on above analysis it can be concluded that theproposed static control model is effective and feasible the hitrate canmeet the requirements of the real productions for lowcarbon steel For other types of steels the proposed modelis still suitable for use Firstly the specific samples of theheats should be obtained and preprocessed and the analysisof the influence factors on the specific type of steel shouldbe carried out to obtain the input variables of the modelThe output variables are the same as the relative proposedmodels Secondly the end-point criteria should be specifiedwhich is determined by the type of steelThen the parametersof the relative models can be regulated and determined toachieve the best criteria Finally the BOF control model for

Complexity 15

the specific type of steel is established to guide the productionin the plant

BOF steelmaking is a complex physicochemical processthe proposed static control model can be established basedon the real samples collected from the plant However theremust be some undetected factors during the steelmakingprocess which will affect the accuracy of the calculations ofV and W To solve this problem the following strategies canbe introduced at the early stage of the oxygen blowing theproposed control model is used to calculate the relative Vand W and guide the BOF production Then the sublancetechnology is adopted at the late stage of oxygen blowingbecause the physical and chemical reactions tend to be stablein this stage Hence the information of the melt liquid canbe collected by the sublance Therefore another dynamiccontrol model can be established with the sublance samplesto achieve a higher end-point hit rate For medium and smallsteel plants the proposed static model is a suitable choice toreduce the consumption and save the cost

Remark 3 Although this paper is mainly based on staticcontrol model the prediction scheme is also compatiblefor other datasets over the globe Especially the proposedalgorithm is competitive for the regression of time sequencedatasets such as the prediction of the blast furnace processand continuous casting process in the metallurgical industry

5 Conclusion

In this paper a WTWTSVR control model has been pro-posed The new weighted matrix and the coefficient vectorhave been determined by the wavelet transform theory andadded into the objective function of TSVR to improve theperformance of the algorithm The simulation results haveshown that the proposed models are effective and feasibleThe prediction error bound with 0005 in C and 10∘C inT can achieve a hit rate of 92 and 96 respectively Inaddition the double hit rate of 90 is the best result bycomparing with other three existing methods The controlerror bound with 800 Nm3 in V and 55 tons in W canachieve the hit rate of 90 and 88 respectively Thereforethe proposed method can provide a significant reference forreal BOF applications For the further work on the basis ofthe proposed control model a dynamic control model couldbe established to improve the end-point double hit rate ofBOF up to 90 or higher

Data Availability

The Excel data (Research Dataxls) used to support thefindings of this study is included within the SupplementaryMaterials (available here)



Acknowledgments

This work was supported by Liaoning Province PhD Start-Up Fund (No 201601291) and Liaoning Province Ministry ofEducation Scientific Study Project (No 2017LNQN11)


The supplementary material contains the Excel research datafor our simulations There are 220 numbers of preprocessedsamples collected from one steel plant in China Due to thelimitations of confidentiality agreement we cannot providethe name of the company The samples include the historicalinformation of hot metal total oxygen volume total weightof auxiliary raw materials and end-point information Inthe excel table the variables from column A to K aremainly used to establish the static control model for BOFColumns H I J and K are the output variables for C modelT model V model andW model in the manuscript respec-tively The input variables of the relative models are listedin Tables 1 and 2 By following the design procedure ofthe manuscript the proposed algorithm and other existingalgorithms can be evaluated by using the provided researchdata (Supplementary Materials)

References

[1] C Blanco and M Dıaz ldquoModel of Mixed Control for Carbonand Silicon in a Steel ConverterrdquoTransactions of the Iron amp SteelInstitute of Japan vol 33 pp 757ndash763 2007

[2] I J Cox R W Lewis R S Ransing H Laszczewski andG Berni ldquoApplication of neural computing in basic oxygensteelmakingrdquo Journal of Materials Processing Technology vol120 no 1-3 pp 310ndash315 2002

[3] A M Bigeev and V V Baitman ldquoAdapting a mathematicalmodel of the end of the blow of a converter heat to existingconditions in the oxygen-converter shop at the MagnitogorskMetallurgical CombinerdquoMetallurgist vol 50 no 9-10 pp 469ndash472 2006

[4] M Bramming B Bjorkman and C Samuelsson ldquoBOF ProcessControl and Slopping Prediction Based on Multivariate DataAnalysisrdquo Steel Research International vol 87 no 3 pp 301ndash3102016

[5] Z Wang F Xie B Wang et al ldquoThe control and predictionof end-point phosphorus content during BOF steelmakingprocessrdquo Steel Research International vol 85 no 4 pp 599ndash6062014

[6] M Han and C Liu ldquoEndpoint prediction model for basicoxygen furnace steel-making based on membrane algorithmevolving extreme learning machinerdquo Applied Soft Computingvol 19 pp 430ndash437 2014

[7] X Wang M Han and J Wang ldquoApplying input variablesselection technique on input weighted support vector machinemodeling for BOF endpoint predictionrdquo Engineering Applica-tions of Artificial Intelligence vol 23 no 6 pp 1012ndash1018 2010

[8] M Han and Z Cao ldquoAn improved case-based reasoningmethod and its application in endpoint prediction of basicoxygen furnacerdquoNeurocomputing vol 149 pp 1245ndash1252 2015

[9] L X Kong P D Hodgson and D C Collinson ldquoModellingthe effect of carbon content on hot strength of steels using a

16 Complexity

modified artificial neural networkrdquo ISIJ International vol 38no 10 pp 1121ndash1129 1998

[10] A M F Fileti T A Pacianotto and A P Cunha ldquoNeuralmodeling helps the BOS process to achieve aimed end-pointconditions in liquid steelrdquo Engineering Applications of ArtificialIntelligence vol 19 no 1 pp 9ndash17 2006

[11] S M Xie J Tao and T Y Chai ldquoBOF steelmaking endpointcontrol based on neural networkrdquo Control Theory Applicationsvol 20 no 6 pp 903ndash907 2003

[12] J Jimenez JMochon J S de Ayala and FObeso ldquoBlast furnacehot metal temperature prediction through neural networks-based modelsrdquo ISIJ International vol 44 no 3 pp 573ndash5802004

[13] C Kubat H Taskin R Artir and A Yilmaz ldquoBofy-fuzzylogic control for the basic oxygen furnace (BOF)rdquo Robotics andAutonomous Systems vol 49 no 3-4 pp 193ndash205 2004

[14] Jayadeva R Khemchandani and S Chandra ldquoTwin supportvector machines for pattern classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 29 no 5 pp905ndash910 2007

[15] X Peng ldquoTSVR an efficient Twin Support Vector Machine forregressionrdquo Neural Networks vol 23 no 3 pp 365ndash372 2010

[16] D Gupta ldquoTraining primal K-nearest neighbor based weightedtwin support vector regression via unconstrained convex mini-mizationrdquo Applied Intelligence vol 47 no 3 pp 962ndash991 2017

[17] R Rastogi P Anand and S Chandra ldquoA ]-twin support vectormachine based regression with automatic accuracy controlrdquoApplied Intelligence vol 46 pp 1ndash14 2016

[18] M Tanveer K Shubham M Aldhaifallah and K S Nisar ldquoAnefficient implicit regularized Lagrangian twin support vectorregressionrdquoApplied Intelligence vol 44 no 4 pp 831ndash848 2016

[19] Y Xu and L Wang ldquoK-nearest neighbor-based weighted twinsupport vector regressionrdquoApplied Intelligence vol 41 no 1 pp299ndash309 2014

[20] Y Xu X Li X Pan and Z Yang ldquoAsymmetric v-twin supportvector regressionrdquo Neural Computing and Applications vol no2 pp 1ndash16 2017

[21] N Parastalooi A Amiri and P Aliheidari ldquoModified twinsupport vector regressionrdquo Neurocomputing vol 211 pp 84ndash972016

[22] Y-F Ye L Bai X-Y Hua Y-H Shao Z Wang and N-YDeng ldquoWeighted Lagrange 120576-twin support vector regressionrdquoNeurocomputing vol 197 pp 53ndash68 2016

[23] C Gao M Shen and L Wang ldquoEnd-point Prediction ofBOF Steelmaking Based onWavelet Transform BasedWeightedTSVRrdquo in Proceedings of the 2018 37th Chinese Control Confer-ence (CCC) pp 3200ndash3204 Wuhan July 2018

[24] J Shen and G Strang ldquoAsymptotics of Daubechies filtersscaling functions and waveletsrdquo Applied and ComputationalHarmonic Analysis vol 5 no 3 pp 312ndash331 1998


Complexity





Editorial Board




Contents
















1 Introduction












2 Complexity


3 Hybrid Models















Complexity 3







10 Conclusion






References

















1 Introduction





2 Complexity







Complexity 3





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (1)


120576푡 = sum퐴푅퐸119905(푖)gt훿

119863푡 (119894) (2)



(3)





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119865푡 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (5)


119864푘 = sum퐴푅퐸119896(푖)gt훿

119863푡 (119894) (6)

4 Complexity




119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖











3 Experiments


Complexity 5





1198902 (119894))12 (8)






accuracy = 119873푠119873푤 (9)




6 Complexity







Drying slurry

size

trough


2 4 6 8 10 12 1401

02

03

04

05

06

The number of ELMs

TrainingTest

RMSE

Mea

n



2 4 6 8 10 12 1403

035

04

045

05

055

06

The number of ELMs


RMSE

Mea

n







Complexity 7

0 5 10 15 20 25 30 35 40 45

times


10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e







4 Conclusions


8 Complexity





S1 sim S8 S10 S11 S907386 00868 03773


S1 sim S7 S9 sim S10 S11 S808405 00772 02962


S1 sim S6 S8 sim S10 S11 S70936 00655 04255


S1 sim S5 S7 sim S10 S11 S607342 00618 03451


S1 simS4S6 simS10 S11 S508533 00872 03546


S1 sim S3 S5 sim S10 S11 S407752 00749 03249


S1 sim S2 S4 sim S10 S11 S308430 00823 03281


S1 S3 sim S10 S11 S209127 01104 04302


S2 sim S10 S11 S107905 00910 02245

AdaBoostR 05045 00436 01773ILES 03145 00191 00756




x2x4)2 + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]


x1) + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]




Complexity 9



Friedman1ELM 08278 07662 47880

AdaBoostR 07987 06536 38817ILES 03099 03073 27132

Friedman2ELM 06338 09975 179870

AdaBoostR 04190 04495 73544ILES 02079 03683 72275

Friedman3ELM 11409 14314 149348

AdaBoostR 08959 13461 115991ILES 04624 03930 52719

0 10 20 30 40 50 60 70 80 90 100

times

80

60

40

20

minus20

0

y

minus40



y


2500

2000

1500

1000

500

0

minus2500

minus2000

minus1500

minus1000

minus500

0 10 20 30 40 50 60 70 80 90 100

times


10 Complexity

y

0 10 20 30 40 50 60 70 80 90 100

times


15

10

5

0

minus5

minus10



Appendix




120573푖119892푖 (119909푗) = 푁sum푖=1


(A1)


H120573 = T (A2)

where

H = [[[[[


]]]]]푁times푁

(A3)

120573 = [[[[[

120573푇1120573푇푁

]]]]]푁times푚

(A4)

and

T = [[[[[

119905푇1119905푇푁]]]]]푁times푚

(A5)


Complexity 11


1006704120573 = HdaggerT (A6)







Data Availability




Acknowledgments


References






















12 Complexity












1 Introduction





2 Complexity


Comminution



Flotation

Drying

Anodes (995 Cu)


Drying


smelting

Matte (50-70Cu)

Converting

Blister Cu (99 Cu)

Electrorefining

Cathodes (9999 Cu)








Complexity 3





2 Related Work





3 Methodology



4 Complexity


AUTHOR RESEARCH
















Complexity 5


0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012


05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012




2012 2013 2014 2015 2016 2017



2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)




6 Complexity













Complexity 7




119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000












8 Complexity

















Complexity 9


119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000














10 Complexity











6 Conclusions





Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014 2016 2018


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014456789


Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018


405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011








12 Complexity



Appendix

A Models


119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)






E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)



119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)










119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13


B Model Calibration



(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)


119898 = 1119899119899sum119905=1


119904 = radic 1119899 minus 1119899sum119894=1


119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)


119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)


= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)



119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)


0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)


120572 lt 4120573(1 + 120573)2 (B14)


14 Complexity





119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)


119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)



119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)



119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability






References






Complexity 15








































16 Complexity





















1 Introduction






2 Complexity









119896119899119899 (119896) = 119899sumℎ=1

ℎ119875 (ℎ | 119896) (1)


119875 (119896) ge 0and 119875 (ℎ | 119896) ge 0 (2)


119899sum119896=1

119875 (119896) = 1and119899sumℎ=1

119875 (ℎ | 119896) = 1(3)


ℎ119875 (119896 | ℎ) 119875 (ℎ) = 119896119875 (ℎ | 119896) 119875 (119896)forallℎ 119896 = 119894 = 1 119899 (4)


119902119896 = (119896 + 1)sum119899119895=1 119895119875 (119895)119875 (119896 + 1) (5)


119903 = 11205902119902119899minus1sum119896ℎ=0

119896ℎ (119890119896ℎ minus 119902119896119902ℎ) (6)


1198962119902119896 minus ( 119899sum119896=1

119896119902119896)2 (7)


119875 (ℎ | 119896) = 119890119896ℎsum119899119895=1 119890119896119895 forallℎ 119896 = 119894 = 1 119899 (8)


119890ℎ119896 = 119875 (ℎ | 119896) 119896119875 (119896)sum119899119895=1 119895119875 (119895) (9)

Complexity 3



119875 (119896) = 2120573 (120573 + 1)119896 (119896 + 1) (119896 + 2) (10)




) (11)


119861119898119895 = 119898119895 (119898 minus 119895) (12)


119875119899 (ℎ | 119896) = 119875 (ℎ | 119896)119862119896 (13)





119889119865 (119905)119889119905 = [1 minus 119865 (119905)] [119901 + 119902119865 (119905)] (14)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (119905)]119894 = 1 119899

(15)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901119894 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (t)]119894 = 1 119899

(16)





4 Complexity

t t

ftot

002

004

006

008

002

004

006

008

010

012

20 30 40 5010 20 30 40 5010

fi


q = 03 q = 048

= 1

= 2

= 3

= 5

Assortative


= 4

DIFFUSION TIME

BASS q PARAMETER

2 4 6 8 10

40

45

50

55

60







Complexity 5




119875 (119896) = 1198881198963 (17)



119875 (ℎ | 119896) = ℎ119875 (ℎ)⟨ℎ⟩ (18)


119896119899119899 (119896) = ⟨ℎ2⟩⟨ℎ⟩ (19)






119875 (ℎ | 119896) = 1(119862119898119886119909 minus 1)1198751 (ℎ | 119896) for ℎ 119896 = 1 119899 (21)



(22)




4 Discussion



6 Complexity

k_nn

k20 40 60 80 100

10

20

30

40

50

60

70


k

k_nn

20 40 60 80 100

1115

1120

1125

1130











Complexity 7

2 4 6 8 10n1000

minus0025

minus0020

minus0015

minus0010

minus0005

r


0 20 40 60 80 100k

36

37

38

39

40

k_nn









⟨1198962⟩ = 119899sum119896=1

119896119875 (119896)119870 (119896 119899) (25)

8 Complexity

k

0 200 400 600 800 1000

56

58

60

62

64knn



119896=1

1198962 4119896 (119896 + 1) (119896 + 2) = 4 ln (119899) + 119900 (119899) (26)



119896119875 (119896)119870 (119896 119899) = 119899sum119896=1

4(119896 + 1) (119896 + 2)119870 (119896 119899) (27)






119889119865119894 (119905)119889119905 = [1 minus 119865119894 (119905)](119901 + 119902 119873sum119895=1

119886119894119895119865119895 (119905)) 119894 = 1 119873

(28)


Complexity 9










10 Complexity

BASS q PARAMETER


Assortative


BA

q = 048q = 03

2 4 6 8 10

4

5

6

7

8

9

10


7 Conclusions






Appendix




2 119899sum119895=1

1119895120574 minusinfinsum119895=1

11198952 ge 2 minus 1205876 gt 0 (A2)


is equal to

Complexity 11



(sum119899119895=1 (1119895120574))2le 1

(A3)


(119896119889 + ℎ119889)sum119899119895=1 (11198952+119889)1198962+119889ℎ2+119889sum119899119895=1 (11198952) (sum119899119895=1 (11198952+119889))2

le (119896119889 + ℎ119889)119896119889ℎ119889 11198962ℎ2 le 1(A4)

Data Availability




Acknowledgments


References






























12 Complexity













1 Introduction






2 Complexity








119877 (119905) = 119873119905=0 minus 119891 (119905)119873119905=0 (1)







Complexity 3


119875 (120585119896119895 (119860120578119894)) =120585119895 (119860120578119894)

sum119872119895=1 120585119895 (119860120578119894)(2)



120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (3)


120585119895 (119860120578119894) =119876

sum119896=1

120585119895119896 (119860120578119894) (4)


=


(5)







120585119895 (119860120578119894) (119905 + 120591) = 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (6)




120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120574120585119895 (119860120578119894) (7)





4 Complexity




Yes



i=i+1

i=1






doubling strategy

End

Start


No

Yes

NltN1 NgtN2

N1⩽N⩽N2



Rout1

Rout1

Rout

x1

x2

xm



119864 (120578119894119895 1205781015840119902119896 120572119895 120573119896) = 12119875

sum119901=1

119892

sum119896=1

(119877119901119896minus 119877119901119900119906119905119896)






Complexity 5



Data normalization











119910 = 05 (119910119894 + 1) (119910max minus 119910min) + 119910min (10)









119901 = int10119901ℎ( 119901119899119894) 119889119901 (11)

119891119894 = 119899119894 (1 minus 119901) (12)

6 Complexity



1 293 40 3 02 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 010 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 116 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 128 308 50 20 329 308 55 2 030 308 55 5 131 308 55 15 232 308 55 20 2





119901119894 =int119901119894+1119901119895

119901119891119895+1+1 (1 minus 119901)119904119895+1 119889119901int119901119894+1119901119895

119901119891119895+1 (1 minus 119901)119904119895+1 119889119901 (13)



Complexity 7







ants i=1goto begin


return resulte












8 Complexity



1 293 40 3 07142862 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 071428610 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 216 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 228 308 50 20 329 308 55 2 071428630 308 55 5 131 308 55 15 232 308 55 20 2




119872119878119864 = 1119873119873

sum119896=1

10038161003816100381610038161003816119877119896 minus 119900119906119905119896 10038161003816100381610038161003816 (14)

119872119860119875119864 = 1119873119873

sum119896=1

1003816100381610038161003816100381610038161003816100381610038161003816119877119896 minus 119900119906119905119896

1198771198961003816100381610038161003816100381610038161003816100381610038161003816times 100 (15)



Complexity 9


Group [NY]- 1

Group [NY]-1

Group [NY]-1

Group [NY]-1

Group 2[NY]

Group (Y-1) [NY]

Group [NY]

Group Y[NY]

1

N

Train Test

Group (YY









10 Complexity











6 Conclusion


Complexity 11

IACO-BPACO-BP

PSO-BPBP

0

02

04

06

08

1

12

14

MSE



IACO-BPACO-BPPSO-BP

0001002003004005006007008

MSE



065

07

08

09

1

Relia

bilit

y

IACO-BPACO-BPPSO-BP




12 Complexity


Data Availability




Acknowledgments


References

























Complexity 13













1 Introduction









2 Complexity










2 Literature Review











Complexity 3


(GO)


(EVM)


(EI)


(IC)


(BGI)















4 Complexity


















Complexity 5


















6 Complexity





1)


2)



1)


2)


3)



1)


2)



1)


2)



1)


2)



1)


2)


3)





4 Results








Complexity 7








Compliance(ECC)


(EVM)


(GAN)

GAN1 GAN2

ECC2

EI2

EI1 Environmental

Impact

(EI)

Innovation and

Competitiveness

(IC)

Green

Opportunities

(GO)

Balanced Green

Investment

(BGI)

IC2

IC3

IC1

GO2

GO1

BGI2

BGI1

ECC1

ECC3

EVM2EVM1

GO3


8 Complexity






p-Value Supported






(ECC)0487

(EVM)

0779(GAN)

GAN1 GAN2

0888

07770780

ECC2

EI2

EI10532(EI)

0356(IC)

0598(GO)

0609(BGI)

0772

IC2

IC3

IC1

GO2

GO3

GO1

BGI2

BGI1

ECC1

ECC3

0697

EVM2EVM1

0815

0887

0852

0798

08180729

0867

0597

0730

0773

0417

09790740

07820844 0882

0539

0793 0876







Complexity 9















10 Complexity





n=54k=2 (31)

n=51k=3 (307)

n=47MGAk1 vs k2

MGAk1 vs k3

MGAk2 vs k3













Complexity 11














0

5

10

15

20

25

minus25

0minus2

25

minus20

0minus1

75

minus15

0minus1

25

minus10

0minus0

75

minus05

0minus0

25

000

025

050

075

100

125

150

175

200

225

250

275

300

GAN









12 Complexity

000 010 020 030 040 050 060 070 080 090 100




EVMECC

EVMECC

EVMECC

0

20

40

60

80

100

GA

N

010 020 030 040 050 060 070 080 090 100000Total-Effects

Total-Effects

Total-Effects

0

20

40

60

80

100

GA

N

000 010 020 030 040 050 060 070 080 090 1000

20

40

60

80

100

GA

N




5 Discussion













Complexity 13


5





6 EVM997888rarr

EI 120573 = 0730 p value le 0001)



6 Conclusion













14 Complexity



Data Availability






Acknowledgments


References




























Complexity 15




































16 Complexity


































Complexity 17














1 Introduction





2 Complexity






Complexity 3

50

40

30

20

10

15

20


1 5 10Time

Am

plitu

de


10864


minus10

2

2

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

TimeAm

plitu

de




2 Preliminaries





119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 [119905] 119896 = 2 119870 (1)



119909119894 [119905] = 119909 [119905] + 119908119894 [119905] (2)



119868119872119865119896 = 1119873119873sum119894=1

119868119872119865119894119896 [119905] (3)

4 Complexity



1198681198721198652 = 1119873119873sum119894=1

1198641 (1199031 [119905] + 12057611198641 (119908119894 [119905])) (4)

1199032 [119905] = 1199031 [119905] minus 1198681198721198652 (5)



119868119872119865119896 = 1119873119873sum119894=1


119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 (7)


R [119905] = x [119905] minus 119870sum119896=1

119868119872119865119896 (8)



1199031 [119905] = 1119873119873sum119894=1

119872(119909119894 [119905]) (9)


1198681198721198651 = 119909 [119905] minus 1199031 [119905] (10)


119903119896 [119905] = 1119873119873sum119894=1

119872(119903119896minus1 [119905] + 120576119896minus1119864k (119908119894 [119905])) (11)

119868119872119865119896 = 119903119896minus1 [119905] minus 119903119896 [119905] (12)



119910119894 = 0 (119909119894) = 119870sum119896=1

119891119896 (119909119894) (13)




119871 (0) = 119899sum119894=1

119897 (119910119894 119910119894) + 119870sum119896=1

Ω(119891119896) (14)

Ω(119891119896) = 120574119879 + 12120572 1205962 (15)



119871119905 = 119899sum119894=1

119897 (119910119894 119910119894(119905minus1) + 119891119905 (119909119894)) + Ω (119891119905) (16)



119892119894 = 120597119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (17)

Complexity 5

ℎ119894 = 1205972119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (18)

119871 119905 = 119899sum119894=1

[119897 (119910119894 119910119894(119905minus1)) + 119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)]+ Ω (119891119905)

(19)


(119905) = 119899sum119894=1

[119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)] + 120574119879 + 12120572119879sum119895=1

1205962119895= 119879sum119895=1

[[(sum119894isin119868119895

119892119894)120596119895 + 12 (sum119894isin119868119895

ℎ119894 + 119886)1205962119895]] + 120574119879(20)




(119905) (119902) = minus12119879sum119895=1



119866119886119894119899= 12 [[[












6 Complexity

X[n]

CEEMDAN



Decomposition


Ensemble

XGBoost

x[n]

R[n]IMF1IMF2

IMF3IMFk

k

i=1IMFi +

R[n]sum



Date

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Pric

es (U

SDb

arre

l)




1199091015840119905 = 119909119905 minus 120583120590 (25)


Complexity 7


RMSE = radic 1119873119873sum119894=1

(119910119905 minus 119910119905)2 (26)

MAE = 1119873 ( 119873sum119894=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816) (27)

Dstat = sum119873119894=2 119889119894119873 minus 1 119889119894 =


(28)





119871119895119905 = 1119879ℎ+119879sum119905=ℎ+1



119889119895119896119905 = 119871119895119905 minus 119871119896119905 (30)





1198670 119864 (119889119895119896119905) = 0 forall119898119895 119898119896 isin 119872 sub M0 (32)




119879119877 = max119895119896isin119872

1003816100381610038161003816100381611990511989511989610038161003816100381610038161003816 (33)

119879119878119876 = max119895119896isinM

1199052119895119896 (34)

119905119895119896 = 119889119895119896radicvar (119889119895119896) (35)

119889119895119896 = 1119879ℎ+119879sum119905=ℎ+1

119889119895119896119905 (36)



8 Complexity











1

XGBOOST 12640 09481 04827SVR 12899 09651 04826FNN 13439 09994 04837

ARIMA 12692 09520 04883

3

XGBOOST 20963 16159 04839SVR 22444 17258 05080FNN 21503 16512 04837

ARIMA 21056 16177 04901

6

XGBOOST 29269 22945 05158SVR 31048 24308 05183FNN 30803 24008 05028

ARIMA 29320 22912 05151




Complexity 9

minus505

IMF1

minus4minus2

024

IMF2

minus505

IMF3

minus505

IMF4

minus4minus2

024

IMF5

minus505

IMF6

minus200

20

IMF7

minus100

1020

IMF8

minus100

10

IMF9

minus20minus10

010

IMF1

0

minus200

20

IMF1

1


Resid

ue







10 Complexity

minus505

IMF1

minus505

IMF2

minus505

IMF3

minus505

IMF4

minus505

IMF5

minus100

10

IMF6

minus100

10

IMF7

minus20

0

20

IMF8

minus200

20

IMF9

minus200

20

IMF1

0


Resid

ue







Complexity 11



1



EEMD-FNN 26835 19932 07361

3



EEMD-FNN 12046 08637 06959

6



EEMD-FNN 27786 20495 06337








12 Complexity


CEEMDAN-XGBOOST

EEMD-XGBOOST

CEEMDAN-SVR

EEMD-SVR

CEEMDAN-FNN

EEMD-FNN



EEMD-FNN 00726 00556 86135e-09 00007 81427e-196 1








5 Conclusions



Data Availability


Complexity 13

RMSEMAEDstat

Dsta

t

RMSE

MA

E

070

065

060

050

055

045

040

035

03010 25 50 500 750 1000 75 100 250

088

087

086

085

084

083

082

081

080



RMSEMAEDstat

Dsta

t

RMSE

MA

E

2 4 6 8 10lag order

088

086

084

082

080

078

076

074

12

10

08

06

04


RMSEMAEDstat

Dsta

t


088

086

084

082

080

078

076

074

10

08

09

06

07

04

05

03

001 002 003 004 005 006 007

RMSE

MA

E


14 Complexity



Acknowledgments


References






























Complexity 15



























1 Introduction




2 Complexity




Complexity 3

Decompose-

D-step


WAEMD-based method


CEEMDAN


step

IMF1

ARMA

RGMDH

RBFNN

IMF2 IMFl

ARMA

RGMDH

RBFNN

ARMA

RGMDH

RBFNN

P-step

Select the best

Predicted IMF

Component




2 Proposed Methods








4 Complexity



119886119895119896 =2119873minus119895minus1

sum119896=0


and

119889119895119896 =119869

sum119895=1


sum119896=0



1198891015840119895119896 =

119889119895119896 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(3)

and

1198891015840119895119896 =

sgn (119889119895119896) (1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 minus 119879119895)

1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(4)


119909 (119905) =2119873minus119895minus1

sum119896=0


+119869

sum119895=1


sum119896=0


(5)






ℎ (119905) = 119909 (119905) minus 119898 (119905) (6)




119909 (119905) =119898

sum119894

ℎ119894 (119905) + 119903 (119905) (7)


119909 (119905) =119898minus2

sum119894=1

ℎ119894 (119905) +119898

sum119894=119898minus2

ℎ119894 (119905) + 119903 (119905) (8)



Complexity 5









23 P-Step






Transfer Functions





119910119905 = 119886 +119903

sum119894=1

119887119894119909119894 +119903

sum119894=1

119903

sum119895=1

119888119894119895119909119894119909119895 (10)



6 Complexity


Selected

Unselected

８1

８2

８3

８4


Input X Output Y

x(1)

x(2)

x(n)

K(X )

K(X )

K(X )

Bias b



0119894 (119909) = 11 + exp (119887119879119909 minus 1198870) (11)



Complexity 7

AFGHANISTAN

INDIA

Arabian Sea

Sindh

Punjab

Kabul R










119872119877119864 = 1119899sum119899119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816

119891 (119905)(12)

119872119860119864 = 1119899119899

sum119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816 (13)

and

119872119878119864 = 1119899119899

sum119905=1

(119891 (119905) minus 119891 (119905))2 (14)


4 Results


8 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow








Complexity 9



EMD 805931 873925 39118 01275 367636Wavelet 802267 861632 38188 00987 229626




minus20

020

minus40

24

6minus1

00

1020

minus30

minus10

10minus2

00

20

0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200

minus20

020

minus20

020

minus60

040

minus100

050

minus100

010

0

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

15

minus50

510

minus50

510

minus10

05

minus15

minus55

1030

50

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time Time Time Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual




10 Complexity

minus20

020

minus40

24

6minus1

00

1020

minus20

020

minus20

020

0 200 400 600 800 1200

minus20

010

minus20

020

minus60

minus20

2060

minus100

050

minus100

010

0

0 200 400 600 800 1200

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

0 200 400 600 800 1200

minus50

5minus5

05

minus10

05

10minus1

5minus5

510

3050

0 200 400 600 800 1200

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time









Complexity 11







WA-RGMDH 26208 00773 377954WA-RBFN 98608 07714 1807443




WA-RGMDH 20794 00729 210612WA-RBFN 16565 00997 133554




WA-RGMDH 36147 00943 816493WA-RBFN 41424 02757 476184





5 Conclusion


12 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl





2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow





Complexity 13


Data Availability




Acknowledgments


References



























14 Complexity






















1 Introduction





2 Complexity








Complexity 3



2 Background









119891 (x) = 12 (1198911 (x) + 1198912 (x))= 12119870 (x119879A119879) (1205961 + 1205962) + 12 (1198871 + 1198872)

(3)


min1205961 1198871120585


and

min1205962 1198872120574



max120572


st 0e le 120572 le 1198881e(6)

4 Complexity

and

max120573


st 0e le 120573 le 1198882e(7)


[12059611198871 ] = (G119879G)minus1G119879 (g minus 120572) (8)

and

[12059621198872 ] = (G119879G)minus1G119879 (h + 120573) (9)




min1205961 11988711205761120585

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus (119870 (AA119879)1205961 + 1198871e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198881V11205761+ 1119897 1198881e119879120585


(10)

and

min1205962 11988721205762 120574



(11)


min1205961 11988711205851205761


(12)

and

min1205962 1198872120585

lowast 1205762


(13)



Complexity 5



(14)



(15)


+ 11988811198871 = 0 (16)

120597119871120597120585 = 1198882D2 minus 120572 minus 120573 = 0 (17)

1205971198711205971205761 = 1198882V1 minus e119879120572 minus 120574 = 0 (18)



120572119879 (y minus (119870 (AA119879)1205961 + 1198871e) + 1205761e + 120585) = 0120572 ge 0e

120573119879120585 = 0120573 ge 0e

1205741205761 = 0 120574 ge 0

(19)


minus [[119870(AA119879)119879

e119879]]D1 (y minus (119870 (AA119879)1205961 + 1198871e))

+ [[119870(AA119879)119879

e119879]]120572 + 1198881 [

12059611198871 ] = 0(20)


minusH119879D1y + (H119879D1H + 1198881I) u1 +H119879120572 = 0 (21)




e119879120572 le 1198882V1 0e le 120572 le 1198882D2 (23)


min120572

12120572119879H (H119879D1H + 1198881I)minus1H119879120572 + y119879120572


(24)


min120578


119904119905 0e le 120578 le 1198884D2 e119879120578 le 1198884V2(25)



and

[12059621198872 ] = (H119879D1H + 1198883I)minus1H119879 (D1y + 120578) (27)



6 Complexity



Sl119896 (119899) = 119897119896minus1sum119898=1

120601 (119898 minus 2119899) Sl119896minus1 (119898) (28)

Sh119896 (119899) = 119897119896minus1sum119898=1

120593 (119898 minus 2119899) Sl119896minus1 (119898) (29)


Sl119896 (119899) = P sdot Sl119896minus1 (119898) (30)

Sh119896 (119899) = Q sdot Sl119896minus1 (119898) (31)




S

Sh1

Sl1Sh2

Sl2

Slk

Shk


SlowastShlowastk-1

Sllowastk-1

Sllowast1

Sllowastk

Shlowastk

Shlowast

1



119889119894 = exp(minus 11990321198942120590lowast2) 119894 = 1 2 119897 (33)



Complexity 7

WTWTSVRVM


R2 +

WTWTSVRCT

Prediction Model

R1 +

+

-

+

-

V

W

x1

x7

CP

TP

Tr

CrCgTg







119891119862119879 (x) = 12119870 (x119879A119879) (1205961 + 1205962)119879 + 12 (1198871 + 1198872) (34)


8 Complexity














119891119881119882 (x) = 12119870 (x119879A119879) (1205963 + 1205964)119879 + 12 (1198873 + 1198874) (35)





Complexity 9








119877119872119878119864 = radic 1119899119899sum119894=1

(119910119894 minus 119910119894)2 (36)

119872119860119864 = 1119899119899sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (37)








119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 012) (41)

119910119894 = sin 119909119894119909119894 + 120579119894119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 022) (42)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 01] (43)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 02] (44)


10 Complexity


002040608

112

Outlier

OutlierOutlier

Outlier









100381610038161003816100381610038161003816100381610038162(10038161003816100381610038161003816100381610038161003816x minus x119894100381610038161003816100381610038161003816100381610038162 + 120590)




Complexity 11



Type A



Type B



Type C



Type D





Type A



Type B



Type C



Type D





12 Complexity

1670 1675 1680 1685 1690 1695 17001670

1675

1680

1685

1690

1695

1700WTWTSVR

plusmn

125 13 135 14 145 15 155 16125

13

135

14

145

15

155

16 x WTWTSVR

40 45 50 55 60 65 70

WTWTSVR

Actual value (tons)

40

45

50

55

60

65

70

Pred

icte

d va

lue (

tons

)

plusmn55t

003 0035 004 0045 005003

0035

004

0045

005WTWTSVR

Actual value ()

Pred

icte

d va

lue (

)

plusmn000510

∘CPred

icte

d va

lue (

∘C)

Actual value (∘C)

Pred

icte

d va

lue (

NＧ

3)


plusmn800NＧ3

104

x 104








Complexity 13

WTWTSVR

Hit rate is 92

plusmn0005

WTWTSVR

Hit rate is 96

plusmn


5

10

15WTWTSVR

Freq

uenc

y

Hit rate is 90

0

5

10

15WTWTSVR

Freq

uenc

y

plusmn55t

Hit rate is 88

0

5

10

15Fr

eque

ncy

0

5

10

15

Freq

uenc

y




plusmn800NＧ3



10∘C










14 Complexity








0 5 10 15 20 25 30 35 40 45 50Heat


Ｒ 104

125

13

135

14

145

15

155

16

Oxy

gen

volu

me (

NＧ

3)




35

40

45

50

55

60

65

70W

eigh

t of a

uxili

ary

raw

mat

eria

ls (to

ns)

5 10 15 20 25 30 35 40 45 500Heat




Complexity 15




5 Conclusion


Data Availability




Acknowledgments




References










16 Complexity

















Complexity





Editorial Board




Contents
















1 Introduction












2 Complexity


3 Hybrid Models















Complexity 3







10 Conclusion






References

















1 Introduction





2 Complexity







Complexity 3





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (1)


120576푡 = sum퐴푅퐸119905(푖)gt훿

119863푡 (119894) (2)



(3)





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119865푡 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (5)


119864푘 = sum퐴푅퐸119896(푖)gt훿

119863푡 (119894) (6)

4 Complexity




119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖











3 Experiments


Complexity 5





1198902 (119894))12 (8)






accuracy = 119873푠119873푤 (9)




6 Complexity







Drying slurry

size

trough


2 4 6 8 10 12 1401

02

03

04

05

06

The number of ELMs

TrainingTest

RMSE

Mea

n



2 4 6 8 10 12 1403

035

04

045

05

055

06

The number of ELMs


RMSE

Mea

n







Complexity 7

0 5 10 15 20 25 30 35 40 45

times


10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e







4 Conclusions


8 Complexity





S1 sim S8 S10 S11 S907386 00868 03773


S1 sim S7 S9 sim S10 S11 S808405 00772 02962


S1 sim S6 S8 sim S10 S11 S70936 00655 04255


S1 sim S5 S7 sim S10 S11 S607342 00618 03451


S1 simS4S6 simS10 S11 S508533 00872 03546


S1 sim S3 S5 sim S10 S11 S407752 00749 03249


S1 sim S2 S4 sim S10 S11 S308430 00823 03281


S1 S3 sim S10 S11 S209127 01104 04302


S2 sim S10 S11 S107905 00910 02245

AdaBoostR 05045 00436 01773ILES 03145 00191 00756




x2x4)2 + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]


x1) + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]




Complexity 9



Friedman1ELM 08278 07662 47880

AdaBoostR 07987 06536 38817ILES 03099 03073 27132

Friedman2ELM 06338 09975 179870

AdaBoostR 04190 04495 73544ILES 02079 03683 72275

Friedman3ELM 11409 14314 149348

AdaBoostR 08959 13461 115991ILES 04624 03930 52719

0 10 20 30 40 50 60 70 80 90 100

times

80

60

40

20

minus20

0

y

minus40



y


2500

2000

1500

1000

500

0

minus2500

minus2000

minus1500

minus1000

minus500

0 10 20 30 40 50 60 70 80 90 100

times


10 Complexity

y

0 10 20 30 40 50 60 70 80 90 100

times


15

10

5

0

minus5

minus10



Appendix




120573푖119892푖 (119909푗) = 푁sum푖=1


(A1)


H120573 = T (A2)

where

H = [[[[[


]]]]]푁times푁

(A3)

120573 = [[[[[

120573푇1120573푇푁

]]]]]푁times푚

(A4)

and

T = [[[[[

119905푇1119905푇푁]]]]]푁times푚

(A5)


Complexity 11


1006704120573 = HdaggerT (A6)







Data Availability




Acknowledgments


References






















12 Complexity












1 Introduction





2 Complexity


Comminution



Flotation

Drying

Anodes (995 Cu)


Drying


smelting

Matte (50-70Cu)

Converting

Blister Cu (99 Cu)

Electrorefining

Cathodes (9999 Cu)








Complexity 3





2 Related Work





3 Methodology



4 Complexity


AUTHOR RESEARCH
















Complexity 5


0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012


05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012




2012 2013 2014 2015 2016 2017



2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)




6 Complexity













Complexity 7




119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000












8 Complexity

















Complexity 9


119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000














10 Complexity











6 Conclusions





Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014 2016 2018


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014456789


Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018


405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011








12 Complexity



Appendix

A Models


119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)






E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)



119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)










119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13


B Model Calibration



(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)


119898 = 1119899119899sum119905=1


119904 = radic 1119899 minus 1119899sum119894=1


119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)


119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)


= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)



119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)


0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)


120572 lt 4120573(1 + 120573)2 (B14)


14 Complexity





119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)


119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)



119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)



119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability






References






Complexity 15








































16 Complexity





















1 Introduction






2 Complexity









119896119899119899 (119896) = 119899sumℎ=1

ℎ119875 (ℎ | 119896) (1)


119875 (119896) ge 0and 119875 (ℎ | 119896) ge 0 (2)


119899sum119896=1

119875 (119896) = 1and119899sumℎ=1

119875 (ℎ | 119896) = 1(3)


ℎ119875 (119896 | ℎ) 119875 (ℎ) = 119896119875 (ℎ | 119896) 119875 (119896)forallℎ 119896 = 119894 = 1 119899 (4)


119902119896 = (119896 + 1)sum119899119895=1 119895119875 (119895)119875 (119896 + 1) (5)


119903 = 11205902119902119899minus1sum119896ℎ=0

119896ℎ (119890119896ℎ minus 119902119896119902ℎ) (6)


1198962119902119896 minus ( 119899sum119896=1

119896119902119896)2 (7)


119875 (ℎ | 119896) = 119890119896ℎsum119899119895=1 119890119896119895 forallℎ 119896 = 119894 = 1 119899 (8)


119890ℎ119896 = 119875 (ℎ | 119896) 119896119875 (119896)sum119899119895=1 119895119875 (119895) (9)

Complexity 3



119875 (119896) = 2120573 (120573 + 1)119896 (119896 + 1) (119896 + 2) (10)




) (11)


119861119898119895 = 119898119895 (119898 minus 119895) (12)


119875119899 (ℎ | 119896) = 119875 (ℎ | 119896)119862119896 (13)





119889119865 (119905)119889119905 = [1 minus 119865 (119905)] [119901 + 119902119865 (119905)] (14)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (119905)]119894 = 1 119899

(15)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901119894 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (t)]119894 = 1 119899

(16)





4 Complexity

t t

ftot

002

004

006

008

002

004

006

008

010

012

20 30 40 5010 20 30 40 5010

fi


q = 03 q = 048

= 1

= 2

= 3

= 5

Assortative


= 4

DIFFUSION TIME

BASS q PARAMETER

2 4 6 8 10

40

45

50

55

60







Complexity 5




119875 (119896) = 1198881198963 (17)



119875 (ℎ | 119896) = ℎ119875 (ℎ)⟨ℎ⟩ (18)


119896119899119899 (119896) = ⟨ℎ2⟩⟨ℎ⟩ (19)






119875 (ℎ | 119896) = 1(119862119898119886119909 minus 1)1198751 (ℎ | 119896) for ℎ 119896 = 1 119899 (21)



(22)




4 Discussion



6 Complexity

k_nn

k20 40 60 80 100

10

20

30

40

50

60

70


k

k_nn

20 40 60 80 100

1115

1120

1125

1130











Complexity 7

2 4 6 8 10n1000

minus0025

minus0020

minus0015

minus0010

minus0005

r


0 20 40 60 80 100k

36

37

38

39

40

k_nn









⟨1198962⟩ = 119899sum119896=1

119896119875 (119896)119870 (119896 119899) (25)

8 Complexity

k

0 200 400 600 800 1000

56

58

60

62

64knn



119896=1

1198962 4119896 (119896 + 1) (119896 + 2) = 4 ln (119899) + 119900 (119899) (26)



119896119875 (119896)119870 (119896 119899) = 119899sum119896=1

4(119896 + 1) (119896 + 2)119870 (119896 119899) (27)






119889119865119894 (119905)119889119905 = [1 minus 119865119894 (119905)](119901 + 119902 119873sum119895=1

119886119894119895119865119895 (119905)) 119894 = 1 119873

(28)


Complexity 9










10 Complexity

BASS q PARAMETER


Assortative


BA

q = 048q = 03

2 4 6 8 10

4

5

6

7

8

9

10


7 Conclusions






Appendix




2 119899sum119895=1

1119895120574 minusinfinsum119895=1

11198952 ge 2 minus 1205876 gt 0 (A2)


is equal to

Complexity 11



(sum119899119895=1 (1119895120574))2le 1

(A3)


(119896119889 + ℎ119889)sum119899119895=1 (11198952+119889)1198962+119889ℎ2+119889sum119899119895=1 (11198952) (sum119899119895=1 (11198952+119889))2

le (119896119889 + ℎ119889)119896119889ℎ119889 11198962ℎ2 le 1(A4)

Data Availability




Acknowledgments


References






























12 Complexity













1 Introduction






2 Complexity








119877 (119905) = 119873119905=0 minus 119891 (119905)119873119905=0 (1)







Complexity 3


119875 (120585119896119895 (119860120578119894)) =120585119895 (119860120578119894)

sum119872119895=1 120585119895 (119860120578119894)(2)



120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (3)


120585119895 (119860120578119894) =119876

sum119896=1

120585119895119896 (119860120578119894) (4)


=


(5)







120585119895 (119860120578119894) (119905 + 120591) = 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (6)




120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120574120585119895 (119860120578119894) (7)





4 Complexity




Yes



i=i+1

i=1






doubling strategy

End

Start


No

Yes

NltN1 NgtN2

N1⩽N⩽N2



Rout1

Rout1

Rout

x1

x2

xm



119864 (120578119894119895 1205781015840119902119896 120572119895 120573119896) = 12119875

sum119901=1

119892

sum119896=1

(119877119901119896minus 119877119901119900119906119905119896)






Complexity 5



Data normalization











119910 = 05 (119910119894 + 1) (119910max minus 119910min) + 119910min (10)









119901 = int10119901ℎ( 119901119899119894) 119889119901 (11)

119891119894 = 119899119894 (1 minus 119901) (12)

6 Complexity



1 293 40 3 02 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 010 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 116 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 128 308 50 20 329 308 55 2 030 308 55 5 131 308 55 15 232 308 55 20 2





119901119894 =int119901119894+1119901119895

119901119891119895+1+1 (1 minus 119901)119904119895+1 119889119901int119901119894+1119901119895

119901119891119895+1 (1 minus 119901)119904119895+1 119889119901 (13)



Complexity 7







ants i=1goto begin


return resulte












8 Complexity



1 293 40 3 07142862 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 071428610 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 216 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 228 308 50 20 329 308 55 2 071428630 308 55 5 131 308 55 15 232 308 55 20 2




119872119878119864 = 1119873119873

sum119896=1

10038161003816100381610038161003816119877119896 minus 119900119906119905119896 10038161003816100381610038161003816 (14)

119872119860119875119864 = 1119873119873

sum119896=1

1003816100381610038161003816100381610038161003816100381610038161003816119877119896 minus 119900119906119905119896

1198771198961003816100381610038161003816100381610038161003816100381610038161003816times 100 (15)



Complexity 9


Group [NY]- 1

Group [NY]-1

Group [NY]-1

Group [NY]-1

Group 2[NY]

Group (Y-1) [NY]

Group [NY]

Group Y[NY]

1

N

Train Test

Group (YY









10 Complexity











6 Conclusion


Complexity 11

IACO-BPACO-BP

PSO-BPBP

0

02

04

06

08

1

12

14

MSE



IACO-BPACO-BPPSO-BP

0001002003004005006007008

MSE



065

07

08

09

1

Relia

bilit

y

IACO-BPACO-BPPSO-BP




12 Complexity


Data Availability




Acknowledgments


References

























Complexity 13













1 Introduction









2 Complexity










2 Literature Review











Complexity 3


(GO)


(EVM)


(EI)


(IC)


(BGI)















4 Complexity


















Complexity 5


















6 Complexity





1)


2)



1)


2)


3)



1)


2)



1)


2)



1)


2)



1)


2)


3)





4 Results








Complexity 7








Compliance(ECC)


(EVM)


(GAN)

GAN1 GAN2

ECC2

EI2

EI1 Environmental

Impact

(EI)

Innovation and

Competitiveness

(IC)

Green

Opportunities

(GO)

Balanced Green

Investment

(BGI)

IC2

IC3

IC1

GO2

GO1

BGI2

BGI1

ECC1

ECC3

EVM2EVM1

GO3


8 Complexity






p-Value Supported






(ECC)0487

(EVM)

0779(GAN)

GAN1 GAN2

0888

07770780

ECC2

EI2

EI10532(EI)

0356(IC)

0598(GO)

0609(BGI)

0772

IC2

IC3

IC1

GO2

GO3

GO1

BGI2

BGI1

ECC1

ECC3

0697

EVM2EVM1

0815

0887

0852

0798

08180729

0867

0597

0730

0773

0417

09790740

07820844 0882

0539

0793 0876







Complexity 9















10 Complexity





n=54k=2 (31)

n=51k=3 (307)

n=47MGAk1 vs k2

MGAk1 vs k3

MGAk2 vs k3













Complexity 11














0

5

10

15

20

25

minus25

0minus2

25

minus20

0minus1

75

minus15

0minus1

25

minus10

0minus0

75

minus05

0minus0

25

000

025

050

075

100

125

150

175

200

225

250

275

300

GAN









12 Complexity

000 010 020 030 040 050 060 070 080 090 100




EVMECC

EVMECC

EVMECC

0

20

40

60

80

100

GA

N

010 020 030 040 050 060 070 080 090 100000Total-Effects

Total-Effects

Total-Effects

0

20

40

60

80

100

GA

N

000 010 020 030 040 050 060 070 080 090 1000

20

40

60

80

100

GA

N




5 Discussion













Complexity 13


5





6 EVM997888rarr

EI 120573 = 0730 p value le 0001)



6 Conclusion













14 Complexity



Data Availability






Acknowledgments


References




























Complexity 15




































16 Complexity


































Complexity 17














1 Introduction





2 Complexity






Complexity 3

50

40

30

20

10

15

20


1 5 10Time

Am

plitu

de


10864


minus10

2

2

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

TimeAm

plitu

de




2 Preliminaries





119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 [119905] 119896 = 2 119870 (1)



119909119894 [119905] = 119909 [119905] + 119908119894 [119905] (2)



119868119872119865119896 = 1119873119873sum119894=1

119868119872119865119894119896 [119905] (3)

4 Complexity



1198681198721198652 = 1119873119873sum119894=1

1198641 (1199031 [119905] + 12057611198641 (119908119894 [119905])) (4)

1199032 [119905] = 1199031 [119905] minus 1198681198721198652 (5)



119868119872119865119896 = 1119873119873sum119894=1


119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 (7)


R [119905] = x [119905] minus 119870sum119896=1

119868119872119865119896 (8)



1199031 [119905] = 1119873119873sum119894=1

119872(119909119894 [119905]) (9)


1198681198721198651 = 119909 [119905] minus 1199031 [119905] (10)


119903119896 [119905] = 1119873119873sum119894=1

119872(119903119896minus1 [119905] + 120576119896minus1119864k (119908119894 [119905])) (11)

119868119872119865119896 = 119903119896minus1 [119905] minus 119903119896 [119905] (12)



119910119894 = 0 (119909119894) = 119870sum119896=1

119891119896 (119909119894) (13)




119871 (0) = 119899sum119894=1

119897 (119910119894 119910119894) + 119870sum119896=1

Ω(119891119896) (14)

Ω(119891119896) = 120574119879 + 12120572 1205962 (15)



119871119905 = 119899sum119894=1

119897 (119910119894 119910119894(119905minus1) + 119891119905 (119909119894)) + Ω (119891119905) (16)



119892119894 = 120597119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (17)

Complexity 5

ℎ119894 = 1205972119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (18)

119871 119905 = 119899sum119894=1

[119897 (119910119894 119910119894(119905minus1)) + 119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)]+ Ω (119891119905)

(19)


(119905) = 119899sum119894=1

[119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)] + 120574119879 + 12120572119879sum119895=1

1205962119895= 119879sum119895=1

[[(sum119894isin119868119895

119892119894)120596119895 + 12 (sum119894isin119868119895

ℎ119894 + 119886)1205962119895]] + 120574119879(20)




(119905) (119902) = minus12119879sum119895=1



119866119886119894119899= 12 [[[












6 Complexity

X[n]

CEEMDAN



Decomposition


Ensemble

XGBoost

x[n]

R[n]IMF1IMF2

IMF3IMFk

k

i=1IMFi +

R[n]sum



Date

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Pric

es (U

SDb

arre

l)




1199091015840119905 = 119909119905 minus 120583120590 (25)


Complexity 7


RMSE = radic 1119873119873sum119894=1

(119910119905 minus 119910119905)2 (26)

MAE = 1119873 ( 119873sum119894=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816) (27)

Dstat = sum119873119894=2 119889119894119873 minus 1 119889119894 =


(28)





119871119895119905 = 1119879ℎ+119879sum119905=ℎ+1



119889119895119896119905 = 119871119895119905 minus 119871119896119905 (30)





1198670 119864 (119889119895119896119905) = 0 forall119898119895 119898119896 isin 119872 sub M0 (32)




119879119877 = max119895119896isin119872

1003816100381610038161003816100381611990511989511989610038161003816100381610038161003816 (33)

119879119878119876 = max119895119896isinM

1199052119895119896 (34)

119905119895119896 = 119889119895119896radicvar (119889119895119896) (35)

119889119895119896 = 1119879ℎ+119879sum119905=ℎ+1

119889119895119896119905 (36)



8 Complexity











1

XGBOOST 12640 09481 04827SVR 12899 09651 04826FNN 13439 09994 04837

ARIMA 12692 09520 04883

3

XGBOOST 20963 16159 04839SVR 22444 17258 05080FNN 21503 16512 04837

ARIMA 21056 16177 04901

6

XGBOOST 29269 22945 05158SVR 31048 24308 05183FNN 30803 24008 05028

ARIMA 29320 22912 05151




Complexity 9

minus505

IMF1

minus4minus2

024

IMF2

minus505

IMF3

minus505

IMF4

minus4minus2

024

IMF5

minus505

IMF6

minus200

20

IMF7

minus100

1020

IMF8

minus100

10

IMF9

minus20minus10

010

IMF1

0

minus200

20

IMF1

1


Resid

ue







10 Complexity

minus505

IMF1

minus505

IMF2

minus505

IMF3

minus505

IMF4

minus505

IMF5

minus100

10

IMF6

minus100

10

IMF7

minus20

0

20

IMF8

minus200

20

IMF9

minus200

20

IMF1

0


Resid

ue







Complexity 11



1



EEMD-FNN 26835 19932 07361

3



EEMD-FNN 12046 08637 06959

6



EEMD-FNN 27786 20495 06337








12 Complexity


CEEMDAN-XGBOOST

EEMD-XGBOOST

CEEMDAN-SVR

EEMD-SVR

CEEMDAN-FNN

EEMD-FNN



EEMD-FNN 00726 00556 86135e-09 00007 81427e-196 1








5 Conclusions



Data Availability


Complexity 13

RMSEMAEDstat

Dsta

t

RMSE

MA

E

070

065

060

050

055

045

040

035

03010 25 50 500 750 1000 75 100 250

088

087

086

085

084

083

082

081

080



RMSEMAEDstat

Dsta

t

RMSE

MA

E

2 4 6 8 10lag order

088

086

084

082

080

078

076

074

12

10

08

06

04


RMSEMAEDstat

Dsta

t


088

086

084

082

080

078

076

074

10

08

09

06

07

04

05

03

001 002 003 004 005 006 007

RMSE

MA

E


14 Complexity



Acknowledgments


References






























Complexity 15



























1 Introduction




2 Complexity




Complexity 3

Decompose-

D-step


WAEMD-based method


CEEMDAN


step

IMF1

ARMA

RGMDH

RBFNN

IMF2 IMFl

ARMA

RGMDH

RBFNN

ARMA

RGMDH

RBFNN

P-step

Select the best

Predicted IMF

Component




2 Proposed Methods








4 Complexity



119886119895119896 =2119873minus119895minus1

sum119896=0


and

119889119895119896 =119869

sum119895=1


sum119896=0



1198891015840119895119896 =

119889119895119896 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(3)

and

1198891015840119895119896 =

sgn (119889119895119896) (1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 minus 119879119895)

1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(4)


119909 (119905) =2119873minus119895minus1

sum119896=0


+119869

sum119895=1


sum119896=0


(5)






ℎ (119905) = 119909 (119905) minus 119898 (119905) (6)




119909 (119905) =119898

sum119894

ℎ119894 (119905) + 119903 (119905) (7)


119909 (119905) =119898minus2

sum119894=1

ℎ119894 (119905) +119898

sum119894=119898minus2

ℎ119894 (119905) + 119903 (119905) (8)



Complexity 5









23 P-Step






Transfer Functions





119910119905 = 119886 +119903

sum119894=1

119887119894119909119894 +119903

sum119894=1

119903

sum119895=1

119888119894119895119909119894119909119895 (10)



6 Complexity


Selected

Unselected

８1

８2

８3

８4


Input X Output Y

x(1)

x(2)

x(n)

K(X )

K(X )

K(X )

Bias b



0119894 (119909) = 11 + exp (119887119879119909 minus 1198870) (11)



Complexity 7

AFGHANISTAN

INDIA

Arabian Sea

Sindh

Punjab

Kabul R










119872119877119864 = 1119899sum119899119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816

119891 (119905)(12)

119872119860119864 = 1119899119899

sum119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816 (13)

and

119872119878119864 = 1119899119899

sum119905=1

(119891 (119905) minus 119891 (119905))2 (14)


4 Results


8 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow








Complexity 9



EMD 805931 873925 39118 01275 367636Wavelet 802267 861632 38188 00987 229626




minus20

020

minus40

24

6minus1

00

1020

minus30

minus10

10minus2

00

20

0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200

minus20

020

minus20

020

minus60

040

minus100

050

minus100

010

0

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

15

minus50

510

minus50

510

minus10

05

minus15

minus55

1030

50

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time Time Time Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual




10 Complexity

minus20

020

minus40

24

6minus1

00

1020

minus20

020

minus20

020

0 200 400 600 800 1200

minus20

010

minus20

020

minus60

minus20

2060

minus100

050

minus100

010

0

0 200 400 600 800 1200

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

0 200 400 600 800 1200

minus50

5minus5

05

minus10

05

10minus1

5minus5

510

3050

0 200 400 600 800 1200

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time









Complexity 11







WA-RGMDH 26208 00773 377954WA-RBFN 98608 07714 1807443




WA-RGMDH 20794 00729 210612WA-RBFN 16565 00997 133554




WA-RGMDH 36147 00943 816493WA-RBFN 41424 02757 476184





5 Conclusion


12 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl





2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow





Complexity 13


Data Availability




Acknowledgments


References



























14 Complexity






















1 Introduction





2 Complexity








Complexity 3



2 Background









119891 (x) = 12 (1198911 (x) + 1198912 (x))= 12119870 (x119879A119879) (1205961 + 1205962) + 12 (1198871 + 1198872)

(3)


min1205961 1198871120585


and

min1205962 1198872120574



max120572


st 0e le 120572 le 1198881e(6)

4 Complexity

and

max120573


st 0e le 120573 le 1198882e(7)


[12059611198871 ] = (G119879G)minus1G119879 (g minus 120572) (8)

and

[12059621198872 ] = (G119879G)minus1G119879 (h + 120573) (9)




min1205961 11988711205761120585

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus (119870 (AA119879)1205961 + 1198871e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198881V11205761+ 1119897 1198881e119879120585


(10)

and

min1205962 11988721205762 120574



(11)


min1205961 11988711205851205761


(12)

and

min1205962 1198872120585

lowast 1205762


(13)



Complexity 5



(14)



(15)


+ 11988811198871 = 0 (16)

120597119871120597120585 = 1198882D2 minus 120572 minus 120573 = 0 (17)

1205971198711205971205761 = 1198882V1 minus e119879120572 minus 120574 = 0 (18)



120572119879 (y minus (119870 (AA119879)1205961 + 1198871e) + 1205761e + 120585) = 0120572 ge 0e

120573119879120585 = 0120573 ge 0e

1205741205761 = 0 120574 ge 0

(19)


minus [[119870(AA119879)119879

e119879]]D1 (y minus (119870 (AA119879)1205961 + 1198871e))

+ [[119870(AA119879)119879

e119879]]120572 + 1198881 [

12059611198871 ] = 0(20)


minusH119879D1y + (H119879D1H + 1198881I) u1 +H119879120572 = 0 (21)




e119879120572 le 1198882V1 0e le 120572 le 1198882D2 (23)


min120572

12120572119879H (H119879D1H + 1198881I)minus1H119879120572 + y119879120572


(24)


min120578


119904119905 0e le 120578 le 1198884D2 e119879120578 le 1198884V2(25)



and

[12059621198872 ] = (H119879D1H + 1198883I)minus1H119879 (D1y + 120578) (27)



6 Complexity



Sl119896 (119899) = 119897119896minus1sum119898=1

120601 (119898 minus 2119899) Sl119896minus1 (119898) (28)

Sh119896 (119899) = 119897119896minus1sum119898=1

120593 (119898 minus 2119899) Sl119896minus1 (119898) (29)


Sl119896 (119899) = P sdot Sl119896minus1 (119898) (30)

Sh119896 (119899) = Q sdot Sl119896minus1 (119898) (31)




S

Sh1

Sl1Sh2

Sl2

Slk

Shk


SlowastShlowastk-1

Sllowastk-1

Sllowast1

Sllowastk

Shlowastk

Shlowast

1



119889119894 = exp(minus 11990321198942120590lowast2) 119894 = 1 2 119897 (33)



Complexity 7

WTWTSVRVM


R2 +

WTWTSVRCT

Prediction Model

R1 +

+

-

+

-

V

W

x1

x7

CP

TP

Tr

CrCgTg







119891119862119879 (x) = 12119870 (x119879A119879) (1205961 + 1205962)119879 + 12 (1198871 + 1198872) (34)


8 Complexity














119891119881119882 (x) = 12119870 (x119879A119879) (1205963 + 1205964)119879 + 12 (1198873 + 1198874) (35)





Complexity 9








119877119872119878119864 = radic 1119899119899sum119894=1

(119910119894 minus 119910119894)2 (36)

119872119860119864 = 1119899119899sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (37)








119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 012) (41)

119910119894 = sin 119909119894119909119894 + 120579119894119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 022) (42)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 01] (43)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 02] (44)


10 Complexity


002040608

112

Outlier

OutlierOutlier

Outlier









100381610038161003816100381610038161003816100381610038162(10038161003816100381610038161003816100381610038161003816x minus x119894100381610038161003816100381610038161003816100381610038162 + 120590)




Complexity 11



Type A



Type B



Type C



Type D





Type A



Type B



Type C



Type D





12 Complexity

1670 1675 1680 1685 1690 1695 17001670

1675

1680

1685

1690

1695

1700WTWTSVR

plusmn

125 13 135 14 145 15 155 16125

13

135

14

145

15

155

16 x WTWTSVR

40 45 50 55 60 65 70

WTWTSVR

Actual value (tons)

40

45

50

55

60

65

70

Pred

icte

d va

lue (

tons

)

plusmn55t

003 0035 004 0045 005003

0035

004

0045

005WTWTSVR

Actual value ()

Pred

icte

d va

lue (

)

plusmn000510

∘CPred

icte

d va

lue (

∘C)

Actual value (∘C)

Pred

icte

d va

lue (

NＧ

3)


plusmn800NＧ3

104

x 104








Complexity 13

WTWTSVR

Hit rate is 92

plusmn0005

WTWTSVR

Hit rate is 96

plusmn


5

10

15WTWTSVR

Freq

uenc

y

Hit rate is 90

0

5

10

15WTWTSVR

Freq

uenc

y

plusmn55t

Hit rate is 88

0

5

10

15Fr

eque

ncy

0

5

10

15

Freq

uenc

y




plusmn800NＧ3



10∘C










14 Complexity








0 5 10 15 20 25 30 35 40 45 50Heat


Ｒ 104

125

13

135

14

145

15

155

16

Oxy

gen

volu

me (

NＧ

3)




35

40

45

50

55

60

65

70W

eigh

t of a

uxili

ary

raw

mat

eria

ls (to

ns)

5 10 15 20 25 30 35 40 45 500Heat




Complexity 15




5 Conclusion


Data Availability




Acknowledgments




References










16 Complexity



















Editorial Board




Contents
















1 Introduction












2 Complexity


3 Hybrid Models















Complexity 3







10 Conclusion






References

















1 Introduction





2 Complexity







Complexity 3





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (1)


120576푡 = sum퐴푅퐸119905(푖)gt훿

119863푡 (119894) (2)



(3)





119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119865푡 (119909푖) minus 119910푖119910푖

100381610038161003816100381610038161003816100381610038161003816 (5)


119864푘 = sum퐴푅퐸119896(푖)gt훿

119863푡 (119894) (6)

4 Complexity




119860119877119864푡 (119894) = 100381610038161003816100381610038161003816100381610038161003816119891 (119909푖) minus 119910푖119910푖











3 Experiments


Complexity 5





1198902 (119894))12 (8)






accuracy = 119873푠119873푤 (9)




6 Complexity







Drying slurry

size

trough


2 4 6 8 10 12 1401

02

03

04

05

06

The number of ELMs

TrainingTest

RMSE

Mea

n



2 4 6 8 10 12 1403

035

04

045

05

055

06

The number of ELMs


RMSE

Mea

n







Complexity 7

0 5 10 15 20 25 30 35 40 45

times


10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e



0 5 10 15 20 25 30 35 40 45

times

10

15

20

Sizi

ng p

erce

ntag

e







4 Conclusions


8 Complexity





S1 sim S8 S10 S11 S907386 00868 03773


S1 sim S7 S9 sim S10 S11 S808405 00772 02962


S1 sim S6 S8 sim S10 S11 S70936 00655 04255


S1 sim S5 S7 sim S10 S11 S607342 00618 03451


S1 simS4S6 simS10 S11 S508533 00872 03546


S1 sim S3 S5 sim S10 S11 S407752 00749 03249


S1 sim S2 S4 sim S10 S11 S308430 00823 03281


S1 S3 sim S10 S11 S209127 01104 04302


S2 sim S10 S11 S107905 00910 02245

AdaBoostR 05045 00436 01773ILES 03145 00191 00756




x2x4)2 + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]


x1) + 120575 x1 sim 119880[0 100] x2 sim 119880[40120587 560120587]




Complexity 9



Friedman1ELM 08278 07662 47880

AdaBoostR 07987 06536 38817ILES 03099 03073 27132

Friedman2ELM 06338 09975 179870

AdaBoostR 04190 04495 73544ILES 02079 03683 72275

Friedman3ELM 11409 14314 149348

AdaBoostR 08959 13461 115991ILES 04624 03930 52719

0 10 20 30 40 50 60 70 80 90 100

times

80

60

40

20

minus20

0

y

minus40



y


2500

2000

1500

1000

500

0

minus2500

minus2000

minus1500

minus1000

minus500

0 10 20 30 40 50 60 70 80 90 100

times


10 Complexity

y

0 10 20 30 40 50 60 70 80 90 100

times


15

10

5

0

minus5

minus10



Appendix




120573푖119892푖 (119909푗) = 푁sum푖=1


(A1)


H120573 = T (A2)

where

H = [[[[[


]]]]]푁times푁

(A3)

120573 = [[[[[

120573푇1120573푇푁

]]]]]푁times푚

(A4)

and

T = [[[[[

119905푇1119905푇푁]]]]]푁times푚

(A5)


Complexity 11


1006704120573 = HdaggerT (A6)







Data Availability




Acknowledgments


References






















12 Complexity












1 Introduction





2 Complexity


Comminution



Flotation

Drying

Anodes (995 Cu)


Drying


smelting

Matte (50-70Cu)

Converting

Blister Cu (99 Cu)

Electrorefining

Cathodes (9999 Cu)








Complexity 3





2 Related Work





3 Methodology



4 Complexity


AUTHOR RESEARCH
















Complexity 5


0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012


05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012




2012 2013 2014 2015 2016 2017



2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012


2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)




6 Complexity













Complexity 7




119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000












8 Complexity

















Complexity 9


119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000














10 Complexity











6 Conclusions





Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014 2016 2018


2004 2006 2008 2010 2012 2014


2004 2006 2008 2010 2012 2014456789


Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018


405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011








12 Complexity



Appendix

A Models


119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)






E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)



119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)










119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13


B Model Calibration



(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)


119898 = 1119899119899sum119905=1


119904 = radic 1119899 minus 1119899sum119894=1


119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)


119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)


= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)



119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)


0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)


120572 lt 4120573(1 + 120573)2 (B14)


14 Complexity





119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)


119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)



119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)



119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability






References






Complexity 15








































16 Complexity





















1 Introduction






2 Complexity









119896119899119899 (119896) = 119899sumℎ=1

ℎ119875 (ℎ | 119896) (1)


119875 (119896) ge 0and 119875 (ℎ | 119896) ge 0 (2)


119899sum119896=1

119875 (119896) = 1and119899sumℎ=1

119875 (ℎ | 119896) = 1(3)


ℎ119875 (119896 | ℎ) 119875 (ℎ) = 119896119875 (ℎ | 119896) 119875 (119896)forallℎ 119896 = 119894 = 1 119899 (4)


119902119896 = (119896 + 1)sum119899119895=1 119895119875 (119895)119875 (119896 + 1) (5)


119903 = 11205902119902119899minus1sum119896ℎ=0

119896ℎ (119890119896ℎ minus 119902119896119902ℎ) (6)


1198962119902119896 minus ( 119899sum119896=1

119896119902119896)2 (7)


119875 (ℎ | 119896) = 119890119896ℎsum119899119895=1 119890119896119895 forallℎ 119896 = 119894 = 1 119899 (8)


119890ℎ119896 = 119875 (ℎ | 119896) 119896119875 (119896)sum119899119895=1 119895119875 (119895) (9)

Complexity 3



119875 (119896) = 2120573 (120573 + 1)119896 (119896 + 1) (119896 + 2) (10)




) (11)


119861119898119895 = 119898119895 (119898 minus 119895) (12)


119875119899 (ℎ | 119896) = 119875 (ℎ | 119896)119862119896 (13)





119889119865 (119905)119889119905 = [1 minus 119865 (119905)] [119901 + 119902119865 (119905)] (14)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (119905)]119894 = 1 119899

(15)


119889119866119894 (119905)119889119905 = [1 minus 119866119894 (119905)] [119901119894 + 119894119902 119899sumℎ=1

119875 (ℎ | 119894) 119866ℎ (t)]119894 = 1 119899

(16)





4 Complexity

t t

ftot

002

004

006

008

002

004

006

008

010

012

20 30 40 5010 20 30 40 5010

fi


q = 03 q = 048

= 1

= 2

= 3

= 5

Assortative


= 4

DIFFUSION TIME

BASS q PARAMETER

2 4 6 8 10

40

45

50

55

60







Complexity 5




119875 (119896) = 1198881198963 (17)



119875 (ℎ | 119896) = ℎ119875 (ℎ)⟨ℎ⟩ (18)


119896119899119899 (119896) = ⟨ℎ2⟩⟨ℎ⟩ (19)






119875 (ℎ | 119896) = 1(119862119898119886119909 minus 1)1198751 (ℎ | 119896) for ℎ 119896 = 1 119899 (21)



(22)




4 Discussion



6 Complexity

k_nn

k20 40 60 80 100

10

20

30

40

50

60

70


k

k_nn

20 40 60 80 100

1115

1120

1125

1130











Complexity 7

2 4 6 8 10n1000

minus0025

minus0020

minus0015

minus0010

minus0005

r


0 20 40 60 80 100k

36

37

38

39

40

k_nn









⟨1198962⟩ = 119899sum119896=1

119896119875 (119896)119870 (119896 119899) (25)

8 Complexity

k

0 200 400 600 800 1000

56

58

60

62

64knn



119896=1

1198962 4119896 (119896 + 1) (119896 + 2) = 4 ln (119899) + 119900 (119899) (26)



119896119875 (119896)119870 (119896 119899) = 119899sum119896=1

4(119896 + 1) (119896 + 2)119870 (119896 119899) (27)






119889119865119894 (119905)119889119905 = [1 minus 119865119894 (119905)](119901 + 119902 119873sum119895=1

119886119894119895119865119895 (119905)) 119894 = 1 119873

(28)


Complexity 9










10 Complexity

BASS q PARAMETER


Assortative


BA

q = 048q = 03

2 4 6 8 10

4

5

6

7

8

9

10


7 Conclusions






Appendix




2 119899sum119895=1

1119895120574 minusinfinsum119895=1

11198952 ge 2 minus 1205876 gt 0 (A2)


is equal to

Complexity 11



(sum119899119895=1 (1119895120574))2le 1

(A3)


(119896119889 + ℎ119889)sum119899119895=1 (11198952+119889)1198962+119889ℎ2+119889sum119899119895=1 (11198952) (sum119899119895=1 (11198952+119889))2

le (119896119889 + ℎ119889)119896119889ℎ119889 11198962ℎ2 le 1(A4)

Data Availability




Acknowledgments


References






























12 Complexity













1 Introduction






2 Complexity








119877 (119905) = 119873119905=0 minus 119891 (119905)119873119905=0 (1)







Complexity 3


119875 (120585119896119895 (119860120578119894)) =120585119895 (119860120578119894)

sum119872119895=1 120585119895 (119860120578119894)(2)



120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (3)


120585119895 (119860120578119894) =119876

sum119896=1

120585119895119896 (119860120578119894) (4)


=


(5)







120585119895 (119860120578119894) (119905 + 120591) = 120585119895 (119860120578119894) (119905) + 120585119895 (119860120578119894) (6)




120585119895 (119860120578119894) (119905 + 120591) = (1 minus 120588) 120585119895 (119860120578119894) (119905) + 120574120585119895 (119860120578119894) (7)





4 Complexity




Yes



i=i+1

i=1






doubling strategy

End

Start


No

Yes

NltN1 NgtN2

N1⩽N⩽N2



Rout1

Rout1

Rout

x1

x2

xm



119864 (120578119894119895 1205781015840119902119896 120572119895 120573119896) = 12119875

sum119901=1

119892

sum119896=1

(119877119901119896minus 119877119901119900119906119905119896)






Complexity 5



Data normalization











119910 = 05 (119910119894 + 1) (119910max minus 119910min) + 119910min (10)









119901 = int10119901ℎ( 119901119899119894) 119889119901 (11)

119891119894 = 119899119894 (1 minus 119901) (12)

6 Complexity



1 293 40 3 02 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 010 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 116 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 128 308 50 20 329 308 55 2 030 308 55 5 131 308 55 15 232 308 55 20 2





119901119894 =int119901119894+1119901119895

119901119891119895+1+1 (1 minus 119901)119904119895+1 119889119901int119901119894+1119901119895

119901119891119895+1 (1 minus 119901)119904119895+1 119889119901 (13)



Complexity 7







ants i=1goto begin


return resulte












8 Complexity



1 293 40 3 07142862 293 40 5 13 293 40 8 14 293 40 12 25 293 45 5 16 293 45 8 17 293 45 15 28 293 45 20 29 298 35 3 071428610 298 35 6 111 298 35 12 212 298 35 25 313 298 40 5 114 298 40 10 215 298 40 12 216 298 40 20 317 303 40 7 118 303 40 10 119 303 40 12 120 303 40 20 221 303 45 5 122 303 45 10 123 303 45 15 224 303 45 25 325 308 50 8 126 308 50 12 227 308 50 15 228 308 50 20 329 308 55 2 071428630 308 55 5 131 308 55 15 232 308 55 20 2




119872119878119864 = 1119873119873

sum119896=1

10038161003816100381610038161003816119877119896 minus 119900119906119905119896 10038161003816100381610038161003816 (14)

119872119860119875119864 = 1119873119873

sum119896=1

1003816100381610038161003816100381610038161003816100381610038161003816119877119896 minus 119900119906119905119896

1198771198961003816100381610038161003816100381610038161003816100381610038161003816times 100 (15)



Complexity 9


Group [NY]- 1

Group [NY]-1

Group [NY]-1

Group [NY]-1

Group 2[NY]

Group (Y-1) [NY]

Group [NY]

Group Y[NY]

1

N

Train Test

Group (YY









10 Complexity











6 Conclusion


Complexity 11

IACO-BPACO-BP

PSO-BPBP

0

02

04

06

08

1

12

14

MSE



IACO-BPACO-BPPSO-BP

0001002003004005006007008

MSE



065

07

08

09

1

Relia

bilit

y

IACO-BPACO-BPPSO-BP




12 Complexity


Data Availability




Acknowledgments


References

























Complexity 13













1 Introduction









2 Complexity










2 Literature Review











Complexity 3


(GO)


(EVM)


(EI)


(IC)


(BGI)















4 Complexity


















Complexity 5


















6 Complexity





1)


2)



1)


2)


3)



1)


2)



1)


2)



1)


2)



1)


2)


3)





4 Results








Complexity 7








Compliance(ECC)


(EVM)


(GAN)

GAN1 GAN2

ECC2

EI2

EI1 Environmental

Impact

(EI)

Innovation and

Competitiveness

(IC)

Green

Opportunities

(GO)

Balanced Green

Investment

(BGI)

IC2

IC3

IC1

GO2

GO1

BGI2

BGI1

ECC1

ECC3

EVM2EVM1

GO3


8 Complexity






p-Value Supported






(ECC)0487

(EVM)

0779(GAN)

GAN1 GAN2

0888

07770780

ECC2

EI2

EI10532(EI)

0356(IC)

0598(GO)

0609(BGI)

0772

IC2

IC3

IC1

GO2

GO3

GO1

BGI2

BGI1

ECC1

ECC3

0697

EVM2EVM1

0815

0887

0852

0798

08180729

0867

0597

0730

0773

0417

09790740

07820844 0882

0539

0793 0876







Complexity 9















10 Complexity





n=54k=2 (31)

n=51k=3 (307)

n=47MGAk1 vs k2

MGAk1 vs k3

MGAk2 vs k3













Complexity 11














0

5

10

15

20

25

minus25

0minus2

25

minus20

0minus1

75

minus15

0minus1

25

minus10

0minus0

75

minus05

0minus0

25

000

025

050

075

100

125

150

175

200

225

250

275

300

GAN









12 Complexity

000 010 020 030 040 050 060 070 080 090 100




EVMECC

EVMECC

EVMECC

0

20

40

60

80

100

GA

N

010 020 030 040 050 060 070 080 090 100000Total-Effects

Total-Effects

Total-Effects

0

20

40

60

80

100

GA

N

000 010 020 030 040 050 060 070 080 090 1000

20

40

60

80

100

GA

N




5 Discussion













Complexity 13


5





6 EVM997888rarr

EI 120573 = 0730 p value le 0001)



6 Conclusion













14 Complexity



Data Availability






Acknowledgments


References




























Complexity 15




































16 Complexity


































Complexity 17














1 Introduction





2 Complexity






Complexity 3

50

40

30

20

10

15

20


1 5 10Time

Am

plitu

de


10864


minus10

2

2

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000

TimeAm

plitu

de




2 Preliminaries





119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 [119905] 119896 = 2 119870 (1)



119909119894 [119905] = 119909 [119905] + 119908119894 [119905] (2)



119868119872119865119896 = 1119873119873sum119894=1

119868119872119865119894119896 [119905] (3)

4 Complexity



1198681198721198652 = 1119873119873sum119894=1

1198641 (1199031 [119905] + 12057611198641 (119908119894 [119905])) (4)

1199032 [119905] = 1199031 [119905] minus 1198681198721198652 (5)



119868119872119865119896 = 1119873119873sum119894=1


119903119896 [119905] = 119903119896minus1 [119905] minus 119868119872119865119896 (7)


R [119905] = x [119905] minus 119870sum119896=1

119868119872119865119896 (8)



1199031 [119905] = 1119873119873sum119894=1

119872(119909119894 [119905]) (9)


1198681198721198651 = 119909 [119905] minus 1199031 [119905] (10)


119903119896 [119905] = 1119873119873sum119894=1

119872(119903119896minus1 [119905] + 120576119896minus1119864k (119908119894 [119905])) (11)

119868119872119865119896 = 119903119896minus1 [119905] minus 119903119896 [119905] (12)



119910119894 = 0 (119909119894) = 119870sum119896=1

119891119896 (119909119894) (13)




119871 (0) = 119899sum119894=1

119897 (119910119894 119910119894) + 119870sum119896=1

Ω(119891119896) (14)

Ω(119891119896) = 120574119879 + 12120572 1205962 (15)



119871119905 = 119899sum119894=1

119897 (119910119894 119910119894(119905minus1) + 119891119905 (119909119894)) + Ω (119891119905) (16)



119892119894 = 120597119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (17)

Complexity 5

ℎ119894 = 1205972119897 (119910119894 119910119894(119905minus1))119910119894(119905minus1) (18)

119871 119905 = 119899sum119894=1

[119897 (119910119894 119910119894(119905minus1)) + 119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)]+ Ω (119891119905)

(19)


(119905) = 119899sum119894=1

[119892119894119891119905 (119909119894) + 12ℎ1198941198912119905 (119909119894)] + 120574119879 + 12120572119879sum119895=1

1205962119895= 119879sum119895=1

[[(sum119894isin119868119895

119892119894)120596119895 + 12 (sum119894isin119868119895

ℎ119894 + 119886)1205962119895]] + 120574119879(20)




(119905) (119902) = minus12119879sum119895=1



119866119886119894119899= 12 [[[












6 Complexity

X[n]

CEEMDAN



Decomposition


Ensemble

XGBoost

x[n]

R[n]IMF1IMF2

IMF3IMFk

k

i=1IMFi +

R[n]sum



Date

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Pric

es (U

SDb

arre

l)




1199091015840119905 = 119909119905 minus 120583120590 (25)


Complexity 7


RMSE = radic 1119873119873sum119894=1

(119910119905 minus 119910119905)2 (26)

MAE = 1119873 ( 119873sum119894=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816) (27)

Dstat = sum119873119894=2 119889119894119873 minus 1 119889119894 =


(28)





119871119895119905 = 1119879ℎ+119879sum119905=ℎ+1



119889119895119896119905 = 119871119895119905 minus 119871119896119905 (30)





1198670 119864 (119889119895119896119905) = 0 forall119898119895 119898119896 isin 119872 sub M0 (32)




119879119877 = max119895119896isin119872

1003816100381610038161003816100381611990511989511989610038161003816100381610038161003816 (33)

119879119878119876 = max119895119896isinM

1199052119895119896 (34)

119905119895119896 = 119889119895119896radicvar (119889119895119896) (35)

119889119895119896 = 1119879ℎ+119879sum119905=ℎ+1

119889119895119896119905 (36)



8 Complexity











1

XGBOOST 12640 09481 04827SVR 12899 09651 04826FNN 13439 09994 04837

ARIMA 12692 09520 04883

3

XGBOOST 20963 16159 04839SVR 22444 17258 05080FNN 21503 16512 04837

ARIMA 21056 16177 04901

6

XGBOOST 29269 22945 05158SVR 31048 24308 05183FNN 30803 24008 05028

ARIMA 29320 22912 05151




Complexity 9

minus505

IMF1

minus4minus2

024

IMF2

minus505

IMF3

minus505

IMF4

minus4minus2

024

IMF5

minus505

IMF6

minus200

20

IMF7

minus100

1020

IMF8

minus100

10

IMF9

minus20minus10

010

IMF1

0

minus200

20

IMF1

1


Resid

ue







10 Complexity

minus505

IMF1

minus505

IMF2

minus505

IMF3

minus505

IMF4

minus505

IMF5

minus100

10

IMF6

minus100

10

IMF7

minus20

0

20

IMF8

minus200

20

IMF9

minus200

20

IMF1

0


Resid

ue







Complexity 11



1



EEMD-FNN 26835 19932 07361

3



EEMD-FNN 12046 08637 06959

6



EEMD-FNN 27786 20495 06337








12 Complexity


CEEMDAN-XGBOOST

EEMD-XGBOOST

CEEMDAN-SVR

EEMD-SVR

CEEMDAN-FNN

EEMD-FNN



EEMD-FNN 00726 00556 86135e-09 00007 81427e-196 1








5 Conclusions



Data Availability


Complexity 13

RMSEMAEDstat

Dsta

t

RMSE

MA

E

070

065

060

050

055

045

040

035

03010 25 50 500 750 1000 75 100 250

088

087

086

085

084

083

082

081

080



RMSEMAEDstat

Dsta

t

RMSE

MA

E

2 4 6 8 10lag order

088

086

084

082

080

078

076

074

12

10

08

06

04


RMSEMAEDstat

Dsta

t


088

086

084

082

080

078

076

074

10

08

09

06

07

04

05

03

001 002 003 004 005 006 007

RMSE

MA

E


14 Complexity



Acknowledgments


References






























Complexity 15



























1 Introduction




2 Complexity




Complexity 3

Decompose-

D-step


WAEMD-based method


CEEMDAN


step

IMF1

ARMA

RGMDH

RBFNN

IMF2 IMFl

ARMA

RGMDH

RBFNN

ARMA

RGMDH

RBFNN

P-step

Select the best

Predicted IMF

Component




2 Proposed Methods








4 Complexity



119886119895119896 =2119873minus119895minus1

sum119896=0


and

119889119895119896 =119869

sum119895=1


sum119896=0



1198891015840119895119896 =

119889119895119896 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(3)

and

1198891015840119895119896 =

sgn (119889119895119896) (1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 minus 119879119895)

1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 ge 119879119895

0 1003816100381610038161003816100381611988911989511989610038161003816100381610038161003816 lt 119879119895

(4)


119909 (119905) =2119873minus119895minus1

sum119896=0


+119869

sum119895=1


sum119896=0


(5)






ℎ (119905) = 119909 (119905) minus 119898 (119905) (6)




119909 (119905) =119898

sum119894

ℎ119894 (119905) + 119903 (119905) (7)


119909 (119905) =119898minus2

sum119894=1

ℎ119894 (119905) +119898

sum119894=119898minus2

ℎ119894 (119905) + 119903 (119905) (8)



Complexity 5









23 P-Step






Transfer Functions





119910119905 = 119886 +119903

sum119894=1

119887119894119909119894 +119903

sum119894=1

119903

sum119895=1

119888119894119895119909119894119909119895 (10)



6 Complexity


Selected

Unselected

８1

８2

８3

８4


Input X Output Y

x(1)

x(2)

x(n)

K(X )

K(X )

K(X )

Bias b



0119894 (119909) = 11 + exp (119887119879119909 minus 1198870) (11)



Complexity 7

AFGHANISTAN

INDIA

Arabian Sea

Sindh

Punjab

Kabul R










119872119877119864 = 1119899sum119899119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816

119891 (119905)(12)

119872119860119864 = 1119899119899

sum119905=1

10038161003816100381610038161003816119891 (119905) minus 119891 (119905)10038161003816100381610038161003816 (13)

and

119872119878119864 = 1119899119899

sum119905=1

(119891 (119905) minus 119891 (119905))2 (14)


4 Results


8 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow








Complexity 9



EMD 805931 873925 39118 01275 367636Wavelet 802267 861632 38188 00987 229626




minus20

020

minus40

24

6minus1

00

1020

minus30

minus10

10minus2

00

20

0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200 0 200 400 600 800 1200

minus20

020

minus20

020

minus60

040

minus100

050

minus100

010

0

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

15

minus50

510

minus50

510

minus10

05

minus15

minus55

1030

50

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time Time Time Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual




10 Complexity

minus20

020

minus40

24

6minus1

00

1020

minus20

020

minus20

020

0 200 400 600 800 1200

minus20

010

minus20

020

minus60

minus20

2060

minus100

050

minus100

010

0

0 200 400 600 800 1200

minus20

020

minus40

48

minus15

minus55

15minus1

5minus5

515

minus10

05

0 200 400 600 800 1200

minus50

5minus5

05

minus10

05

10minus1

5minus5

510

3050

0 200 400 600 800 1200

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time

IMF

1IM

F 2

IMF

3IM

F 4

IMF

5

Time

IMF

6IM

F 7

IMF

8IM

F 9

Resid

ual

Time









Complexity 11







WA-RGMDH 26208 00773 377954WA-RBFN 98608 07714 1807443




WA-RGMDH 20794 00729 210612WA-RBFN 16565 00997 133554




WA-RGMDH 36147 00943 816493WA-RBFN 41424 02757 476184





5 Conclusion


12 Complexity

2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl





2015 2016 2017 2018

010

020

030

040

0

Time

Indu

s_Ri

ver_

Inflo

w


2015 2016 2017 2018

050

100

150

Time

Jhelu

m_R

iver

_Infl

ow





Complexity 13


Data Availability




Acknowledgments


References



























14 Complexity






















1 Introduction





2 Complexity








Complexity 3



2 Background









119891 (x) = 12 (1198911 (x) + 1198912 (x))= 12119870 (x119879A119879) (1205961 + 1205962) + 12 (1198871 + 1198872)

(3)


min1205961 1198871120585


and

min1205962 1198872120574



max120572


st 0e le 120572 le 1198881e(6)

4 Complexity

and

max120573


st 0e le 120573 le 1198882e(7)


[12059611198871 ] = (G119879G)minus1G119879 (g minus 120572) (8)

and

[12059621198872 ] = (G119879G)minus1G119879 (h + 120573) (9)




min1205961 11988711205761120585

12 1003816100381610038161003816100381610038161003816100381610038161003816y minus (119870 (AA119879)1205961 + 1198871e)10038161003816100381610038161003816100381610038161003816100381610038162 + 1198881V11205761+ 1119897 1198881e119879120585


(10)

and

min1205962 11988721205762 120574



(11)


min1205961 11988711205851205761


(12)

and

min1205962 1198872120585

lowast 1205762


(13)



Complexity 5



(14)



(15)


+ 11988811198871 = 0 (16)

120597119871120597120585 = 1198882D2 minus 120572 minus 120573 = 0 (17)

1205971198711205971205761 = 1198882V1 minus e119879120572 minus 120574 = 0 (18)



120572119879 (y minus (119870 (AA119879)1205961 + 1198871e) + 1205761e + 120585) = 0120572 ge 0e

120573119879120585 = 0120573 ge 0e

1205741205761 = 0 120574 ge 0

(19)


minus [[119870(AA119879)119879

e119879]]D1 (y minus (119870 (AA119879)1205961 + 1198871e))

+ [[119870(AA119879)119879

e119879]]120572 + 1198881 [

12059611198871 ] = 0(20)


minusH119879D1y + (H119879D1H + 1198881I) u1 +H119879120572 = 0 (21)




e119879120572 le 1198882V1 0e le 120572 le 1198882D2 (23)


min120572

12120572119879H (H119879D1H + 1198881I)minus1H119879120572 + y119879120572


(24)


min120578


119904119905 0e le 120578 le 1198884D2 e119879120578 le 1198884V2(25)



and

[12059621198872 ] = (H119879D1H + 1198883I)minus1H119879 (D1y + 120578) (27)



6 Complexity



Sl119896 (119899) = 119897119896minus1sum119898=1

120601 (119898 minus 2119899) Sl119896minus1 (119898) (28)

Sh119896 (119899) = 119897119896minus1sum119898=1

120593 (119898 minus 2119899) Sl119896minus1 (119898) (29)


Sl119896 (119899) = P sdot Sl119896minus1 (119898) (30)

Sh119896 (119899) = Q sdot Sl119896minus1 (119898) (31)




S

Sh1

Sl1Sh2

Sl2

Slk

Shk


SlowastShlowastk-1

Sllowastk-1

Sllowast1

Sllowastk

Shlowastk

Shlowast

1



119889119894 = exp(minus 11990321198942120590lowast2) 119894 = 1 2 119897 (33)



Complexity 7

WTWTSVRVM


R2 +

WTWTSVRCT

Prediction Model

R1 +

+

-

+

-

V

W

x1

x7

CP

TP

Tr

CrCgTg







119891119862119879 (x) = 12119870 (x119879A119879) (1205961 + 1205962)119879 + 12 (1198871 + 1198872) (34)


8 Complexity














119891119881119882 (x) = 12119870 (x119879A119879) (1205963 + 1205964)119879 + 12 (1198873 + 1198874) (35)





Complexity 9








119877119872119878119864 = radic 1119899119899sum119894=1

(119910119894 minus 119910119894)2 (36)

119872119860119864 = 1119899119899sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (37)








119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 012) (41)

119910119894 = sin 119909119894119909119894 + 120579119894119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119873(0 022) (42)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 01] (43)

119910119894 = sin 119909119894119909119894 + 120579119894 119909 sim 119880 [minus4120587 4120587] 120579119894 sim 119880 [0 02] (44)


10 Complexity


002040608

112

Outlier

OutlierOutlier

Outlier









100381610038161003816100381610038161003816100381610038162(10038161003816100381610038161003816100381610038161003816x minus x119894100381610038161003816100381610038161003816100381610038162 + 120590)




Complexity 11



Type A



Type B



Type C



Type D





Type A



Type B



Type C



Type D





12 Complexity

1670 1675 1680 1685 1690 1695 17001670

1675

1680

1685

1690

1695

1700WTWTSVR

plusmn

125 13 135 14 145 15 155 16125

13

135

14

145

15

155

16 x WTWTSVR

40 45 50 55 60 65 70

WTWTSVR

Actual value (tons)

40

45

50

55

60

65

70

Pred

icte

d va

lue (

tons

)

plusmn55t

003 0035 004 0045 005003

0035

004

0045

005WTWTSVR

Actual value ()

Pred

icte

d va

lue (

)

plusmn000510

∘CPred

icte

d va

lue (

∘C)

Actual value (∘C)

Pred

icte

d va

lue (

NＧ

3)


plusmn800NＧ3

104

x 104








Complexity 13

WTWTSVR

Hit rate is 92

plusmn0005

WTWTSVR

Hit rate is 96

plusmn


5

10

15WTWTSVR

Freq

uenc

y

Hit rate is 90

0

5

10

15WTWTSVR

Freq

uenc

y

plusmn55t

Hit rate is 88

0

5

10

15Fr

eque

ncy

0

5

10

15

Freq

uenc

y




plusmn800NＧ3



10∘C










14 Complexity








0 5 10 15 20 25 30 35 40 45 50Heat


Ｒ 104

125

13

135

14

145

15

155

16

Oxy

gen

volu

me (

NＧ

3)




35

40

45

50

55

60

65

70W

eigh

t of a

uxili

ary

raw

mat

eria

ls (to

ns)

5 10 15 20 25 30 35 40 45 500Heat




Complexity 15




5 Conclusion


Data Availability




Acknowledgments




References










16 Complexity

















Documents

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