An EMD-recursive ARIMA method to predict wind speedfor railway strong wind warning system

Hui Liu a,b,n, Hong-qi Tian a, Yan-fei Li a

a Key Laboratory of Traffic Safety on Track of Ministry of Education, School of Traffic and Transportation Engineering, Central South University,Changsha 410075, Hunan, Chinab Institute of Automation, Faculty of Computer Science and Electrical Engineering, University of Rostock, Rostock 18119, Mecklenburg-Vorpommern, Germany

a r t i c l e i n f o

Article history:Received 12 September 2014Received in revised form18 February 2015Accepted 26 February 2015Available online 19 March 2015

Keywords:Qinghai–Tibet railwayStrong windWind speed forecastingWind speed predictionWarning systemEmpirical Mode DecompositionRecursive ARIMANeural networks

a b s t r a c t

To protect running trains against the strong crosswind along Chinese Qinghai–Tibet railway, a strongwind warning system is developed. As one of the most important technologies of the developed system,a new short-term wind speed forecasting method is proposed by adopting the Empirical ModeDecomposition (EMD) and the improved Recursive Autoregressive Integrated Moving Average (RARIMA)model. The proposed forecasting method consists of three computational steps as: (a) use the EMDmethod to decompose the original wind speed data into a group of wind speed sub-layers; (b) build theforecasting models for all the decomposed sub-layers by utilizing the RARIMA algorithm; (c) employ thebuilt RARIMA models to predict the wind speed in the sub-layers; and (d) summarize the predictedresults of the wind speed sub-layers to get the final forecasting results for the original wind speed. Sincethe wind speed forecasting method is proposed for the real-time warning system, the forecastingaccuracy and the time performance of the forecasting computation are both considered. Two experi-ments show that: (a) the proposed method has better forecasting performance than the traditionalAutoregressive Integrated Moving Average (ARIMA) model, the Persistent Random Walk Model (PRWM)and the Back Propagation (BP) neural networks; and (b) the proposed method has satisfactoryperformance in both of the accuracy and the time performance.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

In recent decade, there are several derailment accidents hap-pened in the world caused by strong crosswind. A train consisting ofsix vehicles was derailed between Akita station and Niigata station inUetsu railway line on December 25, 2005 in Japan (Train death tollrises in Japan, 2005). The investigators from the Department ofTraffic Safety of East Japan Railway Company confirmed that the trainderailment was caused by the strong wind along the Uetsu line.Eleven vehicles of a train were blown over in Xinjiang railway onFebruary 28, 2007 in China (Chen et al., 2010; Strong wind topplesChinese train, 2007). The reason of the derailment proposed by theMinistry of Railway of China was also the strong crosswind. Unfortu-nately, similar train derailment accidents also happened in Canada(Freight train derails in southern Saskatchewan, 2014) and USA(Shust et al., 2010).

Aimed at the issue of the train derailment under the strongcrosswind, some important results have been studied and published.A numerical-experimental procedurewas provided for the aerodynamicoptimization of a new train named EMUV250 in terms of behaviors tothe crosswind (Cheli et al., 2010). In the numerical experiment, twodifferent train shapes were proposed by modifying the train's roof andnose. A new framework was proposed for the consideration of theeffects of crosswind on trains based on correcting the current CENmethodology (Baker, 2013). In the framework, an improved methodol-ogy was presented, which can be used for the train authorization andthe route risk analysis. A numerical calculation was conducted toinvestigate the safe domain of a train in crosswind caused by differentattitudes (Cui et al., 2014). A methodology was put forward for thesafety assessment to the risk of a train overturning in strong cross-wind(Andersson et al., 2004). The proposed methodology had been appliedin a high-speed railway named Botniabanan line with a maximumspeed of 250 km/h in the northeast coastal region of Sweden.

Besides the upper important studies, some scientist presentedthat developing railway strong wind warning systems is also a feasi-ble option to protect the running trains (Hoppmann et al., 2002;Kobayashi and Shimamura, 2003; Liu et al., 2009; Pan et al., 2008). Inthese warning systems, the short-termwind speed prediction is one of

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Journal of Wind Engineeringand Industrial Aerodynamics

http://dx.doi.org/10.1016/j.jweia.2015.02.0040167-6105/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author at: Key Laboratory of Traffic Safety on Track of Ministry ofEducation, School of Traffic and Transportation Engineering, Central SouthUniversity, Changsha 410075, Hunan, China.Tel.: þ86 73182655294; fax: þ86 73182656374.

E-mail address: csuliuhui@csu.edu.cn (H. Liu).

J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38

the most important technologies. However, since the wind speedsignal is non-stationary and stochastic, it is difficult to predict itaccurately. To study the internal characters of stochastic wind speedsignals, a simulation algorithm was proposed to generate samplefunctions of a stationary, multivariate stochastic process according toits prescribed cross-spectral density matrix (Deodatis, 1996). Currently,the wind speed forecasting methods for railway applications can beclassified as three types: physical methods, statistical methods andintelligent methods. The physical models adopt the physical para-meters (e.g., terrains, obstacles, pressures and temperatures, etc) toestimate the future three-dimensional railway wind speed. Due to thetime performance of the physical methods, they are suitable for theoff-line estimation but not the real-time decision in the warningsystems. A representatively physical method was proposed by theUniversity of Genoa to study the wind hazard of the Rome–NaplesHigh Speed (HS)/High Capacity (HC) railway line. In this physicalmethods, the wind numerical simulation and the probabilistic assess-ment of the simulated wind speed results along the line werecompleted (Burlando et al., 2010; Freda and Solari, 2010). The secondtype is the statistical methods, which utilize the statistical models todescribe the changing law of the wind speed for the future prediction.A wind speed statistical forecasting model was proposed for the‘Nowcasting’ warning system developed by Deutsche Bahn AG inGermany (Hoppmann et al., 2002). In this method a linear extra-polated algorithmwas used to get the future wind speed prediction bycombining the average values of the historical wind speed data, theestimated gust supplement and the error supplement. Although thestatistical methods are always simple and good at the real-timeperformance, their accuracy can be further improved. The intelligentmethods focusing on the intelligent forecasting models. A wind speedintelligent predicting method was presented using Kalman Filtertheory for the ‘Windas’ system from Japanese East Railway Company(Kobayashi and Shimamura, 2003). Although the intelligent methodscan have better accuracy than the physical and statistical ones,sometimes they have the problems in the computational convergence.From the upper references (Hoppmann et al., 2002; Kobayashi andShimamura, 2003), it can also be found that currently in the strongwind warning systems only the wind speed signals but not the winddirection signals are processed and used to present the warningcommands. The reasons of this condition can be explained as follows:

(a) compared to the wind speed prediction, the wind directionprediction is more difficult. Because the wind direction signals arealways more non-stationary than the wind speed signals; (b) althoughthe wind direction data at a position can be measured conveniently,the real angles between the running trains and the real-time winddirection signals cannot be calculated accurately. Because the trainreal-time postures are dynamic and nonlinear; and (c) it is not difficultfor these warning systems to measure both the wind speed data andthe wind direction data at a large number of wind stations at the sametime, but it can be a challenge for these systems to calculate the windspeed predictions then to propose the critical safety velocities un-der huge different combinations of wind speed and wind directionsynchronously.

In this study, we focus on the wind speed high-accuracy predic-tions for the railway wind warning systems. By considering both theforecasting accuracy and the real-time performance, the statisticalmethods have been selected in this study. A new hybrid statisticalwind speed forecasting method is proposed by combing the EmpiricalMode Decomposition (EMD) and the improved recursive ARIMA(RARIMA) model. In the proposed hybrid method, not only a newRARIMA model is presented to calculate the wind speed predictions,but also it is the first time to adopt the EMD algorithm to process anddecompose the original non-stationary wind speed data. Additionally,a comparison of the forecasting performance made by differentmodels is provided. The comparing models include the proposedEMD-RARIMA model, the standard Autoregressive Integrated MovingAverage (ARIMA) model, the Back Propagation (BP) neural networkand the Persistent Random Walk Model (PRWM). The paper isorganized as follows: Section 2 presents the architecture of the strongwind warning system; Section 3 explains the detailed steps of thewind speed prediction; Section 4 provides the experimental results oftwo cases; and Section 5 concludes this study.

2. Strong wind warning system for Qinghai–Tibet railway

2.1. System architecture

The Chinese Qinghai–Tibet railway is the highest railway line inthe world, which covers a big plateau area from the Geermu city in

Train and Track DatabaseWind Database

Strong Wind Warning System

Wireless network

Client Computers

Train Dispatching Command System Of Qinghai Tibet Railway

Company

Train InformationTrack Information

Wind speedstation

Fibre network

Wind speedstation

Drivers of Running Trains

GSM R (Global System for Mobile Communications

Railway)

NetworkRouter

Network Hub

Fig. 1. Architecture of the strong wind warning system for the Qinghai–Tibet railway.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3828

Qinghai province to the Lhasa city in Tibet province. The lengthand altitude of the Qinghai–Tibet railway are 1142 km and5072 km, respectively. Due to its high altitude, the strong cross-wind phenomenon happens frequently along the Qinghai–Tibetrailway. To protect the safety of the running trains, an intelligentwarning system has been developed, as the system architecturedemonstrated in Fig. 1.

From Fig. 1, it can be seen that: (a) all the real-time informationof trains and tracks are sampled from the Train DispatchingCommand System (TDSC) of Qinghai–Tibet Company and sent intothe Train and Track Database (TTD) of the developed warningsystem; (b) the wind speed data at the dangerous places along theline are measured and stored in the Wind Database (WD) of thewarning system; (c) in the TTD, all the critical speeds of varioustrains under different wind speed and track parameters arecalculated in advance based on the results from the ComputationalFluid Dynamics (CFD) and the wind tunnel experiments; and(d) once the warning system proposes a warning message, themessage will be sent to the corresponding driver of the monitoredtrain automatically by the wireless Global System for MobileCommunications-Railway (GSM-R) technology. The monitoringGUI of the running wind warning system is demonstrated in Fig. 2.

2.2. Network of wind speed measurement

To measure the original wind speed data along the Qinghai–Tibetrailway, 52 wind speed stations are installed, including two types:the wireless stations and the wired stations. The wireless stations usethe GPRS/GSM-R communicating channel and the wired stationsadopt the LAN TCP/IP one. In these wind speed stations, the wind cupanemometers are adopted to sample the raw wind speed data. Thesampling frequency of the wind speed stations is ten times perminute. The positions of those wind speed stations are decided byconsidering four aspects as: (a) the historical derailment accidents;(b) the historical meteorological data; (c) the CFD computational

results; and (d) the railway construction conditions. By consideringthe upper aspects, the mileage positions of those wind speed stationsare selected as illustrated in Fig. 3. As shown in Fig. 3, the blue linerepresents the Qinghai–Tibet railway and the red points are theselected mileage positions of the wind speed stations. As shown inFig. 4, every wind speed station consists of two anemometers, twosolar power plates, two batteries, an on-board Micro-PC and somemechanical structures. The working process of a wireless wind speedstation can be explained as follows: (a) once a wind speed station isturned on, it will connect to the remote wind warning systemautomatically by the GPRS/GSM-R dual redundant channels;(b) once the wireless data transmitting channel is built, the windspeed station will start to measure the wind speed data and sentthem to the remote server synchronously; (c) to insure the measuredwind speed data are credible, an error-checking strategy is proposedbased on the installed two anemometers. If the measured results ofthe two anemometers are significantly different (like 10%), the micro-pc of the wind speed station will terminate the measuring processand sent out an error message; and (d) the sampling frequency ofevery wind speed station can be revised remotely by the server of thewarning system.

Fig. 2. Monitoring GUI of the strong wind warning system for the Qinghai–Tibet railway.

Fig. 3. Mileage positions of the 52 wind speed stations along the Qinghai–Tibetrailway. (For interpretation of the references to color in this figure, the reader isreferred to the web version of this article.)

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38 29

2.3. Application of wind speed predictions

The concept of the application of the proposed wind speedpredictions in the warning system is given in Fig. 5.

As shown in Fig. 5, the working process can be explained asfollows:

(a) The warning system reads all the train dispatching data fromthe Qinghai–Tibet Railway Company (see Fig. 1), so it knowsthe detailed dispatching information of all trains (e.g., thearriving time and the leaving time to a train station, themileage distances between the running trains and theinstalled wind speed stations, the train real-time runningvelocities, etc).

(b) When a train reaches the first train station 1, the warningsystem will start to calculate the wind speed predictions. Inthe warning system, three time parameters ‘t1’, ‘t2’ and ‘t3’ aredefined to manage the predicted wind speed results. Theparameter ‘t1’ is proposed to validate the built forecastingmodel. In the ‘t1’ time, the wind speed predicted results willnot be used to dispatch the trains but only to estimate the

forecasting performance. If the results of a forecasting modelcannot meet the requirements of the accuracy and the com-putational time performance in this period, this forecastingmodel will be terminated. The parameter ‘t2’ is presented fordispatching the trains. Once a model enters the ‘t2’ timeperiod, it means this model is qualified and the forecastingresults of this model will be adopted to generate variouswarning commands. The parameter ‘t3’ is provided for thetrains' drivers to process a coming warning command.

(c) When the train enters the areas managed by the time period‘t2’, the warning system will start to output the multi-stepwind speed predictions. When the trains enters areas mana-ged by the time period ‘t3’, the warning system will use thelatest multi-step wind speed forecasting value to search forthe critical safety velocity of the train. If the current trainrunning velocity is bigger than the expected critical safetyvelocity, a warning message will be sent by the system to thedriver of the train.

(d) In this study, the parameters ‘t2’ and ‘t3’ are equal to fiveminutes and two minutes, respectively. They are both pro-posed by the Qinghai–Tibet Railway Company. They can berevised referring to the specific parameters of a running train.In the period ‘t2’, there are two standards adopted for thevalidation of the wind speed forecasting models. The firstvalidating standard is the forecasting accuracy. The currentaccuracy in the warning system is required as: the MAE (MeanAbsolute Error) is less than 2 m/s and the MAPE (MeanAbsolute Percentage Error) is less than 10%. The secondvalidating standard is the real-time forecasting performance.It estimates how fast the models can output the multi-stepwind speed predictions. The current requirement of theforecasting elapsed time is less than 3 s.

(e) There is a protecting strategy in the developed warningsystem. If the wind speed forecasted results cannot meet theaccuracy or the time requirements, the real-time wind speedmeasured data will be used to replace the multi-step forecast-ing results to propose the critical safety velocities of themonitored trains.

3. Wind speed forecasting

3.1. Framework of EMD-RARIMA forecasting method

In this study, a new hybrid EMD-RARIMA method is proposedto predict the wind speed. The computational framework of theproposed method is given in Fig. 6. As shown in Fig. 6, the contentsof the method include: (a) use the EMD to decompose the original

Fig. 4. Wireless wind speed station at Fenghuoshan Mountain.

3t2t1t

1L 2L

4t

Fig. 5. Concept of the application of the wind speed predictions in the warningsystem.

Start:Original Railway Wind Speed Data

Empirical Mode Decomposition

(EMD)

Decomposed Wind Speed Sub layer

Recursive ARIMA Modeling Predicted Results in the EMD Decomposed Series

Summarizing Calculation

End:Final Wind Speed

Predictions

Fig. 6. Framework of the proposed EMD-RARIMA forecasting method.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3830

wind speed data into a number of wind speed sub-layers; (b) buildthe recursive ARIMA models for all the sub-layers and adoptthe built models to forecast the multi-step predictions; and(c) summarize the wind speed multi-step predictions of the sub-layers to get the final predictions for the original wind speed.

3.2. Original wind speed data

Two groups of real wind speed series measured by twodifferent wind speed stations (i.e., the 30th station and the 32ndstation) along the Qinghai–Tibet railway are adopted to demon-strate the forecasting performance of the proposed EMD-RARIMAmethod, as shown in Figs. 7 and 8. The time interval of twoconsecutive data points is one minute. The mileage distancebetween the two wind speed stations is 41 km. They are namedX1tf g series and X2tf g series, respectively. The 1st–300th ones ofthem are used to establish the forecasting models and the 301st–400th ones are utilized to check the built models.

3.3. Wind speed decomposition by EMD

As explained in Section 3.1, the Empirical Mode Decomposition(EMD) is adopted to decompose the originally non-stationarywind speed samples. The EMD is a mathematical time domaindecomposing method presented by scientists N. E. Huang in 1998(Huang et al., 1998), which can convert a group of time series intolocally narrow band components, named Intrinsic Mode Functions(IMFs). The purpose of executing the EMD decomposition in thestudy is to convert the original non-stationary wind speed seriesinto a number of relatively stable wind speed sub-layers, whichwill decrease the difficulty to realize the high-precision predic-tions for the original wind speed. Different to the wavelet decom-position and the wavelet packet decomposition (Liu et al., 2013a,2013b) which both need to select the decomposing parameters

manually, the computational process of the EMD decomposition isadaptive and automatic (Liu et al., 2015).

The EMD considers any signal XðtÞ can be described using theequation as follows:

XðtÞ ¼Xni ¼ 1

CiðtÞþRðtÞ ð1Þ

where CiðtÞ� �

; ði¼ 1;2;⋯;nÞ is the IMFs in different decomposi-tions, RðtÞ� �

is the residue and ‘n’ is the number of the IMFs.The computational steps of the EMD are given as follows

(Huang et al., 1998):

Step 1: Identify all the local extrema of series XðtÞ� �, including

local maxima and local minima.Step 2: Connect all the local maxima by a cubic spline line togenerate its upper envelop XupðtÞ

� �. Similarly the lower

envelop XlowðtÞ� �

is made with all the local minima.Step 3: Compute the mean envelop MðtÞ� �

from the upper andlower envelops as follows:

MðtÞ ¼ XupðtÞþXlowðtÞ� �

=2 ð2Þ

Step 4: Extract the details as follows:

ZðtÞ ¼ XðtÞ�MðtÞ ð3Þ

Step 5: Check whether ZðtÞ� �is an IMF: (a) if ZðtÞ� �

is an IMFthen set CðtÞ ¼ ZðtÞ and meantime replace XðtÞ� �

with theresidual RðtÞ ¼ XðtÞ�CðtÞ ; (b) if ZðtÞ� �

is not an IMF, replaceXðtÞ� �

with ZðtÞ� �then repeat Steps 2–4 until the termination

criterion is satisfied. The following equation can be regarded asthe termination condition of this iterative calculation:

Xmt ¼ 1

Zj�1ðtÞ�ZjðtÞ� �2

Zj�1ðtÞ� �2 rδ ðj¼ 1;2;⋯; t ¼ 1;2;⋯;mÞ ð4Þ

where ‘m’ is the number of the wind speed data points, ‘δ’ is theterminated parameter, and ‘j’ denotes the times of iterativecalculation. In this study, the ‘δ’ is equal to 0.2.Step 6: The procedure of Steps 1–5 is repeated until all the IMFsare found.

The EMD decomposed results of the original wind speed seriesX1tf g are given in Fig. 9. From Fig. 9, it can be seen that the X1tf gseries have been converted into a group of wind speed sub-layerssuccessfully. A number of RARIMA models will be built in thesedecomposed wind speed sub-layers to complete the multi-stepforecasting computation. It is obvious that it is easier for the builtRARIMA models to obtain the accurate forecasting results by usingthe decomposed sub-layers than adopting the original windspeed data.

In this study, all of the decomposed wind speed sub-layers willbe checked their stability by the Run Sequence Method (RSM)(Hentati-Kaffel and de Peretti, 2015). The RSM is a non-parametricstatistical test to check the stability of a group of time series data.The contents of the RSM algorithm are given as follows:

Step 1: Calculate the average value X of an original series Xtf g.Step 2: Convert the original series Xtf g to a symbol series Stf gusing the criterion as below: set XtðiÞ is the ‘ith’ wind speedsample in the series Xtf g; if XtðiÞZX, the symbol ‘þ ’ will beadopted to replace the position of this sample; Oppositely, ifXtðiÞoX, the symbol ‘� ’ will be used to replace the position ofthis sample.Step 3: Calculate the number of the positive runs Nþ and thenumber of the negative runs N� using the standard as below:

Fig. 7. Original wind speed time series X1tf g.

Fig. 8. Original wind speed time series X2tf g.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38 31

Fig. 9. Decomposed results of the original wind speed series X1tf g. (a) XD1tf g series, (b) XD2tf g series, (c) XD3tf g series, (d) XD4tf g series, (e) XD5tf g series, (f) XD6tf g series,(g) XD7tf g series, (h) XD8tf g series, (i) XRtf g series.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3832

in the converted symbol series Stf g, a sequence of the con-secutive ‘þ ’ symbols is named a positive run, and a sequence ofthe consecutive ‘� ’ symbols is named a negative run. Forexample, the Nþ is equal to three and the N� is equal to twoin a symbol series like “þþþ��þ�þþ”.Step 4: The RSM considers that if the time series Xtf g is stable,its corresponding symbol series Stf g should meet the normaldistribution. So the following statistical equations can be usedto check whether the series Xtf g is stable as:

Z ¼ ðγ�μγÞ=σγ ð5Þ

where:

σγ ¼ ½2N1N2ð2N1N2�NÞN2ðN�1Þ

�1=2 ð6Þ

μγ ¼2N1N2

Nþ1 ð7Þ

N¼N1þN2 ð8ÞThen, if Zj jr1:96, the series Xtf g is stable under the signifi-

cant level parameter α¼ 0:05. Otherwise, the series Xtf g isunstable.

3.4. Wind speed forecasting by RARIMA

Time series method is a statistical algorithmwhich can build anexplicit equation to describe the potential changing law of asection of time series data (Wei, 1994). It has five standard models,including the Auto-Regressive (AR), the Moving Average (MA), theAuto-Regressive Moving Average (ARMA) and the Auto-RegressiveIntegrated Moving Average (ARIMA). The ARIMA models candescribe all the other ones.

3.4.1. Autoregressive Integrated Moving Average (ARIMA) modelThe equations of the ARIMA (p, d, q) are defined as follows:

ϕðBÞUKd UXðtÞ ¼ θðBÞUaðtÞ ð9Þwhere:

B¼ Xðt�1Þ=XðtÞK ¼ 1�B

ϕðBÞ ¼ 1�ϕ1B�ϕ2B2�ϕ3B

3�⋯�ϕp�1Bp�1�ϕpB

p

θðBÞ ¼ 1�θ1B�θ2B2�θ3B

3�⋯�θq�1Bq�1�θqB

q

8>>>><>>>>:

ð10Þ

where XðtÞ� �is the wind speed time series, aðtÞ� �

is the random errorseries assumed to be a white noise with a mean of zero and equalvariance, ‘B’ is the backward shift operator, ‘K’ is the difference

operator, ‘p’ is the order of the auto-regressive part in the ARIMAmodel, ‘d’ is the order of the difference computation operator, ‘q’ is theorder of the moving average part in the ARIMA model, ϕp

n ois the

parameter of the auto-regressive part, and θq� �

is the parameter ofthe moving average part.

As shown in Eq. (9), the ARIMA (p, d, q) model is actually acombination of the AR(p) model, the MA(q) model and thecomponent of ‘d’ times Difference Computation (DC). To demon-strate the derivation of the ARIMA (p, d, q) model, an ARIMA(2, 2,2) model is taken as an example as follows:

ð1�ϕ1B�ϕ2B2Þ

ARð2Þ U ð1�BÞ2DC UXðtÞ ¼ ð1�θ1B�θ2B2Þ UaðtÞ

MAð2Þ

) ð1�ϕ1B�ϕ2B2ÞU ð1þB2�2BÞUXðtÞ ¼ ð1�θ1B�θ2B

2ÞUaðtÞ) ½1�ð2þϕ1ÞBþð1þ2ϕ1�ϕ2ÞB2�ðϕ1�2ϕ2ÞB3�ϕ2B

4Þ�UXðtÞ

¼ ð1�θ1B�θ2B2ÞUaðtÞ

) XðtÞ�ð2þϕ1ÞXðt�1Þþð1þ2ϕ1�ϕ2ÞXðt�2Þ�ðϕ1�2ϕ2Þ�Xðt�3Þ�ϕ2Xðt�4Þ ¼ aðtÞ�θ1aðt�1Þ�θ2aðt�2Þ

) XðtÞ ¼ ð2þϕ1ÞXðt�1Þ�ð1þ2ϕ1�ϕ2ÞXðt�2Þþðϕ1�2ϕ2Þ�Xðt�3Þþϕ2Xðt�4ÞþaðtÞ�θ1aðt�1Þ�θ2aðt�2Þ ð11Þ

3.4.2. Box–Jenkins methodologyThe Box–Jenkins methodology, named after statisticians George

Box and Gwilym Jenkins, is presented to establish the suitabletime series models. The reason to choose the Box–Jenkins meth-odology in the study is that it has good real-time performance dueto its simple computational equations.

The Box–Jenkins methodology consists of three modeling contents:the model identification, the selection of model order and theparameter estimation. The details of the Box–Jenkins methodologycan be explained as: (a) in the step of model identification, theAutocorrelation function (ACF) and partial autocorrelation function(PACF) are always utilized to decide whether an Auto-Regressive (AR)component or a Moving Average (MA) one should be included in theARIMA models. If the ACF coefficients show smearing effect, the ARcomponent will be included. If the PACF coefficients indicate censoringeffect, the MA component will be included. Additionally, the RunSequence Method (RSM) is employed to identify the stability of theEMD decomposed sub-layers. If a sub-layer has been found that it isunstable, the Difference Computation (DC) will be executed on thissub-layer; (b) in the step of selection of model order, some criterions(e.g., Final Prediction Error, Akaike Information Criterion, Bayes Infor-mation Criterion, etc) will be adopted to select the best orders of theidentified ARIMA models; and (c) in the step of parameter estimation,some methods (e.g., Least Square Estimation, Yule–Walker Estimation,Greatly Relieved Estimate, etc) will be utilized to calculate the optimal

Fig. 10. Calculated results of the ACF/PACF coefficients for the XD1tf g series. (a) ACF results, (b) PACF results.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38 33

parameters of the identified ARIMA models. In this study, the FinalPrediction Error (FPE) criterion is used to choose the orders of theARIMA models and the Yule–Walker algorithm is adopted to calculatethe parameters of the ARIMA models. The detailed procedure of theBox–Jenkins methodology can be found in Pankratz (1983).

The results of the ACF and PACF coefficients for the decom-posed XD1tf g series are shown in Fig. 10. From Fig. 10, it can be seenthat the ACF results have smearing phenomenon while the PACFresults have censoring effect, which can be concluded that theXD1tf g series meet the ARIMA (p, 0, 0) model.

To select the best order ‘p’ for the recognized ARIMA (p, 0, 0),the FPE coefficients of the XD1tf g series are calculated using thefollowing equations as follows:

FPEðpÞ ¼ ð1þp=NÞð1�p=NÞðR̂0�Xp

j ¼ 1

ϕ̂jR̂jÞ

R̂k ¼ 1N

XN�k

t ¼ 1

XðtÞXðtþkÞ

8>>>>><>>>>>:

ð12Þ

where ‘p’ is the order of the ARIMA(p, 0, 0) model, ‘N’ is thenumber of the series XðtÞ� �

, ϕ̂j

n ois the estimated AR operating

series, and R̂k

n ois the estimated autocorrelation coefficients.

Based on the computational results completed by Eq. (12), thebest model for the XD1tf g series is selected as ARIMA(7,0,0).

To estimate the parameters of the identified ARIMA(7,0,0), theYule–Walker computation is executed using the following equa-tion as follows:

ϕ̂1

ϕ̂2

⋮ϕ̂p

2666664

3777775¼

R̂0 R̂1 ⋯ R̂p�1

R̂1 R̂0 ⋯ R̂p�2

⋮ ⋮ ⋮ ⋮R̂p�1 R̂p�2 ⋯ R̂0

2666664

3777775

�1R̂1

R̂2

⋮R̂p

266664

377775

ð13Þ

Based on Eq. (13), the final equation of the ARIMA(7,0,0) modelfor the XD1tf g series can be obtained as follows:

XD1tðtÞ ¼ 0:1879XD1tðt�1Þ�0:1910XD1tðt�2Þþ0:1988XD1tðt�3Þþ0:2706XD1tðt�4Þþ0:1377XD1tðt�5Þþ0:3088XD1tðt�6Þþ0:0873XD1tðt�7ÞþaD1t ð14Þ

where ‘t’ is the sampling time, and aD1tf g is a white noise series.

3.4.3. Recursive estimation algorithm for model parametersTo strength the tracking ability of the built time series models

in their multi-step recursive computation, a new recursive algo-rithm is proposed as in Fig. 11.

Based on Fig. 11, the recursive ARIMA computational process ofthe decomposed XD1tf g series can be demonstrated as follows:

(a) In the non-recursive ARIMA forecasting strategy, the three-step predictions can be made by using Eq. (14) as below. It can befound that the parameters of Eqs. (15)–(17) will not be updated.

� One-step forecasting:

X̂D1tð301Þ ¼ 0:1879XD1tð300Þ�0:1910XD1tð299Þþ0:1988XD1tð298Þþ0:2706XD1tð297Þþ0:1377XD1tð296Þþ0:3088XD1tð295Þþ0:0873XD1tð294Þ ð15Þ

� Two-step forecasting:

X̂D1tð302Þ ¼ 0:1879X̂D1tð301Þ�0:1910XD1tð300Þþ0:1988XD1tð299Þþ0:2706XD1tð298Þþ0:1377XD1tð297Þþ0:3088XD1tð296Þþ0:0873XD1tð295Þ ð16Þ

� Three-step forecasting:

X̂D1tð303Þ ¼ 0:1879X̂D1tð302Þ�0:1910X̂D1tð301Þþ0:1988XD1tð300Þþ0:2706XD1tð299Þþ0:1377XD1tð298Þþ0:3088XD1tð297Þþ0:0873XD1tð296Þ ð17Þ

(b) Different to the non-recursive ARIMA computational process,in the proposed recursive ARIMA strategy the parame-ters of the ARIMA models will be updated in real-time byusing the previous forecasted data. For instance, the seriesfX̂D1tð301Þ;XD1tð300Þ;XD1tð299Þ:;XD1tð298Þ;XD1tð297Þ;XD1tð296Þ;XD1tð295Þgare adopted to update the parameters of Eq. (19) beforecalculating the forecasting value X̂D1tð302Þ.

� One-step forecasting:

X̂D1tð301Þ ¼ 0:1879XD1tð300Þ�0:1910XD1tð299Þþ0:1988XD1tð298Þþ0:2706XD1tð297Þþ0:1377XD1tð296Þþ0:3088XD1tð295Þþ0:0873XD1tð294Þ ð18Þ

Fig. 11. Calculation steps of the proposed recursive ARIMA model.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3834

� Two-step forecasting:

X̂D1tð302Þ ¼ 0:1327X̂D1tð301Þ�0:1253XD1tð300Þþ0:1680XD1tð299Þþ0:2837XD1tð298Þþ0:1102XD1tð297Þþ0:3235XD1tð296Þþ0:1071XD1tð295Þ ð19Þ

� Three-step forecasting:

X̂D1tð303Þ ¼ 0:1993X̂D1tð302Þ�0:0945X̂D1tð301Þþ0:1895XD1tð300Þþ0:2710XD1tð299Þþ0:0771XD1tð298Þþ0:2953XD1tð297Þþ0:0623XD1tð296Þ ð20Þ

(c) Since the RARIMAmodels can adopt the latest forecasted resultsto update the parameters of the forecasting equations before starting anew computational process so that the RARIMA models can havebetter forecasting accuracy than the traditional ARIMA models (i.e.,non-recursive ARIMA models).

3.5. Estimation standard for forecasting results

Three error indexes are employed to estimate the accuracyperformance of all the involved forecasting models (Liu et al., 2012),

Fig. 12. Results of the one-step predictions for the original wind speed series X1tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

Fig. 13. Results of the three-step predictions for the original wind speed series X1tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

Fig. 14. Results of the five-step predictions for the original wind speed series X1tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38 35

including the Mean Absolute Error (MAE), the Mean Absolute Percen-tage Error (MAPE) and the Standard Deviation (SD).

� Mean Absolute Error:

MAE¼ 1M

XMi ¼ 1

jXðiÞ� X̂ðiÞj ð21Þ

� Mean Absolute Percentage Error:

MAPE¼ 1M

XMi ¼ 1

jXðiÞ� X̂ðiÞX̂ðiÞ

j ð22Þ

� Standard Deviation:

SD¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

M�1

XMi ¼ 1

½XðiÞ� X̂ðiÞ�2vuut ð23Þ

where XðiÞ� �is the measured wind speed time series, X̂ðiÞ

n ois

the forecasted wind speed time series and ‘M’ is the number ofthe terms in the XðiÞ� �

series.

4. Experimental results and analysis

In this study, two different cases are selected to demonstratethe effectiveness of the proposed wind speed forecasting methodsto protect the safety of the running trains. In the Case One, there isno warning message proposed by the warning system because allthe wind speed do not exceed the safety threshold. In the CaseTwo, there is a warning message presented by the warning systembecause some wind speed date exceed the safety threshold.Additionally, the wind speed data in the Case One and the CaseTwo show the decreasing and increasing trend, respectively.

4.1. Case One

Figs. 12–14 show the prediction results of the 301st–400thoriginal wind speed series X1tf g by the different forecasting meth-ods. The estimated results of the accuracy and the time perfor-mance (with a normal PC with Intel i5 CPU) for these predictions aregiven in Tables 1 and 2, respectively. Based on the original resultsshown in Table 1, the improvement percentages of the accuracies ofthe ARIMA, the BP and the PRWM by the hybrid EMD-RARIMAmodel can also be calculated as the results shown in Tables 3–5.

From Tables 1–5, it can be analyzed that:

(a) When comparing the proposed hybrid EMD-RARIMA modelwith the BP neural network, the former has improved theaccuracy performance of the latter obviously. The promoting

accuracy percentages of the MAE indexes from one-step tofive-step are 63.52%, 75.03% and 82.75%, respectively. Thepromoting accuracy percentages of the MAPE indexes fromone-step to five-step are 63.11%, 75.24% and 83.24%, respec-tively. The promoting accuracy percentages of the SD indexesfrom one-step to five-step are 62.14%, 70.01% and 76.05%,respectively. The reason of this improving phenomenon is thatthe EMD converts the original jumping wind speed series intoa series of relatively stationary wind speed sub-layers.

(b) When comparing the proposed hybrid EMD-RARIMA modelwith the ARIMA model, the former has also improved theperformance of the latter considerably. The promoting accu-racy percentages of the MAE indexes from one-step to five-step are 50.40%, 64.15% and 75.41%, respectively. The promot-ing accuracy percentages of the MAPE indexes from one-stepto five-step are 50.00%, 63.81% and 75.76%, respectively. Thepromoting accuracy percentages of the SD indexes from one-step to five-step are 52.22%, 63.32% and 69.30%, respectively.The reasons of this promoting phenomenon are given as: theadopted EMD decreases the forecasting difficulty of the ARIMA

Table 1Analysis of the accuracies of the predictions given in Figs. 12–14.

Indexes EMD-RARIMA BP neural network

1-step 3-step 5-step 1-step 3-step 5-step

MAE (m/s) 0.1607 0.2763 0.3248 0.4405 1.1067 1.8826MAPE (%) 0.76 1.31 1.51 2.06 5.29 9.01SD (m/s) 0.2343 0.4166 0.5522 0.6188 1.3889 2.3054

Indexes ARIMA model PRWM model

1-step 3-step 5-step 1-step 3-step 5-step

MAE (m/s) 0.3240 0.7708 1.3210 0.5610 – –

MAPE (%) 1.52 3.62 6.23 2.64 – –

SD (m/s) 0.4904 1.1357 1.7988 0.7886 – –

Table 2Analysis of the time performance of the predictions given in Figs. 12–14.

Indexes EMD-RARIMA BP neural network

1-step 3-step 5-step 1-step 3-step 5-step

Time (s) 1.2243 1.8413 1.9743 2.8940 3.1223 3.4475Indexes ARIMA model PRWM model

1-step 3-step 5-step 1-step 3-step 5-stepTime (s) 0.0631 0.0752 0.0937 0.0165 – –

Table 3Improved accuracy percentages of the BP neural network by the proposed hybridEMD-RARIMA model.

Indexes EMD-RARIMA vs. BP neural network

1-step 3-step 5-step

MAE (%) 63.52 75.03 82.75MAPE (%) 63.11 75.24 83.24SD (%) 62.14 70.01 76.05

Table 5Improved accuracy percentages of the PRWM model by the proposed hybrid EMD-RARIMA model.

Indexes EMD-RARIMA vs. PRWM

1-step 3-step 5-step

MAE (%) 71.35 – –

MAPE (%) 71.21 – –

SD (%) 70.29 – –

Table 4Improved accuracy percentages of the ARIMA model by the proposed hybrid EMD-RARIMA model.

Indexes EMD-RARIMA vs. ARIMA

1-step 3-step 5-step

MAE (%) 50.40 64.15 75.41MAPE (%) 50.00 63.81 75.76SD (%) 52.22 63.32 69.30

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3836

models by decomposing the original wind speed series; andThe RARIMA component in the hybrid EMD- RARIMA methodimproves the forecasting performance of the ARIMA model byupdating the equation parameters in every step.

(c) When comparing the proposed hybrid EMD-RARIMA modelwith the PRWM model, the former has still improved theperformance of the latter considerably. The promoting accu-racy percentages of the MAE, MAPE and SD indexes for theone-step predictions are 71.35%, 71.21% and 70.29%.

(d) All the built forecasting models can meet the elapsed timerequirements of the warning system.

4.2. Case Two

Figs. 15–17 show the forecasting results of the 301st–400th originalwind speed series X2tf g by the different forecasting methods. Theresults of the accuracy and the time performance estimation for thesepredictions are given in Tables 6 and 7, respectively.

Fig. 17. Results of the five-step predictions for the original wind speed series X2tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

Fig. 16. Results of the three-step predictions for the original wind speed series X2tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

Fig. 15. Results of the one-step predictions for the original wind speed series X2tf g by the EMD-RARIMA model, the BP neural networks, the ARIMA and the PRWM.

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–38 37

From Tables 6 and 7, the same conclusions can be made to theones proposed in the Case One in Section 4.1 as follows: (a) theproposed hybrid EMD-RARIMA model has the best forecastingaccuracy in all step predictions among all the built forecasting models;and (b) all of the built models can meet the time performancerequirements proposed by the warning system (i.e.,o3 s).

5. Conclusions

In this study a wind warning system is developed to protect therunning trains under strong crosswind along the China Qinghai–Tibet railway. In the warning system, a hybrid EMD-RARIMAmethod is proposed to forecast the multi-step wind speed. Twoexperimental cases validate that: (a) the proposed EMD-RARIMAmethod has satisfactory performance, which meets all the require-ments of the warning system either in the forecasting accuracy orin the real-time computation; (b) compared to the ARIMA model,the BP neural network and the PRWM model, the proposed EMD-RARIMA method improves all their forecasting accuracy consider-ably; and (c) in the current warning system, the forecasted windspeed data are used to select the trains' critical safety velocities topropose warning messages. In the future work, the wind directionsignals will also be included in the system decision to optimize theproposal of the warning messages.

Acknowledgments

The authors would like to thank the referees for their preciousreviewing. This study is fully supported by the National NaturalScience Foundation of China (Grant no. 51308553), the ScientificResearch Fund of Hunan Provincial Education Department (Prin-ciple Investigator: Dr. Hui Liu) and the Shenghua Yu-ying Talents

Program of the Central South University (Principle Investigator: Dr.Hui Liu). This study is partly supported by the National NaturalScience Foundation of China (Grant nos. U1134203, U1334205).

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Table 6Analysis of the accuracies of the predictions given in Figs. 15–17.

Indexes EMD-RARIMA BP neural network

1-step 3-step 5-step 1-step 3-step 5-step

MAE 0.2103 0.3408 0.6620 0.7582 1.6859 2.4703MAPE 1.53 2.54 4.84 5.03 11.12 16.13SD 0.2747 0.4924 1.0625 1.1185 2.1842 3.2208

Indexes ARIMA model PRWM model

1-step 3-step 5-step 1-step 3-step 5-step

MAE 0.4884 1.2543 1.9998 0.7606 – –

MAPE 3.51 9.32 14.88 5.75 – –

SD 0.7016 1.8286 2.7480 1.0161 – –

Table 7Analysis of the time performance of the predictions given in Figs. 15–17.

Indexes EMD-RARIMA BP neural network

1-step 3-step 5-step 1-step 3-step 5-step

Time (s) 1.3289 1.9023 2.0189 2.8528 3.2109 3.5723

Indexes ARIMA model PRWM model

1-step 3-step 5-step 1-step 3-step 5-step

Time (s) 0.0708 0.0832 0.0998 0.0185 – –

H. Liu et al. / J. Wind Eng. Ind. Aerodyn. 141 (2015) 27–3838