11
Research Article Daily Crude Oil Price Forecasting Using Hybridizing Wavelet and Artificial Neural Network Model Ani Shabri 1 and Ruhaidah Samsudin 2 1 Department of Science Mathematic, Faculty of Science, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia 2 Department of Soſtware Engineering, Faculty of Computing, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor, Malaysia Correspondence should be addressed to Ani Shabri; ani [email protected] Received 24 January 2014; Revised 2 July 2014; Accepted 2 July 2014; Published 16 July 2014 Academic Editor: Marek Lefik Copyright © 2014 A. Shabri and R. Samsudin. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new method based on integrating discrete wavelet transform and artificial neural networks (WANN) model for daily crude oil price forecasting is proposed. e discrete Mallat wavelet transform is used to decompose the crude price series into one approximation series and some details series (DS). e new series obtained by adding the effective one approximation series and DS component is then used as input into the ANN model to forecast crude oil price. e relative performance of WANN model was compared to regular ANN model for crude oil forecasting at lead times of 1 day for two main crude oil price series, West Texas Intermediate (WTI) and Brent crude oil spot prices. In both cases, WANN model was found to provide more accurate crude oil prices forecasts than individual ANN model. 1. Introduction Crude oil prices fluctuations are of significant interest to both financial practitioners and market participants. However, crude oil price is one of the most complex and difficult to model because fluctuation of the crude oil price is rather irregular, nonlinear, nonstationary, and with high volatility. us, accurate forecasting of the crude oil price time series is one of the greatest challenges and among the most important issues facing energy economists towards better decisions in several managerial levels. For this reason, many researchers have devoted considerable effort to the development of different types of models for crude oil price forecasting. e application of the classical time series models such as autoregressive moving average (ARMA) model (Moham- madi and Su [1], Ahmad [2], Wang et al. [3], and Xie et al. [4]) and generalized autoregressive conditional heteroscedastic (GARCH) type models (Morana [5], Sadorsky [6], and Agnolucci [7]) for crude oil price forecasting has received much attention in the last decade. However, the above models can provide good prediction results, when the price series under study is basically linear or near linear, and have a limited ability to capture nonlinearity and nonstationary in crude oil prices data. Artificial neural network (ANN) techniques have shown great ability in modeling and forecasting nonlinear and complex time series. ANN offers an effective approach for handling large amounts of dynamic, nonlinear, and noise data. Numerous papers have already presented successful application of ANN for modeling and forecasting the crude oil price series (Lackes et al. [8], Khazem and Mazouz [9], Mirmirani and Li [10], Kulkarni and Haidar [11], Yu et al. [12], and Hu et al. [13]) as well as for forecasting real world time series. eir experimental results show that the performance of ANN is superior to various traditional statistical models. ANN has an ability to learn complex and nonlinear time series that is difficult to model with conventional models. However, there are some disadvantages of ANN. Although ANN has advantages of accurate forecasting, their perfor- mance in some specific situation is inconsistent (Khashei and Bijari [14]). ANN also oſten suffers from local minima and overfitting, and the network structure of this model is difficult to determine and it is usually determined by using a trial- and-error approach (Kis ¸i [15]). In addition, ANN model has Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 201402, 10 pages http://dx.doi.org/10.1155/2014/201402

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Research ArticleDaily Crude Oil Price Forecasting Using Hybridizing Waveletand Artificial Neural Network Model

Ani Shabri1 and Ruhaidah Samsudin2

1 Department of Science Mathematic Faculty of Science Universiti Teknologi Malaysia 81310 Skudai Johor Malaysia2 Department of Software Engineering Faculty of Computing Universiti Teknologi Malaysia (UTM) 81310 Skudai Johor Malaysia

Correspondence should be addressed to Ani Shabri ani sabrihotmailcom

Received 24 January 2014 Revised 2 July 2014 Accepted 2 July 2014 Published 16 July 2014

Academic Editor Marek Lefik

Copyright copy 2014 A Shabri and R Samsudin 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

A new method based on integrating discrete wavelet transform and artificial neural networks (WANN) model for daily crudeoil price forecasting is proposed The discrete Mallat wavelet transform is used to decompose the crude price series into oneapproximation series and some details series (DS) The new series obtained by adding the effective one approximation series andDS component is then used as input into the ANN model to forecast crude oil price The relative performance of WANN modelwas compared to regular ANNmodel for crude oil forecasting at lead times of 1 day for two main crude oil price series West TexasIntermediate (WTI) and Brent crude oil spot prices In both cases WANN model was found to provide more accurate crude oilprices forecasts than individual ANN model

1 Introduction

Crude oil prices fluctuations are of significant interest to bothfinancial practitioners and market participants Howevercrude oil price is one of the most complex and difficult tomodel because fluctuation of the crude oil price is ratherirregular nonlinear nonstationary and with high volatilityThus accurate forecasting of the crude oil price time series isone of the greatest challenges and among the most importantissues facing energy economists towards better decisions inseveral managerial levels For this reason many researchershave devoted considerable effort to the development ofdifferent types of models for crude oil price forecasting

The application of the classical time series models suchas autoregressive moving average (ARMA) model (Moham-madi and Su [1] Ahmad [2]Wang et al [3] and Xie et al [4])and generalized autoregressive conditional heteroscedastic(GARCH) type models (Morana [5] Sadorsky [6] andAgnolucci [7]) for crude oil price forecasting has receivedmuch attention in the last decade However the abovemodelscan provide good prediction results when the price seriesunder study is basically linear or near linear and have

a limited ability to capture nonlinearity and nonstationary incrude oil prices data

Artificial neural network (ANN) techniques have showngreat ability in modeling and forecasting nonlinear andcomplex time series ANN offers an effective approach forhandling large amounts of dynamic nonlinear and noisedata Numerous papers have already presented successfulapplication of ANN for modeling and forecasting the crudeoil price series (Lackes et al [8] Khazem and Mazouz [9]Mirmirani and Li [10] Kulkarni andHaidar [11] Yu et al [12]and Hu et al [13]) as well as for forecasting real world timeseries Their experimental results show that the performanceof ANN is superior to various traditional statistical modelsANN has an ability to learn complex and nonlinear timeseries that is difficult to model with conventional modelsHowever there are some disadvantages of ANN AlthoughANN has advantages of accurate forecasting their perfor-mance in some specific situation is inconsistent (Khashei andBijari [14]) ANN also often suffers from local minima andoverfitting and the network structure of thismodel is difficultto determine and it is usually determined by using a trial-and-error approach (Kisi [15]) In addition ANN model has

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 201402 10 pageshttpdxdoiorg1011552014201402

2 Mathematical Problems in Engineering

a limitation with nonstationary data and may not be able tohandle nonstationary data if preprocessing of the input datais not done (Adamowski and Sun [16])

In the last decade wavelet transforms have been inves-tigated in diverse fields such as in economics [18] business[19] and hydrology [20] Wavelet transforms have gainedvery high attention and have been found to be very effectivewith nonstationary time series data Recently different hybridmodels based on wavelet transform have been improved inthe field of oil price forecasting For example Qunli et al[21] proposed a hybrid model based on wavelet transformand radial basis function (RBF) neural network to forecastthe future oil price They found that wavelet decomposesthe original price sequence successfully as the input layerof the network RBF neural network model Bao et al [22]developed a hybrid model integrating wavelet and leastsquares support vector machines (LSSVM) for crude oilforecasting Youesfi et al [23] apply a wavelet methodologyto decompose the crude oil price and extended them directlyto make forecasts de Souza e Silva et al [24] apply waveletanalysis to omit high frequency noises and then use hiddenMarkov model to predict future price movement He et al[25] proposed a wavelet decomposed ensemble model toincorporate the Heterogeneous Market Hypothesis into themodeling process However in the crude oil field a hybridwavelet and ANN model have received very little attentionand there are only a few applications of this model to forecastcrude oil price series Shambora and Rossitier [26] developeda hybrid wavelet and ANN model in crude oil forecastingJammazi and Aloui [27] explored the Haar A Trous waveletand backpropagation neural network algorithm to developa crude oil price forecasting model Mingming and Jinliang[28] constructed a multiple wavelet recurrent neural networksimulationmodel to analyze crude oil pricesWavelet analysiswas used to catch multiscale data characteristics and recur-rent neural network model was used to simulate crude oilpricesThe simulation results showed that themodel has highprediction accuracy

The main contribution of this paper is to propose a novelhybrid integrating wavelet transform and ANN model forcrude oil forecasting In order to catch this purpose the dailyWest Texas Intermediate (WTI) andBrent crude oil spot pricewere decomposed into subseries at different scale by Mallatalgorithm Then effective subseries were summed togetherand then used as inputs into the ANN model for crude oilforecasting Finally to evaluate themodel ability the proposedmodel was compared with individual ANN model

2 Artificial Neural Network

An ANN is flexible computing that has been extensivelystudied and used for time series forecasting in many areasof science and engineering since the early 1990s ANN is amathematical model which has a highly connected structuresimilar to brain cells AnANNmodel usually consists of threelayers the first layer is the input layer where the data areintroduced to the network the second layer is the hiddenlayer where data are processed and the last layer is the output

Input layer

Hidden layer

Output layer

Bias unit

(tangent sigmoid transfer function)

(linear transfer function)ytminus1

ytminus2

ytminusp

w0j

wj

wij

yt

Figure 1 Three-layer feedforward ANNmodel

layer where the results of given input are produced Figure 1illustrates the architecture of the proposed ANN for crudeoil price forecastingThe relationship between the output (119910

119905)

and the input (119910119905minus1 119910119905minus2 119910

119905minus119901) is given by [29]

119910119905= 1198870+

119902

sum119895=1

119887119895119891(1199080119895+

119901

sum119894=1

119908119894119895119910119905minus119894) + 120576119905 (1)

where 119887119895(119895 = 0 1 2 119902) and 119908

119894119895(119894 = 0 1 2 119901 119895 =

0 1 2 119902) are the model parameters 119901 is the number ofinput nodes 119902 is the number of hidden nodes and 119891(sdot) is thetransfer function

Generally ANN may have different transfer functionsfor different neurons in the same or different layers Themajority of research uses hyperbolic tangent sigmoid func-tion for hidden neurons and there is no consensus on whichtransfer function should be used for output neurons [30]Thehyperbolic tangent sigmoid is often used as the hidden layertransfer function given by

119891 (119904) =119890119904 minus 119890minus119904

119890119904 + 119890minus119904(2)

and one unit output layer is a pure linear transfer functiongiven by

119891 (1199041015840) = 1199041015840 (3)

where 119904 is the weighted input of the hidden layer 119891(119904)is hyperbolic tangent sigmoid transfer function for hiddenlayer 1199041015840 is the weighted input of the output layer 119891(119904) is purelinear transfer function for output layer

The most popular neural network training method usedis the backpropagation (BP) algorithm which is essentially agradient steepest descent method introduced by Rumelhartet al [31]This algorithm suffers the problems of slow conver-gence inefficiency and lack of robustness [32] Furthermoreit can be very sensitive to the choice of the learning rate It is

Mathematical Problems in Engineering 3

hard to choose a proper learning rate because a low-learningratemay result in long training times and a high learning ratemay cause system instability [33]

To overcome the weakness of BP algorithm manyresearchers have investigated the use of Levenberg-Marquardt (LM) algorithm The LM algorithm is one ofthe most popular and more efficient nonlinear optimisationmethods It is a variation of Newtonrsquos method using Hessian-based algorithm for nonlinear least squares optimizationwhich is used in most optimisation packages The LMalgorithm takes large steps down the gradient where thegradient is small such as near local minima and takes smallsteps when the gradient is large Its faster convergencerobustness and the ability to find good local minima makeit attractive in ANN training This optimisation techniqueis more powerful than the conventional gradient descenttechnique [34ndash36]

3 Wavelet Transform

Thewavelet transforms is a strongmathematical tool that pro-vides a time-frequency representation of an analyzed signalin the time domin This overcomes the basic shortcoming ofFourier analysis which is that the Fourier spectrum containsonly globally averaged information Wavelet transformationsprovide useful decomposition of the original time seriesby capturing useful information on various decompositionlevels Wavelet transform can be divided by two categoriescontinuous wavelet transforms (CWT) and discrete wavelettransforms (DWT) The CWT of the time series 119909(119905) is givenby

119882(120591 119904) =1

radic|119904|intinfin

minusinfin

119909 (119905) 120595 lowast (119905 minus 120591

119904) 119889119905 (4)

where (lowast) indicates the conjugate complex function 119905 standsfor a time 120591 stands for the time step 119904 isin [0infin] standsfor the wavelet scale and 120595(119905) is the transforming functionand is called the mother wavelet The term wavelet meanssmall wave where small refers to the condition that thefunction is of a finite lengthThe wave refers to the conditionthat the function is of a finite length Several families ofwavelets (120595) that have been proven to be useful for variousapplications are described in related references (Mallat [37])The wavelets are chosen based on their shape and theirability to analyse the time series in a particular applicationTypical wavelet families such as Haar Daubechies CoifletSymlet Meyer Morlet and the Mexican Hat can be used asa mother wavelet The CWT is not often used for forecastingdue to its computationally complex and time requirementsto compute [38] Instead successive wavelet is often discretein forecasting applications to simplify the numeric solutionsDWT requires less computation time and is simpler to applyDWT is given by

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

01199041198980

1199041198980

) (5)

where 119898 and 119899 are integers that control the scale and timeThe most common and simplest choice for parameters are1199040= 2 and 120591 = 1 For a discrete time series 119909(119905) the DWT

becomes

119882119898 119899

= 2minus1198982

119873minus1

sum119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (6)

where119882119898 119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to the Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearly independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby

119909 (119905) = 119879 +

119872

sum119898=1

2119872minus119898minus1

sum119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (7)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum119898=1

119863119898(119905) (8)

in which 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905)(119898 = 1 2 119872) is detailed

subseries which can capture small features of interpretationalvalue in the data

4 Application

In this study two main crude oil prices Brent crude oilspot price andWest Texas Intermediate (WTI) crude oil spotprice (in US dollars per barrel) series were chosen as theexperimental sample The main reason for selecting thesecrude oil spot price indicators is that these two crude oilprices are the most famous benchmark prices which arewidely used as the basis of many crude oil price formulae[12 17] For Brent data we use the daily data from May 201987 to September 30 2006 with a total of 4933 observationsData from May 20 1987 to December 31 2002 are used forthe training set (3965 observations) and the remainder isused as the testing set (968 observations)

While for WTI data the daily data from January 11986 to September 30 2006 excluding public holidays witha total of 5237 were employed as experimental data forconvenience of WANN modeling the data from January 11986 toDecember 31 2000 are used for the training set (3800observations) and the remainder is used as the testing set(1437 observations)The graphical representation of the pricefor Brent and WTI crude oil price is given in Figure 2

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study two performance criteria suchas RMSE andMAPEwere used to evaluate the accuracy of themodels These criteria are given below

RMSE = radic 1119899

119899

sum119905=1

(119910119905minus 119910119905)2

MAPE = 1119899

119899

sum119905=1

|BIAS|

(9)

4 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily Brent crude oil (20 May 1987ndash30 Sept 2006) Daily WTI crude oil (1 Jan 1986ndash30 Sept 2006)

Figure 2 Daily Brent and WTI crude oil prices series

where BIAS = (119910119905minus 119910119905)119910119905 119910119905is the actual and 119910

119905is the

forecasted value of period 119905 and 119899 is the number of totalobservations The criteria to judge for the best model arerelatively small RMSE and MAPE found in the training andtesting of the data

5 Fitting ANN Model to the Data

At first the multilayer perceptron (MLP) feedforward ANNmodel without any data preprocessing was used to modelthe crude oil price series One of the most important stepsin the model development process of the ANN model is thedetermination of significant input variables There are nofixed rules for the selection of input variables for developingANN model even though a general framework can befollowed based on previous successful application in crudeoil prices problems [39] For Brent and WTI data six inputcombinations based on previous log return of daily oil pricesare evaluated to estimate current prices value The inputcombinations evaluated in the study are (i) 119910

119905minus1 (ii) 119910

119905minus1 119910119905minus2

(iii) 119910

119905minus1 119910119905minus2

119910119905minus3

(iv) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

(v) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

and (vi) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

119910119905minus6

ANNmodel used in this study is the standard three-layer

feedforward network For improving the network modelingand reducing the chance of being trapped in a local minimadata normalization is applied in ANN input and outputDue to the crude oil prices series for both data collected arenonstationary (see Figure 2) to improve the accuracy of thetraining and testing each data point in the input neuronshad to be preprocessed by normalizing and transformedwithin the range of [minus1 1] There are no fixed rules as towhich standardization approach should be used in particularcircumstances In this study normalization for each raw inputand output data set was calculated by taking log return ofcurrent oil prices as the following formula

119877119905= log(

119910119905

119910119905minus1

) (10)

where 119910119905and 119910

119905minus1are current and one-period lagged prices

To achieve optimal weights of ANN LM algorithm pro-vided by theMATLABneural network toolbox is used to trainthe network In this study the hyperbolic tangent sigmoid

function and a linear function are employed as activationfunctions for the hidden and output layers respectively Theinitial value of learning rate is set to 0001 and a momentumcoefficient of 09 The learning rate is increased by a factor of10 and a decrease step of 01 until the change above resultsin a reduced performance value The number of epochs thatare used to train is set to 1000 The training of ANN will stopwhen the error achieves 10minus5 or when the number of epochsreaches 1000

The hidden layer plays important roles inmany successfulapplications of ANN It has been proven that only one hiddenlayer is sufficient for ANN to approximate any complexnonlinear function with any desired accuracy In the caseof the popular one hidden layer networks several practicalnumbers of neurons in the hidden layer were identified forbetter forecasting accuracy The optimal number of nodesin the hidden layer was identified using several practicalguidelines Berry and Linoff [40] claimed that the numberof hidden nodes should never be more than 2119868 where 119868 isthe number of inputs Hecht-Nielsen [41] claimed that thenumber of hidden neuron is equal to 2119868 + 1 However as far asthe number of hiddenneurons is concerned there is currentlyno theory to determine how many nodes in the hidden layerare optimal In the present study the number of hidden nodeswas progressively increased from 1 to 2119868 + 1

A program code including the wavelet toolbox waswritten in MATLAB language for the development of theANNmodel The optimal complexity of ANNmodel that isthe number of input and hidden nodes was determined by atrial-and-error approach Table 1 shows the best performanceresults of different ANN with different numbers of hiddenneurons from 1 to 2119868 + 1 for both data series The trainingset is used to estimate parameters for any specific ANNarchitecture The testing set is then used to select the bestANN among all numbers of hidden neurons considered

Table 1 shows the performance of ANN varying in accor-dance with the number of neurons in the hidden layer Itcan be seen that the prediction errors (MAPE) by ANNrange from 160 to 228 for Brent data and from 167to 180 for WTI data For Brent data the best architectureaccording RMSE and MAPE criterion for training data andtesting data has 2 input layer neurons 4 hidden layer neuronsand 1 output layer neuron (2-4-1) For this architecture RMSE

Mathematical Problems in Engineering 5

Table 1 Training and testing performance of different network architecture of ANN model

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 1 07281 228 09932 1652 4 05023 160 09910 1643 2 05108 160 09912 1644 2 05059 160 09910 1645 1 05244 161 09928 1656 1 05231 161 09920 165

WTI

1 3 05651 172 09507 1792 4 05590 172 09452 1793 5 05187 169 09496 1794 7 05137 167 09514 1805 1 05626 172 09529 1806 1 05636 172 09493 179

Table 2 Coefficient of determination (1198772) between each of subtime series for Brent and WTI data

Data DWT Coefficient of determination (1198772) with original crude oil prices series Mean119905 minus 1 119905 minus 2 119905 minus 3 119905 minus 4 119905 minus 5 119905 minus 6 119877

2

Brent

D1 00000 00000 00000 00000 00000 00000 00000D2 00000 00000 00001 00002 00000 00000 00001D3 00006 00006 00002 00000 00002 00002 00003A3 09895 09917 09937 09954 09967 09967 09940

WTI

D1 00000 00052 00021 00053 00000 01619 00291D2 00027 00149 00001 00866 01092 00274 00402D3 00353 00622 00472 00060 00107 00750 00394A3 00040 00143 00311 00511 00721 00886 00435

value is 09910 and MAPE value is 164 for testing datawhile forWTI data the best architecture according to RMSE(09452) andMAPE (179) criterion for testing data is 2-4-1

6 Fitting Hybrid Wavelet-ANNModel to the Data

The hybrid wavelet and ANN (WANN) model is obtained bycombining two methods discrete wavelet transform (DWT)andANNmodel InWANN the original crude oil price serieswas decomposed into a certain number of subtime seriescomponents which were entered to ANN in order to improvethe model accuracy In this study the Daubechies waveletone of the most widely used wavelet families is chosenas the wavelet function to decompose the original series(Eynard et al [42] Bagheri et al [43]) When conductingwavelet analysis the number of decomposition levels thatare appropriate for the data must be chosen To choose thenumber of decomposition level the following formula is used[34]

119871 = int [log (119873)] (11)

where 119871 is the level of decomposition and 119873 is the numberof time series data According to this formula the optimalnumber of decomposition levels for both crude oil price

series data in this study would have been 3 For this purposefirstly the original oil price series is decomposed into 3level components (D1 D2 and D3) that stand for differentfrequency components of the original series (details) and anapproximation component (A3) as shown in Figures 3 and 4

Each of Drsquos series plays a distinct role in the original timeseries and has different effects on the original crude priceseries Instead of using each Drsquos component individually asmodel input employment of the added suitable Drsquos compo-nent is more useful and can highly increase forecast per-formances of hybrid models In this study the effectivenessof wavelet components is determined using the coefficientof determination (R2) between each Drsquos subtime series andoriginal dataTheR2 values are given in Table 2 It can be seenthat R2of the D1 is zero for Brent data and 00291 for WTIdata However the wavelet components D2 and D3 showsignificantly higher R2 compared to the D1 for both seriesAccording to the R2 analyses the effective components D2and D3 were selected as the dominant wavelet componentsAfterward the significant wavelet components D2 D3 andapproximation (A3) component were added to each other asANN model input for crude oil forecast Figure 6 shows thestructure of the WANN model Figure 5 shows the originalcrude oil prices and their Ds that is the time series of 2-monthmode (D1) 4-monthmode (D2) 8-monthmode (D3)

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

2 Mathematical Problems in Engineering

a limitation with nonstationary data and may not be able tohandle nonstationary data if preprocessing of the input datais not done (Adamowski and Sun [16])

In the last decade wavelet transforms have been inves-tigated in diverse fields such as in economics [18] business[19] and hydrology [20] Wavelet transforms have gainedvery high attention and have been found to be very effectivewith nonstationary time series data Recently different hybridmodels based on wavelet transform have been improved inthe field of oil price forecasting For example Qunli et al[21] proposed a hybrid model based on wavelet transformand radial basis function (RBF) neural network to forecastthe future oil price They found that wavelet decomposesthe original price sequence successfully as the input layerof the network RBF neural network model Bao et al [22]developed a hybrid model integrating wavelet and leastsquares support vector machines (LSSVM) for crude oilforecasting Youesfi et al [23] apply a wavelet methodologyto decompose the crude oil price and extended them directlyto make forecasts de Souza e Silva et al [24] apply waveletanalysis to omit high frequency noises and then use hiddenMarkov model to predict future price movement He et al[25] proposed a wavelet decomposed ensemble model toincorporate the Heterogeneous Market Hypothesis into themodeling process However in the crude oil field a hybridwavelet and ANN model have received very little attentionand there are only a few applications of this model to forecastcrude oil price series Shambora and Rossitier [26] developeda hybrid wavelet and ANN model in crude oil forecastingJammazi and Aloui [27] explored the Haar A Trous waveletand backpropagation neural network algorithm to developa crude oil price forecasting model Mingming and Jinliang[28] constructed a multiple wavelet recurrent neural networksimulationmodel to analyze crude oil pricesWavelet analysiswas used to catch multiscale data characteristics and recur-rent neural network model was used to simulate crude oilpricesThe simulation results showed that themodel has highprediction accuracy

The main contribution of this paper is to propose a novelhybrid integrating wavelet transform and ANN model forcrude oil forecasting In order to catch this purpose the dailyWest Texas Intermediate (WTI) andBrent crude oil spot pricewere decomposed into subseries at different scale by Mallatalgorithm Then effective subseries were summed togetherand then used as inputs into the ANN model for crude oilforecasting Finally to evaluate themodel ability the proposedmodel was compared with individual ANN model

2 Artificial Neural Network

An ANN is flexible computing that has been extensivelystudied and used for time series forecasting in many areasof science and engineering since the early 1990s ANN is amathematical model which has a highly connected structuresimilar to brain cells AnANNmodel usually consists of threelayers the first layer is the input layer where the data areintroduced to the network the second layer is the hiddenlayer where data are processed and the last layer is the output

Input layer

Hidden layer

Output layer

Bias unit

(tangent sigmoid transfer function)

(linear transfer function)ytminus1

ytminus2

ytminusp

w0j

wj

wij

yt

Figure 1 Three-layer feedforward ANNmodel

layer where the results of given input are produced Figure 1illustrates the architecture of the proposed ANN for crudeoil price forecastingThe relationship between the output (119910

119905)

and the input (119910119905minus1 119910119905minus2 119910

119905minus119901) is given by [29]

119910119905= 1198870+

119902

sum119895=1

119887119895119891(1199080119895+

119901

sum119894=1

119908119894119895119910119905minus119894) + 120576119905 (1)

where 119887119895(119895 = 0 1 2 119902) and 119908

119894119895(119894 = 0 1 2 119901 119895 =

0 1 2 119902) are the model parameters 119901 is the number ofinput nodes 119902 is the number of hidden nodes and 119891(sdot) is thetransfer function

Generally ANN may have different transfer functionsfor different neurons in the same or different layers Themajority of research uses hyperbolic tangent sigmoid func-tion for hidden neurons and there is no consensus on whichtransfer function should be used for output neurons [30]Thehyperbolic tangent sigmoid is often used as the hidden layertransfer function given by

119891 (119904) =119890119904 minus 119890minus119904

119890119904 + 119890minus119904(2)

and one unit output layer is a pure linear transfer functiongiven by

119891 (1199041015840) = 1199041015840 (3)

where 119904 is the weighted input of the hidden layer 119891(119904)is hyperbolic tangent sigmoid transfer function for hiddenlayer 1199041015840 is the weighted input of the output layer 119891(119904) is purelinear transfer function for output layer

The most popular neural network training method usedis the backpropagation (BP) algorithm which is essentially agradient steepest descent method introduced by Rumelhartet al [31]This algorithm suffers the problems of slow conver-gence inefficiency and lack of robustness [32] Furthermoreit can be very sensitive to the choice of the learning rate It is

Mathematical Problems in Engineering 3

hard to choose a proper learning rate because a low-learningratemay result in long training times and a high learning ratemay cause system instability [33]

To overcome the weakness of BP algorithm manyresearchers have investigated the use of Levenberg-Marquardt (LM) algorithm The LM algorithm is one ofthe most popular and more efficient nonlinear optimisationmethods It is a variation of Newtonrsquos method using Hessian-based algorithm for nonlinear least squares optimizationwhich is used in most optimisation packages The LMalgorithm takes large steps down the gradient where thegradient is small such as near local minima and takes smallsteps when the gradient is large Its faster convergencerobustness and the ability to find good local minima makeit attractive in ANN training This optimisation techniqueis more powerful than the conventional gradient descenttechnique [34ndash36]

3 Wavelet Transform

Thewavelet transforms is a strongmathematical tool that pro-vides a time-frequency representation of an analyzed signalin the time domin This overcomes the basic shortcoming ofFourier analysis which is that the Fourier spectrum containsonly globally averaged information Wavelet transformationsprovide useful decomposition of the original time seriesby capturing useful information on various decompositionlevels Wavelet transform can be divided by two categoriescontinuous wavelet transforms (CWT) and discrete wavelettransforms (DWT) The CWT of the time series 119909(119905) is givenby

119882(120591 119904) =1

radic|119904|intinfin

minusinfin

119909 (119905) 120595 lowast (119905 minus 120591

119904) 119889119905 (4)

where (lowast) indicates the conjugate complex function 119905 standsfor a time 120591 stands for the time step 119904 isin [0infin] standsfor the wavelet scale and 120595(119905) is the transforming functionand is called the mother wavelet The term wavelet meanssmall wave where small refers to the condition that thefunction is of a finite lengthThe wave refers to the conditionthat the function is of a finite length Several families ofwavelets (120595) that have been proven to be useful for variousapplications are described in related references (Mallat [37])The wavelets are chosen based on their shape and theirability to analyse the time series in a particular applicationTypical wavelet families such as Haar Daubechies CoifletSymlet Meyer Morlet and the Mexican Hat can be used asa mother wavelet The CWT is not often used for forecastingdue to its computationally complex and time requirementsto compute [38] Instead successive wavelet is often discretein forecasting applications to simplify the numeric solutionsDWT requires less computation time and is simpler to applyDWT is given by

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

01199041198980

1199041198980

) (5)

where 119898 and 119899 are integers that control the scale and timeThe most common and simplest choice for parameters are1199040= 2 and 120591 = 1 For a discrete time series 119909(119905) the DWT

becomes

119882119898 119899

= 2minus1198982

119873minus1

sum119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (6)

where119882119898 119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to the Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearly independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby

119909 (119905) = 119879 +

119872

sum119898=1

2119872minus119898minus1

sum119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (7)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum119898=1

119863119898(119905) (8)

in which 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905)(119898 = 1 2 119872) is detailed

subseries which can capture small features of interpretationalvalue in the data

4 Application

In this study two main crude oil prices Brent crude oilspot price andWest Texas Intermediate (WTI) crude oil spotprice (in US dollars per barrel) series were chosen as theexperimental sample The main reason for selecting thesecrude oil spot price indicators is that these two crude oilprices are the most famous benchmark prices which arewidely used as the basis of many crude oil price formulae[12 17] For Brent data we use the daily data from May 201987 to September 30 2006 with a total of 4933 observationsData from May 20 1987 to December 31 2002 are used forthe training set (3965 observations) and the remainder isused as the testing set (968 observations)

While for WTI data the daily data from January 11986 to September 30 2006 excluding public holidays witha total of 5237 were employed as experimental data forconvenience of WANN modeling the data from January 11986 toDecember 31 2000 are used for the training set (3800observations) and the remainder is used as the testing set(1437 observations)The graphical representation of the pricefor Brent and WTI crude oil price is given in Figure 2

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study two performance criteria suchas RMSE andMAPEwere used to evaluate the accuracy of themodels These criteria are given below

RMSE = radic 1119899

119899

sum119905=1

(119910119905minus 119910119905)2

MAPE = 1119899

119899

sum119905=1

|BIAS|

(9)

4 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily Brent crude oil (20 May 1987ndash30 Sept 2006) Daily WTI crude oil (1 Jan 1986ndash30 Sept 2006)

Figure 2 Daily Brent and WTI crude oil prices series

where BIAS = (119910119905minus 119910119905)119910119905 119910119905is the actual and 119910

119905is the

forecasted value of period 119905 and 119899 is the number of totalobservations The criteria to judge for the best model arerelatively small RMSE and MAPE found in the training andtesting of the data

5 Fitting ANN Model to the Data

At first the multilayer perceptron (MLP) feedforward ANNmodel without any data preprocessing was used to modelthe crude oil price series One of the most important stepsin the model development process of the ANN model is thedetermination of significant input variables There are nofixed rules for the selection of input variables for developingANN model even though a general framework can befollowed based on previous successful application in crudeoil prices problems [39] For Brent and WTI data six inputcombinations based on previous log return of daily oil pricesare evaluated to estimate current prices value The inputcombinations evaluated in the study are (i) 119910

119905minus1 (ii) 119910

119905minus1 119910119905minus2

(iii) 119910

119905minus1 119910119905minus2

119910119905minus3

(iv) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

(v) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

and (vi) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

119910119905minus6

ANNmodel used in this study is the standard three-layer

feedforward network For improving the network modelingand reducing the chance of being trapped in a local minimadata normalization is applied in ANN input and outputDue to the crude oil prices series for both data collected arenonstationary (see Figure 2) to improve the accuracy of thetraining and testing each data point in the input neuronshad to be preprocessed by normalizing and transformedwithin the range of [minus1 1] There are no fixed rules as towhich standardization approach should be used in particularcircumstances In this study normalization for each raw inputand output data set was calculated by taking log return ofcurrent oil prices as the following formula

119877119905= log(

119910119905

119910119905minus1

) (10)

where 119910119905and 119910

119905minus1are current and one-period lagged prices

To achieve optimal weights of ANN LM algorithm pro-vided by theMATLABneural network toolbox is used to trainthe network In this study the hyperbolic tangent sigmoid

function and a linear function are employed as activationfunctions for the hidden and output layers respectively Theinitial value of learning rate is set to 0001 and a momentumcoefficient of 09 The learning rate is increased by a factor of10 and a decrease step of 01 until the change above resultsin a reduced performance value The number of epochs thatare used to train is set to 1000 The training of ANN will stopwhen the error achieves 10minus5 or when the number of epochsreaches 1000

The hidden layer plays important roles inmany successfulapplications of ANN It has been proven that only one hiddenlayer is sufficient for ANN to approximate any complexnonlinear function with any desired accuracy In the caseof the popular one hidden layer networks several practicalnumbers of neurons in the hidden layer were identified forbetter forecasting accuracy The optimal number of nodesin the hidden layer was identified using several practicalguidelines Berry and Linoff [40] claimed that the numberof hidden nodes should never be more than 2119868 where 119868 isthe number of inputs Hecht-Nielsen [41] claimed that thenumber of hidden neuron is equal to 2119868 + 1 However as far asthe number of hiddenneurons is concerned there is currentlyno theory to determine how many nodes in the hidden layerare optimal In the present study the number of hidden nodeswas progressively increased from 1 to 2119868 + 1

A program code including the wavelet toolbox waswritten in MATLAB language for the development of theANNmodel The optimal complexity of ANNmodel that isthe number of input and hidden nodes was determined by atrial-and-error approach Table 1 shows the best performanceresults of different ANN with different numbers of hiddenneurons from 1 to 2119868 + 1 for both data series The trainingset is used to estimate parameters for any specific ANNarchitecture The testing set is then used to select the bestANN among all numbers of hidden neurons considered

Table 1 shows the performance of ANN varying in accor-dance with the number of neurons in the hidden layer Itcan be seen that the prediction errors (MAPE) by ANNrange from 160 to 228 for Brent data and from 167to 180 for WTI data For Brent data the best architectureaccording RMSE and MAPE criterion for training data andtesting data has 2 input layer neurons 4 hidden layer neuronsand 1 output layer neuron (2-4-1) For this architecture RMSE

Mathematical Problems in Engineering 5

Table 1 Training and testing performance of different network architecture of ANN model

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 1 07281 228 09932 1652 4 05023 160 09910 1643 2 05108 160 09912 1644 2 05059 160 09910 1645 1 05244 161 09928 1656 1 05231 161 09920 165

WTI

1 3 05651 172 09507 1792 4 05590 172 09452 1793 5 05187 169 09496 1794 7 05137 167 09514 1805 1 05626 172 09529 1806 1 05636 172 09493 179

Table 2 Coefficient of determination (1198772) between each of subtime series for Brent and WTI data

Data DWT Coefficient of determination (1198772) with original crude oil prices series Mean119905 minus 1 119905 minus 2 119905 minus 3 119905 minus 4 119905 minus 5 119905 minus 6 119877

2

Brent

D1 00000 00000 00000 00000 00000 00000 00000D2 00000 00000 00001 00002 00000 00000 00001D3 00006 00006 00002 00000 00002 00002 00003A3 09895 09917 09937 09954 09967 09967 09940

WTI

D1 00000 00052 00021 00053 00000 01619 00291D2 00027 00149 00001 00866 01092 00274 00402D3 00353 00622 00472 00060 00107 00750 00394A3 00040 00143 00311 00511 00721 00886 00435

value is 09910 and MAPE value is 164 for testing datawhile forWTI data the best architecture according to RMSE(09452) andMAPE (179) criterion for testing data is 2-4-1

6 Fitting Hybrid Wavelet-ANNModel to the Data

The hybrid wavelet and ANN (WANN) model is obtained bycombining two methods discrete wavelet transform (DWT)andANNmodel InWANN the original crude oil price serieswas decomposed into a certain number of subtime seriescomponents which were entered to ANN in order to improvethe model accuracy In this study the Daubechies waveletone of the most widely used wavelet families is chosenas the wavelet function to decompose the original series(Eynard et al [42] Bagheri et al [43]) When conductingwavelet analysis the number of decomposition levels thatare appropriate for the data must be chosen To choose thenumber of decomposition level the following formula is used[34]

119871 = int [log (119873)] (11)

where 119871 is the level of decomposition and 119873 is the numberof time series data According to this formula the optimalnumber of decomposition levels for both crude oil price

series data in this study would have been 3 For this purposefirstly the original oil price series is decomposed into 3level components (D1 D2 and D3) that stand for differentfrequency components of the original series (details) and anapproximation component (A3) as shown in Figures 3 and 4

Each of Drsquos series plays a distinct role in the original timeseries and has different effects on the original crude priceseries Instead of using each Drsquos component individually asmodel input employment of the added suitable Drsquos compo-nent is more useful and can highly increase forecast per-formances of hybrid models In this study the effectivenessof wavelet components is determined using the coefficientof determination (R2) between each Drsquos subtime series andoriginal dataTheR2 values are given in Table 2 It can be seenthat R2of the D1 is zero for Brent data and 00291 for WTIdata However the wavelet components D2 and D3 showsignificantly higher R2 compared to the D1 for both seriesAccording to the R2 analyses the effective components D2and D3 were selected as the dominant wavelet componentsAfterward the significant wavelet components D2 D3 andapproximation (A3) component were added to each other asANN model input for crude oil forecast Figure 6 shows thestructure of the WANN model Figure 5 shows the originalcrude oil prices and their Ds that is the time series of 2-monthmode (D1) 4-monthmode (D2) 8-monthmode (D3)

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

Mathematical Problems in Engineering 3

hard to choose a proper learning rate because a low-learningratemay result in long training times and a high learning ratemay cause system instability [33]

To overcome the weakness of BP algorithm manyresearchers have investigated the use of Levenberg-Marquardt (LM) algorithm The LM algorithm is one ofthe most popular and more efficient nonlinear optimisationmethods It is a variation of Newtonrsquos method using Hessian-based algorithm for nonlinear least squares optimizationwhich is used in most optimisation packages The LMalgorithm takes large steps down the gradient where thegradient is small such as near local minima and takes smallsteps when the gradient is large Its faster convergencerobustness and the ability to find good local minima makeit attractive in ANN training This optimisation techniqueis more powerful than the conventional gradient descenttechnique [34ndash36]

3 Wavelet Transform

Thewavelet transforms is a strongmathematical tool that pro-vides a time-frequency representation of an analyzed signalin the time domin This overcomes the basic shortcoming ofFourier analysis which is that the Fourier spectrum containsonly globally averaged information Wavelet transformationsprovide useful decomposition of the original time seriesby capturing useful information on various decompositionlevels Wavelet transform can be divided by two categoriescontinuous wavelet transforms (CWT) and discrete wavelettransforms (DWT) The CWT of the time series 119909(119905) is givenby

119882(120591 119904) =1

radic|119904|intinfin

minusinfin

119909 (119905) 120595 lowast (119905 minus 120591

119904) 119889119905 (4)

where (lowast) indicates the conjugate complex function 119905 standsfor a time 120591 stands for the time step 119904 isin [0infin] standsfor the wavelet scale and 120595(119905) is the transforming functionand is called the mother wavelet The term wavelet meanssmall wave where small refers to the condition that thefunction is of a finite lengthThe wave refers to the conditionthat the function is of a finite length Several families ofwavelets (120595) that have been proven to be useful for variousapplications are described in related references (Mallat [37])The wavelets are chosen based on their shape and theirability to analyse the time series in a particular applicationTypical wavelet families such as Haar Daubechies CoifletSymlet Meyer Morlet and the Mexican Hat can be used asa mother wavelet The CWT is not often used for forecastingdue to its computationally complex and time requirementsto compute [38] Instead successive wavelet is often discretein forecasting applications to simplify the numeric solutionsDWT requires less computation time and is simpler to applyDWT is given by

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

01199041198980

1199041198980

) (5)

where 119898 and 119899 are integers that control the scale and timeThe most common and simplest choice for parameters are1199040= 2 and 120591 = 1 For a discrete time series 119909(119905) the DWT

becomes

119882119898 119899

= 2minus1198982

119873minus1

sum119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (6)

where119882119898 119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to the Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearly independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby

119909 (119905) = 119879 +

119872

sum119898=1

2119872minus119898minus1

sum119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (7)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum119898=1

119863119898(119905) (8)

in which 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905)(119898 = 1 2 119872) is detailed

subseries which can capture small features of interpretationalvalue in the data

4 Application

In this study two main crude oil prices Brent crude oilspot price andWest Texas Intermediate (WTI) crude oil spotprice (in US dollars per barrel) series were chosen as theexperimental sample The main reason for selecting thesecrude oil spot price indicators is that these two crude oilprices are the most famous benchmark prices which arewidely used as the basis of many crude oil price formulae[12 17] For Brent data we use the daily data from May 201987 to September 30 2006 with a total of 4933 observationsData from May 20 1987 to December 31 2002 are used forthe training set (3965 observations) and the remainder isused as the testing set (968 observations)

While for WTI data the daily data from January 11986 to September 30 2006 excluding public holidays witha total of 5237 were employed as experimental data forconvenience of WANN modeling the data from January 11986 toDecember 31 2000 are used for the training set (3800observations) and the remainder is used as the testing set(1437 observations)The graphical representation of the pricefor Brent and WTI crude oil price is given in Figure 2

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study two performance criteria suchas RMSE andMAPEwere used to evaluate the accuracy of themodels These criteria are given below

RMSE = radic 1119899

119899

sum119905=1

(119910119905minus 119910119905)2

MAPE = 1119899

119899

sum119905=1

|BIAS|

(9)

4 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily Brent crude oil (20 May 1987ndash30 Sept 2006) Daily WTI crude oil (1 Jan 1986ndash30 Sept 2006)

Figure 2 Daily Brent and WTI crude oil prices series

where BIAS = (119910119905minus 119910119905)119910119905 119910119905is the actual and 119910

119905is the

forecasted value of period 119905 and 119899 is the number of totalobservations The criteria to judge for the best model arerelatively small RMSE and MAPE found in the training andtesting of the data

5 Fitting ANN Model to the Data

At first the multilayer perceptron (MLP) feedforward ANNmodel without any data preprocessing was used to modelthe crude oil price series One of the most important stepsin the model development process of the ANN model is thedetermination of significant input variables There are nofixed rules for the selection of input variables for developingANN model even though a general framework can befollowed based on previous successful application in crudeoil prices problems [39] For Brent and WTI data six inputcombinations based on previous log return of daily oil pricesare evaluated to estimate current prices value The inputcombinations evaluated in the study are (i) 119910

119905minus1 (ii) 119910

119905minus1 119910119905minus2

(iii) 119910

119905minus1 119910119905minus2

119910119905minus3

(iv) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

(v) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

and (vi) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

119910119905minus6

ANNmodel used in this study is the standard three-layer

feedforward network For improving the network modelingand reducing the chance of being trapped in a local minimadata normalization is applied in ANN input and outputDue to the crude oil prices series for both data collected arenonstationary (see Figure 2) to improve the accuracy of thetraining and testing each data point in the input neuronshad to be preprocessed by normalizing and transformedwithin the range of [minus1 1] There are no fixed rules as towhich standardization approach should be used in particularcircumstances In this study normalization for each raw inputand output data set was calculated by taking log return ofcurrent oil prices as the following formula

119877119905= log(

119910119905

119910119905minus1

) (10)

where 119910119905and 119910

119905minus1are current and one-period lagged prices

To achieve optimal weights of ANN LM algorithm pro-vided by theMATLABneural network toolbox is used to trainthe network In this study the hyperbolic tangent sigmoid

function and a linear function are employed as activationfunctions for the hidden and output layers respectively Theinitial value of learning rate is set to 0001 and a momentumcoefficient of 09 The learning rate is increased by a factor of10 and a decrease step of 01 until the change above resultsin a reduced performance value The number of epochs thatare used to train is set to 1000 The training of ANN will stopwhen the error achieves 10minus5 or when the number of epochsreaches 1000

The hidden layer plays important roles inmany successfulapplications of ANN It has been proven that only one hiddenlayer is sufficient for ANN to approximate any complexnonlinear function with any desired accuracy In the caseof the popular one hidden layer networks several practicalnumbers of neurons in the hidden layer were identified forbetter forecasting accuracy The optimal number of nodesin the hidden layer was identified using several practicalguidelines Berry and Linoff [40] claimed that the numberof hidden nodes should never be more than 2119868 where 119868 isthe number of inputs Hecht-Nielsen [41] claimed that thenumber of hidden neuron is equal to 2119868 + 1 However as far asthe number of hiddenneurons is concerned there is currentlyno theory to determine how many nodes in the hidden layerare optimal In the present study the number of hidden nodeswas progressively increased from 1 to 2119868 + 1

A program code including the wavelet toolbox waswritten in MATLAB language for the development of theANNmodel The optimal complexity of ANNmodel that isthe number of input and hidden nodes was determined by atrial-and-error approach Table 1 shows the best performanceresults of different ANN with different numbers of hiddenneurons from 1 to 2119868 + 1 for both data series The trainingset is used to estimate parameters for any specific ANNarchitecture The testing set is then used to select the bestANN among all numbers of hidden neurons considered

Table 1 shows the performance of ANN varying in accor-dance with the number of neurons in the hidden layer Itcan be seen that the prediction errors (MAPE) by ANNrange from 160 to 228 for Brent data and from 167to 180 for WTI data For Brent data the best architectureaccording RMSE and MAPE criterion for training data andtesting data has 2 input layer neurons 4 hidden layer neuronsand 1 output layer neuron (2-4-1) For this architecture RMSE

Mathematical Problems in Engineering 5

Table 1 Training and testing performance of different network architecture of ANN model

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 1 07281 228 09932 1652 4 05023 160 09910 1643 2 05108 160 09912 1644 2 05059 160 09910 1645 1 05244 161 09928 1656 1 05231 161 09920 165

WTI

1 3 05651 172 09507 1792 4 05590 172 09452 1793 5 05187 169 09496 1794 7 05137 167 09514 1805 1 05626 172 09529 1806 1 05636 172 09493 179

Table 2 Coefficient of determination (1198772) between each of subtime series for Brent and WTI data

Data DWT Coefficient of determination (1198772) with original crude oil prices series Mean119905 minus 1 119905 minus 2 119905 minus 3 119905 minus 4 119905 minus 5 119905 minus 6 119877

2

Brent

D1 00000 00000 00000 00000 00000 00000 00000D2 00000 00000 00001 00002 00000 00000 00001D3 00006 00006 00002 00000 00002 00002 00003A3 09895 09917 09937 09954 09967 09967 09940

WTI

D1 00000 00052 00021 00053 00000 01619 00291D2 00027 00149 00001 00866 01092 00274 00402D3 00353 00622 00472 00060 00107 00750 00394A3 00040 00143 00311 00511 00721 00886 00435

value is 09910 and MAPE value is 164 for testing datawhile forWTI data the best architecture according to RMSE(09452) andMAPE (179) criterion for testing data is 2-4-1

6 Fitting Hybrid Wavelet-ANNModel to the Data

The hybrid wavelet and ANN (WANN) model is obtained bycombining two methods discrete wavelet transform (DWT)andANNmodel InWANN the original crude oil price serieswas decomposed into a certain number of subtime seriescomponents which were entered to ANN in order to improvethe model accuracy In this study the Daubechies waveletone of the most widely used wavelet families is chosenas the wavelet function to decompose the original series(Eynard et al [42] Bagheri et al [43]) When conductingwavelet analysis the number of decomposition levels thatare appropriate for the data must be chosen To choose thenumber of decomposition level the following formula is used[34]

119871 = int [log (119873)] (11)

where 119871 is the level of decomposition and 119873 is the numberof time series data According to this formula the optimalnumber of decomposition levels for both crude oil price

series data in this study would have been 3 For this purposefirstly the original oil price series is decomposed into 3level components (D1 D2 and D3) that stand for differentfrequency components of the original series (details) and anapproximation component (A3) as shown in Figures 3 and 4

Each of Drsquos series plays a distinct role in the original timeseries and has different effects on the original crude priceseries Instead of using each Drsquos component individually asmodel input employment of the added suitable Drsquos compo-nent is more useful and can highly increase forecast per-formances of hybrid models In this study the effectivenessof wavelet components is determined using the coefficientof determination (R2) between each Drsquos subtime series andoriginal dataTheR2 values are given in Table 2 It can be seenthat R2of the D1 is zero for Brent data and 00291 for WTIdata However the wavelet components D2 and D3 showsignificantly higher R2 compared to the D1 for both seriesAccording to the R2 analyses the effective components D2and D3 were selected as the dominant wavelet componentsAfterward the significant wavelet components D2 D3 andapproximation (A3) component were added to each other asANN model input for crude oil forecast Figure 6 shows thestructure of the WANN model Figure 5 shows the originalcrude oil prices and their Ds that is the time series of 2-monthmode (D1) 4-monthmode (D2) 8-monthmode (D3)

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Page 4: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

4 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily Brent crude oil (20 May 1987ndash30 Sept 2006) Daily WTI crude oil (1 Jan 1986ndash30 Sept 2006)

Figure 2 Daily Brent and WTI crude oil prices series

where BIAS = (119910119905minus 119910119905)119910119905 119910119905is the actual and 119910

119905is the

forecasted value of period 119905 and 119899 is the number of totalobservations The criteria to judge for the best model arerelatively small RMSE and MAPE found in the training andtesting of the data

5 Fitting ANN Model to the Data

At first the multilayer perceptron (MLP) feedforward ANNmodel without any data preprocessing was used to modelthe crude oil price series One of the most important stepsin the model development process of the ANN model is thedetermination of significant input variables There are nofixed rules for the selection of input variables for developingANN model even though a general framework can befollowed based on previous successful application in crudeoil prices problems [39] For Brent and WTI data six inputcombinations based on previous log return of daily oil pricesare evaluated to estimate current prices value The inputcombinations evaluated in the study are (i) 119910

119905minus1 (ii) 119910

119905minus1 119910119905minus2

(iii) 119910

119905minus1 119910119905minus2

119910119905minus3

(iv) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

(v) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

and (vi) 119910119905minus1

119910119905minus2

119910119905minus3

119910119905minus4

119910119905minus5

119910119905minus6

ANNmodel used in this study is the standard three-layer

feedforward network For improving the network modelingand reducing the chance of being trapped in a local minimadata normalization is applied in ANN input and outputDue to the crude oil prices series for both data collected arenonstationary (see Figure 2) to improve the accuracy of thetraining and testing each data point in the input neuronshad to be preprocessed by normalizing and transformedwithin the range of [minus1 1] There are no fixed rules as towhich standardization approach should be used in particularcircumstances In this study normalization for each raw inputand output data set was calculated by taking log return ofcurrent oil prices as the following formula

119877119905= log(

119910119905

119910119905minus1

) (10)

where 119910119905and 119910

119905minus1are current and one-period lagged prices

To achieve optimal weights of ANN LM algorithm pro-vided by theMATLABneural network toolbox is used to trainthe network In this study the hyperbolic tangent sigmoid

function and a linear function are employed as activationfunctions for the hidden and output layers respectively Theinitial value of learning rate is set to 0001 and a momentumcoefficient of 09 The learning rate is increased by a factor of10 and a decrease step of 01 until the change above resultsin a reduced performance value The number of epochs thatare used to train is set to 1000 The training of ANN will stopwhen the error achieves 10minus5 or when the number of epochsreaches 1000

The hidden layer plays important roles inmany successfulapplications of ANN It has been proven that only one hiddenlayer is sufficient for ANN to approximate any complexnonlinear function with any desired accuracy In the caseof the popular one hidden layer networks several practicalnumbers of neurons in the hidden layer were identified forbetter forecasting accuracy The optimal number of nodesin the hidden layer was identified using several practicalguidelines Berry and Linoff [40] claimed that the numberof hidden nodes should never be more than 2119868 where 119868 isthe number of inputs Hecht-Nielsen [41] claimed that thenumber of hidden neuron is equal to 2119868 + 1 However as far asthe number of hiddenneurons is concerned there is currentlyno theory to determine how many nodes in the hidden layerare optimal In the present study the number of hidden nodeswas progressively increased from 1 to 2119868 + 1

A program code including the wavelet toolbox waswritten in MATLAB language for the development of theANNmodel The optimal complexity of ANNmodel that isthe number of input and hidden nodes was determined by atrial-and-error approach Table 1 shows the best performanceresults of different ANN with different numbers of hiddenneurons from 1 to 2119868 + 1 for both data series The trainingset is used to estimate parameters for any specific ANNarchitecture The testing set is then used to select the bestANN among all numbers of hidden neurons considered

Table 1 shows the performance of ANN varying in accor-dance with the number of neurons in the hidden layer Itcan be seen that the prediction errors (MAPE) by ANNrange from 160 to 228 for Brent data and from 167to 180 for WTI data For Brent data the best architectureaccording RMSE and MAPE criterion for training data andtesting data has 2 input layer neurons 4 hidden layer neuronsand 1 output layer neuron (2-4-1) For this architecture RMSE

Mathematical Problems in Engineering 5

Table 1 Training and testing performance of different network architecture of ANN model

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 1 07281 228 09932 1652 4 05023 160 09910 1643 2 05108 160 09912 1644 2 05059 160 09910 1645 1 05244 161 09928 1656 1 05231 161 09920 165

WTI

1 3 05651 172 09507 1792 4 05590 172 09452 1793 5 05187 169 09496 1794 7 05137 167 09514 1805 1 05626 172 09529 1806 1 05636 172 09493 179

Table 2 Coefficient of determination (1198772) between each of subtime series for Brent and WTI data

Data DWT Coefficient of determination (1198772) with original crude oil prices series Mean119905 minus 1 119905 minus 2 119905 minus 3 119905 minus 4 119905 minus 5 119905 minus 6 119877

2

Brent

D1 00000 00000 00000 00000 00000 00000 00000D2 00000 00000 00001 00002 00000 00000 00001D3 00006 00006 00002 00000 00002 00002 00003A3 09895 09917 09937 09954 09967 09967 09940

WTI

D1 00000 00052 00021 00053 00000 01619 00291D2 00027 00149 00001 00866 01092 00274 00402D3 00353 00622 00472 00060 00107 00750 00394A3 00040 00143 00311 00511 00721 00886 00435

value is 09910 and MAPE value is 164 for testing datawhile forWTI data the best architecture according to RMSE(09452) andMAPE (179) criterion for testing data is 2-4-1

6 Fitting Hybrid Wavelet-ANNModel to the Data

The hybrid wavelet and ANN (WANN) model is obtained bycombining two methods discrete wavelet transform (DWT)andANNmodel InWANN the original crude oil price serieswas decomposed into a certain number of subtime seriescomponents which were entered to ANN in order to improvethe model accuracy In this study the Daubechies waveletone of the most widely used wavelet families is chosenas the wavelet function to decompose the original series(Eynard et al [42] Bagheri et al [43]) When conductingwavelet analysis the number of decomposition levels thatare appropriate for the data must be chosen To choose thenumber of decomposition level the following formula is used[34]

119871 = int [log (119873)] (11)

where 119871 is the level of decomposition and 119873 is the numberof time series data According to this formula the optimalnumber of decomposition levels for both crude oil price

series data in this study would have been 3 For this purposefirstly the original oil price series is decomposed into 3level components (D1 D2 and D3) that stand for differentfrequency components of the original series (details) and anapproximation component (A3) as shown in Figures 3 and 4

Each of Drsquos series plays a distinct role in the original timeseries and has different effects on the original crude priceseries Instead of using each Drsquos component individually asmodel input employment of the added suitable Drsquos compo-nent is more useful and can highly increase forecast per-formances of hybrid models In this study the effectivenessof wavelet components is determined using the coefficientof determination (R2) between each Drsquos subtime series andoriginal dataTheR2 values are given in Table 2 It can be seenthat R2of the D1 is zero for Brent data and 00291 for WTIdata However the wavelet components D2 and D3 showsignificantly higher R2 compared to the D1 for both seriesAccording to the R2 analyses the effective components D2and D3 were selected as the dominant wavelet componentsAfterward the significant wavelet components D2 D3 andapproximation (A3) component were added to each other asANN model input for crude oil forecast Figure 6 shows thestructure of the WANN model Figure 5 shows the originalcrude oil prices and their Ds that is the time series of 2-monthmode (D1) 4-monthmode (D2) 8-monthmode (D3)

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 5: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

Mathematical Problems in Engineering 5

Table 1 Training and testing performance of different network architecture of ANN model

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 1 07281 228 09932 1652 4 05023 160 09910 1643 2 05108 160 09912 1644 2 05059 160 09910 1645 1 05244 161 09928 1656 1 05231 161 09920 165

WTI

1 3 05651 172 09507 1792 4 05590 172 09452 1793 5 05187 169 09496 1794 7 05137 167 09514 1805 1 05626 172 09529 1806 1 05636 172 09493 179

Table 2 Coefficient of determination (1198772) between each of subtime series for Brent and WTI data

Data DWT Coefficient of determination (1198772) with original crude oil prices series Mean119905 minus 1 119905 minus 2 119905 minus 3 119905 minus 4 119905 minus 5 119905 minus 6 119877

2

Brent

D1 00000 00000 00000 00000 00000 00000 00000D2 00000 00000 00001 00002 00000 00000 00001D3 00006 00006 00002 00000 00002 00002 00003A3 09895 09917 09937 09954 09967 09967 09940

WTI

D1 00000 00052 00021 00053 00000 01619 00291D2 00027 00149 00001 00866 01092 00274 00402D3 00353 00622 00472 00060 00107 00750 00394A3 00040 00143 00311 00511 00721 00886 00435

value is 09910 and MAPE value is 164 for testing datawhile forWTI data the best architecture according to RMSE(09452) andMAPE (179) criterion for testing data is 2-4-1

6 Fitting Hybrid Wavelet-ANNModel to the Data

The hybrid wavelet and ANN (WANN) model is obtained bycombining two methods discrete wavelet transform (DWT)andANNmodel InWANN the original crude oil price serieswas decomposed into a certain number of subtime seriescomponents which were entered to ANN in order to improvethe model accuracy In this study the Daubechies waveletone of the most widely used wavelet families is chosenas the wavelet function to decompose the original series(Eynard et al [42] Bagheri et al [43]) When conductingwavelet analysis the number of decomposition levels thatare appropriate for the data must be chosen To choose thenumber of decomposition level the following formula is used[34]

119871 = int [log (119873)] (11)

where 119871 is the level of decomposition and 119873 is the numberof time series data According to this formula the optimalnumber of decomposition levels for both crude oil price

series data in this study would have been 3 For this purposefirstly the original oil price series is decomposed into 3level components (D1 D2 and D3) that stand for differentfrequency components of the original series (details) and anapproximation component (A3) as shown in Figures 3 and 4

Each of Drsquos series plays a distinct role in the original timeseries and has different effects on the original crude priceseries Instead of using each Drsquos component individually asmodel input employment of the added suitable Drsquos compo-nent is more useful and can highly increase forecast per-formances of hybrid models In this study the effectivenessof wavelet components is determined using the coefficientof determination (R2) between each Drsquos subtime series andoriginal dataTheR2 values are given in Table 2 It can be seenthat R2of the D1 is zero for Brent data and 00291 for WTIdata However the wavelet components D2 and D3 showsignificantly higher R2 compared to the D1 for both seriesAccording to the R2 analyses the effective components D2and D3 were selected as the dominant wavelet componentsAfterward the significant wavelet components D2 D3 andapproximation (A3) component were added to each other asANN model input for crude oil forecast Figure 6 shows thestructure of the WANN model Figure 5 shows the originalcrude oil prices and their Ds that is the time series of 2-monthmode (D1) 4-monthmode (D2) 8-monthmode (D3)

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D1

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D2

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

D3

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)

Daily (20 May 1987ndash30 Sept 2006)Daily (20 May 1987ndash30 Sept 2006)

Figure 3 Decomposed wavelet subtime series components (Ds) and approximation (As) of Brent crude oil

approximate mode (A3) and the combinations of effectivedetails and approximation components mode (DW = A2 +D2 + D3)

Six different combinations of the new series input dataare used in this study WANN model was trained in thesame way as the ANN model with the exception thatthe inputs were the combinations of effective details andapproximation componentsmode (DW) after the appropriatewavelet decomposition was selectedThe input combinationsfor WANNmodel evaluated in the study are

(i) DW119905minus1

(ii) DW

119905minus1 DW119905minus2

(iii) DW

119905minus1 DW119905minus2

DW119905minus3

(iv) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

(v) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

and(vi) DW

119905minus1 DW119905minus2

DW119905minus3

DW119905minus4

DW119905minus5

DW119905minus6

In all cases the input and output are transformed into thelog return of current oil prices The selection of the optimalnumber of nodes in both the input and hidden layers wasdone in the same way as for the ANN model The modelarchitecture for WANN consists of 1 to 6 nodes in the inputlayer (119868) 1 to 2119868 + 1 nodes in the hidden layer and one neuronin the output layerThe data was partitioned into training andtesting sets in the same manner as the ANNmodel

The performance measures of WANN for both crude oildata are shown in Table 3 According to Table 3 the bestarchitecture of ANN according to RMSE andMAPE criterionis 4-4-1 for Brent data and 5-1-1 for WTI data

For further analysis the best performances are comparedfor the ANN and WANN models in terms of the RMSE andMAPE in the testing phase The statistical results of differentmodels are summarized in Table 4

For Brent data WANN improved ANN forecast about177 and 201 reduction in RMSE and MAPE valuesrespectively ForWTI dataWANN obtained the best value ofRMSE and MAPE during the testing phase which decreasesby 201 and 223 respectively comparing with ANNGenerally it can be observed thatWANNmodel outperformsANN for both data Thus the results indicate that WANN isable to obtain the best result in terms of different evaluationmeasures during the testing phase

In order to further compare in terms of forecastingaccuracy Diebold-Mariano (DM) test statistic is used totest the statistical significance of different forecasting models[44] DM statistic can be defined as

DM =119889

radic119881119889119865 (12)

where 119889119905= sum119865

119905=1(119910119905minus 119910WANN119905)

2minus sum119865

119905=1(119910119905minus 119910ANN119905)

2 119889 =(sum119865

119905=1119889119905)119865 119881

119889= 1205740+ 2sum

infin

119897=1120574119897 and 120574

119897= cov(119889

119905 119889119905minus119897)

119910ANN119905 and119910WANN119905 are forecast values fromANNandWANNmodels respectively The null hypothesis of equal forecastaccuracy is rejected if the test statistic is negative andstatistically significant Table 4 reports the estimate values ofDM test statistic for testing performance ofWANNcomparedwith ANN Table 4 shows that all DM values of the proposedWANN for Brent and WTI data are below minus2326 andthe 119875 values are far more smaller than 1 indicating that

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

Mathematical Problems in Engineering 7

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

A3

-app

roxi

mat

ion

minus4

minus3

minus2

minus1

0

1

2

3

0 1000 2000 3000 4000 5000D1

minus8

minus6

minus4

minus2

0

2

4

0 1000 2000 3000 4000 5000

D2

minus4

minus3

minus2

minus1

0

1

2

3

4

0 1000 2000 3000 4000 5000

D3

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Daily (1 Jan 1986ndash30 Sept 2006)

Figure 4 Decomposed wavelet subtime series components (Ds) and approximation (As) of WTI crude oil

Data collection (yt)

Input time seriesyt = f(ytminus1 ytminus2 ytminusp)

Decompose inputusing DWT

ytminus1 ytminus2ytminusp

middot middot middot

Add theeffects of Ds

A3tminus1 A3tminus2A3tminusp

The effectiveness of DWT asinput for ANN

D1tminus1 D2tminus1 D3tminus3

DWtminus1DWtminus2 DWtminusp

D1tminus2 D2tminus2 D3tminus2D1tminusp D2tminusp D3tminusp

yt = f(DWtminus1DWtminus2 DWtminusp)

Figure 5 The structure of the WANNmodel

the proposed model performs the best in forecasting oil pricein short term at least at the 99 confidence level Now DMtest statistic provides stronger support for WANNmodel

Figure 6 shows the box plots of the BIAS values computedusing ANN andWANN for Brent andWTI data The pictureproduced consists of the most extreme values in the data setthe lower and upper quartiles and the median The criterion

for selecting a suitable model is based on the minimumachieved by median BIAS value and dispersion in BIASFigure 6 shows that the performance ofWANN is better thanthe ANN for Brent and WTI data

Finally in order to evaluate the efficiency of the proposedhybrid model the obtained results were also compared withthe results of ANN andARIMA [12] andGARCH [17]models

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

8 Mathematical Problems in Engineering

WANNANN

010

005

000

minus005

minus010

minus015

minus020

Bias

()

WTI crude oil prices

WANNANN

010

005

000

minus005

minus010

minus015

Bias

()

Brent crude oil prices

Figure 6 Comparison of ANN and WANNmodels for the testing results

Table 3 Training and testing performance of different network architecture of WANNmodel

Data Network architecture Training TestingInput Hidden RMSE MAPE RMSE MAPE

Brent

1 3 04437 146 08939 1492 2 04225 135 08439 1413 7 03756 124 08174 1354 4 03736 120 08151 1315 8 03489 115 08593 1316 8 03431 114 09061 137

WTI

1 1 05108 160 08738 1652 3 04617 146 08216 1543 2 04252 137 07900 1474 3 04171 133 07699 1425 1 04044 129 07549 1396 10 03712 124 09506 149

Table 4 Forecasting performance indices of ANN and WANN

Data ANN WANN DMRMSE MAPE RMSE MAPE WANN versus ANN

Brent 09910 164 08151 131 minus40416lowast

WTI 09452 179 07549 139 minus93756lowastlowastSignificant at 1 level

using the samedataThe comparison has been summarized inTable 5The results are tabulated inTable 5 inwhich the linearARIMA and GARCH models were unable to completelymodel the complex nonlinearity and high irregularity of thecrude oil prices However when ANN is used the modelcould capture both nonlinear and complex features of thecrude oil price series ANN model performs much betterthan the ARIMA and GARCH models which demonstratethe existence and importance of nonlinear and nonstationarybehavior in the crude oil price time series However ANNmodel is less efficient compared to proposed WANN modelIt is observed that the proposed model yields better resultthan the other models for both crude oil price data Thisresult shows that the new input series from discrete wavelet

transforms have significant extremely positive effect on ANNmodel results

7 Conclusions

In this study the wavelet transform and the ANNmodel werecombined in order to develop a hybrid model for modelingand forecasting the crude oil price series In the first modelANN model without any data preprocessing was used tomodel the crude oil price series In the second proposedhybrid model the wavelet transform can capture the multi-scale features of a time series which was used to decomposethe crude oil price series In this study the new series obtainedby adding the effective wavelet components was used as inputto the ANN model to forecast the crude oil price at onetime step ahead It is recommended that the sum of effectivedetails and the approximation component in modeling byANN should be selected rather than using all subtimeseries components Effective wavelet components are selectedbased on correlation between particular component series(approximation series and some details series) and crude oilprice seriesThe performance of the proposedWANNmodelwas compared to ANN model The hybrid model showed

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

Mathematical Problems in Engineering 9

Table 5 The RMSE and MAPE comparisons for different models

Model Brent WTIMAE RMSE MAPE RMSE

ANN 164 09910 179 09452WANN-proposed model 131 01851 139 07449GARCH (Rana and Ani [17]) 18947 09518Yursquo ARIMA (Yu et al [12]) 20350 20350Yursquo ANN (Yu et al [12]) 08410 08410

a great improvement in crude oil price modeling and pro-duced better forecasts than ANN model alone The studyconcludes that the forecasting abilities of WANN model arefound to be improved when the wavelet transformation tech-nique is adopted for the data preprocessingThe decomposedperiodic components obtained from DWT technique arefound to be most effective in yielding accurate forecast whenused as inputs inANNmodelThe accurate forecasting resultsindicate thatWANNmodel provides a superior alternative toother models and a potentially very useful new method forcrude oil price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Universiti TeknologiMalaysia and Ministry Of Science Technology and Inno-vation (MOSTI) for providing research funding (Grant no4F399) to support this piece of research

References

[1] H Mohammadi and L Su ldquoInternational evidence on crudeoil price dynamics applications of ARIMA-GARCH modelsrdquoEnergy Economics vol 32 no 5 pp 1001ndash1008 2010

[2] M I Ahmad ldquoModelling and forecasting Oman Crude OilPrices using Box-Jenkins techniquesrdquo International Journal ofTrade and Global Markets vol 5 pp 24ndash30 2012

[3] S Y Wang L Yu and K K Lai ldquoA novel hybrid AI systemframework for crude oil price forecastingrdquo Lecture Notes inComputer Science vol 3327 pp 233ndash242 2004

[4] W Xie L Yu S Xu and S Wang ldquoA new method for crudeoil price forecasting based on support Vector machinesrdquo inProceedings of the International Conference on ComputationalScience (Part IV) pp 444ndash451 2006

[5] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[6] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[7] P Agnolucci ldquoVolatility in crude oil futures a comparison ofthe predictive ability of GARCH and implied volatility modelsrdquoEnergy Economics vol 31 no 2 pp 316ndash321 2009

[8] R Lackes C Borgermann and M Dirkmorfeld ldquoForecastingthe price development of crude oil with artificial neural net-worksrdquo Distributed Computing Artificial Intelligence Bioinfor-matics Soft Computing and Ambient Assisted Living vol 5518pp 248ndash255 2009

[9] H Khazem and A Mazouz ldquoForecasting the price of crudeoil using artificial neural networksrdquo International Journal ofBusiness Marketing and Decision Sciences vol 6 no 1 pp 119ndash135 2013

[10] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[11] S Kulkarni and I Haidar ldquoForecastingmodel for crude oil priceand commodity futures prices using artificial neural networksrdquoInternational Journal of Computer Science amp Information Secu-rity vol 2 no 1 pp 81ndash88 2009

[12] 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

[13] J W Hu Y Hu and R R Lin ldquoApplying neural networks toprices prediction of crude oil futuresrdquo Mathematical Problemsin Engineering vol 2012 Article ID 959040 12 pages 2012

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

[15] O Kisi ldquoNeural network and wavelet conjunction model formodelling monthly level fluctuations in Turkeyrdquo HydrologicalProcesses vol 23 no 14 pp 2081ndash2092 2009

[16] J Adamowski and K Sun ldquoDevelopment of a coupled wavelettransform and neural network method for flow forecastingof non-perennial rivers in semi-arid watershedsrdquo Journal ofHydrology vol 390 no 1-2 pp 85ndash91 2010

[17] A A Rana and S Ani ldquoDaily crude oil price forecastingmodel using arima generalized autoregressive conditional het-eroscedastic and support vector machinesrdquoAmerican Journal ofApplied Sciences vol 11 no 3 pp 425ndash432 2014

[18] P Hajto ldquoA neural economic time series prediction with the useof a wavelet analysisrdquo Schedae Informaticae vol 11 pp 115ndash1322002

[19] W Tong and Y Li ldquoWavelet method combining BP networksand time series ARMA modeling for data mining forecastingrdquoin Advances in Natural Computation vol 3611 of Lecture Notesin Computer Science pp 123ndash134 Springer NewYork NYUSA2005

[20] V Nourani M Komasi and A Mano ldquoA multivariate ANN-wavelet approach for rainfall-runoffmodelingrdquoWater ResourcesManagement vol 23 no 14 pp 2877ndash2894 2009

[21] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceeding of the International Forum onInformation Technology and Applications vol 1 pp 231ndash2342009

[22] Y Bao X Zhang L Yu K K Lai and SWang ldquoAnmodel usingwavelet decomposition squares support vector machines crudeoil prices forecastingrdquoNewMathematics Computation vol 7 no299 pp 299ndash311 2011

[23] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

10 Mathematical Problems in Engineering

[24] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[25] K He L Yu and K K Lai ldquoCrude oil price analysis and fore-casting using wavelet decomposed ensemble modelrdquo Energyvol 46 no 1 pp 564ndash574 2012

[26] W E Shambora and R Rossiter ldquoAre there exploitable ineffi-ciencies in the futures market for oilrdquo Energy Economics vol29 no 1 pp 18ndash27 2007

[27] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[28] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[29] G Zhang B Eddy Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[30] P Coulibaly and N D Evora ldquoComparison of neural networkmethods for infilling missing daily weather recordsrdquo Journal ofHydrology vol 341 no 1-2 pp 27ndash41 2007

[31] D E Rumelhart J L McClelland and The PDP ResearchGroup Parallel Distributed Processing Explorations in theMicrostructure of Cognition vol 1-2 MIT Press CambridgeMass USA 1986

[32] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[33] H R Maier and G C Dandy ldquoNeural networks for theprediction and forecasting of water resources variables a reviewof modelling issues and applicationsrdquo Environmental Modellingand Software vol 15 no 1 pp 101ndash124 2000

[34] V Nourani M T Alami and M H Aminfar ldquoA combinedneural-wavelet model for prediction of Ligvanchai watershedprecipitationrdquo Engineering Applications of Artificial Intelligencevol 22 no 3 pp 466ndash472 2009

[35] B Khalil T B M J Ouarda and A St-Hilaire ldquoEstimation ofwater quality characteristics at ungauged sites using artificialneural networks and canonical correlation analysisrdquo Journal ofHydrology vol 405 no 3-4 pp 277ndash287 2011

[36] J P S Catalao H M I Pousinho and V M F Mendes ldquoShort-term wind power forecasting in Portugal by neural networksand wavelet transformrdquo Renewable Energy vol 36 no 4 pp1245ndash1251 2011

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] R Gencay F Selcuk and B Whitcher An Introduction toWavelets andOther FilteringMethods in Finance and EconomicsAcademic Press San Diego Calif USA 2002

[39] I Haidar S Kulkarni andH Pan ldquoForecastingmodel for crudeoil prices based on artificial neural networksrdquo in Proceedingsof the International Conference on Intelligent Sensors SensorNetworks and Information Processing pp 103ndash108 2008

[40] M J A Berry and G S Linoff Data Mining Techniques ForMarketing Sales and Customer Support John Wiley amp SonsNew York NY USA 1997

[41] R Hecht-Nielsen Neurocomputing Addison-Wesley ReadingMass USA 1990

[42] J Eynard S Grieu and M Polit ldquoWavelet-based multi-resolution analysis and artificial neural networks for forecastingtemperature and thermal power consumptionrdquo EngineeringApplications of Artificial Intelligence vol 24 no 3 pp 501ndash5162011

[43] A Bagheri H M Peyhani and M Akbari ldquoFinancial fore-casting using ANFIS networks with quantum-behaved particleswarm optimizationrdquo Expert Systems with Applications vol 41no 14 pp 6235ndash6250 2014

[44] F X Diebold and R S Mariano ldquoComparing predictiveaccuracyrdquo Journal of Business and Economic Statistics vol 20no 1 pp 134ndash144 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Daily Crude Oil Price Forecasting Using ...downloads.hindawi.com/journals/mpe/2014/201402.pdf · Research Article Daily Crude Oil Price Forecasting Using Hybridizing

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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