Research Article Crude Oil Price Forecasting Based on...

9
Research Article Crude Oil Price Forecasting Based on Hybridizing Wavelet Multiple Linear Regression Model, Particle Swarm Optimization Techniques, and Principal Component Analysis Ani Shabri 1 and Ruhaidah Samsudin 2 1 Department of Science Mathematic, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor, Malaysia 2 Department of Soſtware Engineering, Faculty of Computing, Universiti Teknologi Malaysia (UTM), 81310 Johor, Malaysia Correspondence should be addressed to Ani Shabri; ani [email protected] Received 31 December 2013; Accepted 18 March 2014; Published 8 May 2014 Academic Editors: S. Balochian, V. Bhatnagar, and Y. Zhang 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. Crude oil prices do play significant role in the global economy and are a key input into option pricing formulas, portfolio allocation, and risk measurement. In this paper, a hybrid model integrating wavelet and multiple linear regressions (MLR) is proposed for crude oil price forecasting. In this model, Mallat wavelet transform is first selected to decompose an original time series into several subseries with different scale. en, the principal component analysis (PCA) is used in processing subseries data in MLR for crude oil price forecasting. e particle swarm optimization (PSO) is used to adopt the optimal parameters of the MLR model. To assess the effectiveness of this model, daily crude oil market, West Texas Intermediate (WTI), has been used as the case study. Time series prediction capability performance of the WMLR model is compared with the MLR, ARIMA, and GARCH models using various statistics measures. e experimental results show that the proposed model outperforms the individual models in forecasting of the crude oil prices series. 1. Introduction Crude oil prices do play significant role in the global economy and constitute an important factor affecting government’s plans and commercial sectors. Forecasting crude oil price is among the most important issues facing energy economists. erefore, proactive knowledge of its future fluctuations can lead to better decisions in several managerial levels. e literature dealing with forecasting crude oil is sub- stantial. e application of the classical time series models such as autoregressive moving average (ARMA) (Yu et al. [1], Mohammadi and Su [2], and Ahmad [3]) and econometric model such as generalized autoregressive conditional het- eroscedasticity (GARCH) type models (Agnolucci [4], Wei et al. [5], Liu and Wan [6]) for crude oil forecasting has received much attention in the last decade. But because the crude oil price has the volatility, nonlinearity, and irregularity, the classical and econometric model can lead to the decrease of the accuracy. Due to the limitations of the classical and econometric models, soſt-computing models, such as neural fuzzy (Ghaf- fari and Zare [7]), artificial neural networks (Kaboudan [8], Mirmirani and Li [9], Shambora and Rossiter [10], and Yu et al. [11]), support vector machines (Xie et al. [12]), and genetic programming (GP), provide powerful solutions to nonlinear crude oil price prediction. Many experiments found that the soſt-computing models oſten had some advantages over statistical-based models. However, these AI models also have their own shortcomings and disadvantages. For example, ANN oſten suffers from local minima and over-fitting, while other soſt-computing models, such as SVM and GP, including ANN, are sensitive to parameter selection [1]. To remedy the above shortcomings, some hybrid meth- ods have been used recently to predict crude oil price and obtain the best performances. In last year, wavelet transform has become a useful method for analyzing such as variations, periodicities, and trends in time series. e hybrid models with wavelet transform processes have been improved for Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 854520, 8 pages http://dx.doi.org/10.1155/2014/854520

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Research ArticleCrude Oil Price Forecasting Based on Hybridizing WaveletMultiple Linear Regression Model Particle Swarm OptimizationTechniques and Principal Component Analysis

Ani Shabri1 and Ruhaidah Samsudin2

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

Correspondence should be addressed to Ani Shabri ani sabrihotmailcom

Received 31 December 2013 Accepted 18 March 2014 Published 8 May 2014

Academic Editors S Balochian V Bhatnagar and Y Zhang

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

Crude oil prices do play significant role in the global economy and are a key input into option pricing formulas portfolio allocationand risk measurement In this paper a hybrid model integrating wavelet and multiple linear regressions (MLR) is proposed forcrude oil price forecasting In this model Mallat wavelet transform is first selected to decompose an original time series into severalsubseries with different scale Then the principal component analysis (PCA) is used in processing subseries data in MLR for crudeoil price forecasting The particle swarm optimization (PSO) is used to adopt the optimal parameters of the MLR model To assessthe effectiveness of this model daily crude oil market West Texas Intermediate (WTI) has been used as the case study Time seriesprediction capability performance of the WMLR model is compared with the MLR ARIMA and GARCH models using variousstatistics measures The experimental results show that the proposed model outperforms the individual models in forecasting ofthe crude oil prices series

1 Introduction

Crude oil prices do play significant role in the global economyand constitute an important factor affecting governmentrsquosplans and commercial sectors Forecasting crude oil price isamong the most important issues facing energy economistsTherefore proactive knowledge of its future fluctuations canlead to better decisions in several managerial levels

The literature dealing with forecasting crude oil is sub-stantial The application of the classical time series modelssuch as autoregressive moving average (ARMA) (Yu et al [1]Mohammadi and Su [2] and Ahmad [3]) and econometricmodel such as generalized autoregressive conditional het-eroscedasticity (GARCH) typemodels (Agnolucci [4]Wei etal [5] Liu andWan [6]) for crude oil forecasting has receivedmuch attention in the last decade But because the crudeoil price has the volatility nonlinearity and irregularity theclassical and econometric model can lead to the decrease ofthe accuracy

Due to the limitations of the classical and econometricmodels soft-computing models such as neural fuzzy (Ghaf-fari and Zare [7]) artificial neural networks (Kaboudan [8]Mirmirani and Li [9] Shambora and Rossiter [10] and Yu etal [11]) support vector machines (Xie et al [12]) and geneticprogramming (GP) provide powerful solutions to nonlinearcrude oil price prediction Many experiments found thatthe soft-computing models often had some advantages overstatistical-based models However these AI models also havetheir own shortcomings and disadvantages For exampleANN often suffers from local minima and over-fitting whileother soft-computingmodels such as SVMandGP includingANN are sensitive to parameter selection [1]

To remedy the above shortcomings some hybrid meth-ods have been used recently to predict crude oil price andobtain the best performances In last year wavelet transformhas become a useful method for analyzing such as variationsperiodicities and trends in time series The hybrid modelswith wavelet transform processes have been improved for

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 854520 8 pageshttpdxdoiorg1011552014854520

2 The Scientific World Journal

forecasting For example wavelet-neural network (Jammaziand Aloui [13] Qunli et al [14] and Yousefi et al [15])wavelet-least square support vector machines (LSVM) (Baoet al [16]) and wavelet-fuzzy neural network (Liu et al [17])have been employed recently on some studies in crude oilforecasting They observed that the wavelet transform fairlyimproves forecasting accuracy

A major drawback of wavelet transform for directionprediction is that the input variables lie in a high-dimensionalfeature space depends on the number of sub-time seriescomponents Because the number of sub-time series compo-nents for wavelet is inadvisable to be too many in this studyprincipal component analysis (PCA) is proposed to reducethe dimensions of sub-time series components

The multiple linear regressions (MLR) model that ismuch easier to interpret is considered as an alternative toANN model In this paper a hybrid wavelet multiple linearregression (WMLR) model integrating wavelet and MLR isproposed for short-term daily crude oil price forecastingThestudy applies particle swarm optimization (PSO) to adoptthe optimal parameters to construct the MLR model Forverification purpose the West Texas Intermediate (WTI)crude oil sport price is used to test the effectiveness of theproposedWMLR ensemble learning methodology Finally toevaluate themodel ability the proposedmodel was comparedwith individual ARIMA and GARCHmodels

2 Methodology

21TheARIMAModel Themost comprehensive of all popu-lar and widely known statistical methods used for time seriesforecasting are Box-Jenkins models (Box and Jenkins [18])It has achieved great success in both academic research andindustrial applications during the last three decades Thegeneral form of ARIMA models can be expressed as

119910119905=

119901

sum

119894=1

120601119894119910119905minus119894+

119902

sum

119894=1

120579119894119890119905minus119894+ 119890119905 (1)

where 119901 is the order of the autoregressive 119902 is the order of themoving average and 119890

119905is the random error The Box-Jenkins

methodology is basically divided into four steps identifica-tion estimation diagnostic checking and forecasting

22 The GARCH Model GARCH models have found exten-sive application in the literature and themost popular volatil-itymodel is GARCH (1 1)model proposed by Bollerslev [19]The standard GARCH (1 1) can be described as follows

119903119905= 120583119905+ 120576119905= 120583119905+ ℎ12

119905120578119905120578119905sim 119873 (0 1)

ℎ119905= 120596 + 120572120576

2

119905minus1+ 120573ℎ119905minus1

(2)

where 120583119905denote the conditional mean and ℎ

119905is the con-

ditional variances and 120578119905is a standardized error and 119903

119905=

ln(119909119905119909119905minus1) is log return

23 Multiple Linear Regressions Multiple linear regressions(MLR) model is one of the modelling techniques to investi-gate the relationship between a dependent variable and sev-eral independent variables Let the MLR have 119901 independentvariables with 119899 observations Thus the MLR can be writtenas

119884 = 1199080+ 1199081119909119894+ 11990821199092+ sdot sdot sdot + 119908

119901119909119901+ 120576119905 (3)

where 119908 are regression coefficients 119884 is dependent variable119909119894are independent varaiables and 120576

119905is fitting errors The

method of least squares is generally used to estimate thecoefficients model In many applications the results of a leastsquares fit are often unacceptable when themodel is wrong orwhen the model is misspecified (Bozdogan and Howe [20])

In this study particle swarm optimization (PSO) methodis presented to determine the optimal parameters of theMLRmodel The PSO methods have proven to be very effectivein solving a variety of difficult global optimization problemsin forecasting (Chen and Kao [21] and Alwee et al [22])heat problem (Ma et al [23] and Tyagi and Pandit [24]) anddynamic environments (Liu et al [25])

The classic solution of MLR model involves the mini-mization of the sum of the square errors between the model-predicted value and the corresponding data value

min119891 (119908) =119899

sum

119894=1

(119884119894minus 119894)2

(4)

where 119899 is the number of training data samples 119884119894is the

actual value and 119894is the forecasted value of train data

The same methodology was used to solve this problem usingPSO algorithms The solution with a smaller fitness 119891(119908) ofthe training data set has a better chance of surviving in thesuccessive generations

24 Particle Swarm Optimization Particle swarm optimiza-tion (PSO) is a population-based heuristic method inspiredby the collective motion of biological organisms such asbird flocking and fish schooling to simulate the seekingbehavior to a food source (Bratton and Kennedy [26]) Thepopulation of PSO is called a swarm and each individual inthe population of PSO is called a particle The PSO beginswith a random population and searchers for fitness optimumjust like genetic algorithm (GA) To find the optimumsolution each particle adjusts the direction through the bestexperience which it has found (119901best) and the best experiencethat has been found by all other members (119892best) Thereforethe particles fly around in a multidimensional space towardsthe better area over the search process

Each particle consists of three vectors the positionfor 119894th individual particle can be denoted as 119883

119894=

(119909(1)

119894 119909(2)

119894 119909

(119863)

119894) the best previous position 119901best that the

119894th particle has searched is 119875119894= (119901(1)

119894 119901(2)

119894 119901

(119863)

119894) and

the fly velocity of the 119894th is 119881119894= (V(1)119894 V(2)119894 V(119863)

119894) The

performance of each particle is measured using a fitness

The Scientific World Journal 3

function varying from problem in hand During the iterativeprocedure the 119894th particle at iteration 119905 is updated by

V119889119894(119905 + 1) = 120596

1015840times V119889119894(119905) + 119888

1times 1205931times [119901119889

119894(119905) minus 119909

119889

119894(119905)]

+ 1198882times 1205932times [119901119889

119892(119905) minus 119909

119889

119894(119905)]

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(5)

where120596 is called inertia weight 1198881and 1198882are acceleration con-

stants and 1205931and 120593

2are stochastic value of [0 1] In a PSO

system particles change their positions at each time step untila relatively unchanging position has been encountered or amaximum number of iterations have been met In generalthe performance of each particle is measured according to afitness function which is problemdependent InMLRmodel(4) is the fitness function under consideration Figure 1 showsthe flowchart of the developed PSO algorithm For furtherdetails regarding PSO please refer to Kennedy and Eberhart[27] and Bratton and Kennedy [26]

25 Wavelet Analysis Wavelet transformations provide use-ful decomposition of original time series by capturing use-ful information on various decomposition levels Discretewavelet transformation (DWT) is preferred in most of theforecasting problems because of its simplicity and ability tocompute with less time The DWT can be defined as

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

0119904119898

0

119904119898

0

) (6)

where119898 and 119899 are integers that control the scale and timeThemost common choices for the parameters 119904

0= 2 and 120591

0= 1

120595(119905) called themother wavelet can be defined asintinfinminusinfin120595(119905)119889119905 =

0For a discrete time series119909(119905)where119909(119905) occurs at discrete

time 119905 the DWT becomes

119882119898119899= 2minus1198982

119873minus1

sum

119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (7)

where119882119898119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearity independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby (Mallat [28])

119909 (119905) = 119879 +

119872

sum

119898=1

2119872minus119898minus1

sum

119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (8)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum

119898=1

119863119898(119905) (9)

where 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905) (119898 = 1 2 119872) are detail

subseries which can capture small features of interpretationalvalue in the data

Generate initial population

Fitnessevaluation

Convergencecheck

Updating

Parametersoptimization

Yes

Training MLR

Cross validationNo

Figure 1 Flowchart of PSO algorithm

26 Principal Component Analysis In an MLR one of maintasks is to determine the model input variables that affect theoutput variables significantly The choice of input variablesis generally based on a priori knowledge of causal variablesinspections of time series plots and statistical analysis ofpotential inputs and outputs PCA is a technique widely usedfor reducing the number of input variables when we havehuge volume of information and we want to have a betterinterpretation of variables (Camdevyren et al [29])

The PCA approach introduces a few combinations formodel input in comparison with the trial and error processGiven a set of centred input vectors 119909

1 1199092 119909

119898and

sum119898

119905=1119909119905= 0 usually 119899 lt 119898 Then the covariance matrix of

vector is given by

119862 =1

119897

119897

sum

119905=1

119909119905119909119879

119905 (10)

The principal components (PCs) are computed by solving theeigenvalue problem of covariance matrix 119862

120582119894119906119894= 119862119906119894 119894 = 1 2 119898 (11)

where 120582119894is one of the eigenvalues of 119862 and 119906

119894is the

corresponding eigenvector Based on the estimated 119906119894 the

components of 119911119905(119894) are then calculated as the orthogonal

transforms of 119909119905

119911119905(119894) = 119906

119879

119894119909119905 119894 = 1 2 119898 (12)

The new components 119911119894(119905) are called principal components

By using only the first several eigenvectors sorted in descend-ing order of the eigenvalues the number of principal com-ponents in 119911

119905can be reduced So PCA has the dimensional

reduction characteristic The principal components of PCAhave the following properties 119911

119905(119894) are linear combinations

of the original variables uncorrelated and have sequentially

4 The Scientific World Journal

maximum variances (Jolliffe [30]) The calculation variancecontribution rate is

119881119894=

120582119894

sum119898

119894=1120582119894

times 100 (13)

The cumulative variance contribution rate is

119881 (119901) =

119901

sum

119894=1

119881119894 (14)

The number of the selected principal components is based onthe cumulative variance contribution rate which as a rule isover 85sim90

3 Computer Simulation

31 An Application In this study the West Texas Intermedi-ate (WTI) crude oil price series was chosen as experimentalsampleThemain reason of selecting theWTI crude oil is thatthese crude oil prices are the most famous benchmark priceswhich are widely used as the basis of many crude oil priceformulae The daily data from January 1 1986 to September30 2006 excluding public holidays with a total of 5237 wasemployed as experimental data For convenience of WMLRmodeling the data from January 1 1986 to December 312000 is used for the training set (3800 observations) andthe remainder is used as the testing set (1437 observations)Figure 2 shows the daily crude oil prices from January 1 1986to September 30

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study three main performancecriteria are used to evaluate the accuracy of themodelsThesecriteria are mean absolute error (MAE) root mean squarederror (RMSE) and119863stat TheMAE and RMSE can be definedby

RMSE = radic 1119899

119899

sum

119905=1

(119910119905minus 119910119905)2

MAE = 1119899

119899

sum

119905=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816

(15)

In crude oil price forecasting improved decisions usuallydepend on correct forecasting of directions of actual price119910119905and forecasted price 119910

119905 The ability to predict movement

direction can bemeasured by a directional statistic (119863stat) (Yuet al [1]) which can be expressed as

119863stat =1

119873

119899

sum

119905=1

119886119905times 100

119886119905=

1 if (119910119905+1minus 119910119905) (119910119905+1minus 119910119905) ge 0

0 otherwise

(16)

32 Application and Result At first the MLR model withoutdata preprocessing was used to model daily oil prices In

0102030405060708090

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily (January 1 1986ndashSeptember 30 2006)

Figure 2 Daily crude oil prices from January 1 198 to September30 2006

the next step the preprocessed data which uses subtimeseries components obtained using discrete wavelet transform(DWT) on original data were entered to the MLR model inorder to improve themodel accuracy For theMLRmodel theoriginal log return time series are decomposed into a certainnumber of subtime series components Deciding the optimaldecomposition level of the time series data in wavelet analysisplays an important role in preserving the information andreducing the distortion of the datasets However there is noexisting theory to tell how many decomposition levels areneeded for any time series

In the present study the previous log return of daily oilprice time series is decomposed into various subtime series(DWs) at different decomposition levels by using DWT toestimate current price value Three decomposition levels (24 and 8 months) were considered for this study For theWTIseries data time series of 2-day mode (DW1) 4-day mode(DW2) and 8-day mode (DW3) and approximate mode arepresented in Figure 3

For the WTI series six input combinations based onprevious log return of daily oil prices are evaluated to esti-mate current prices value The input combinations evaluatedin the study are (i) 119903

119905minus1 (ii) 119903

119905minus1 119903119905minus2

(iii) 119903119905minus1 119903119905minus2 119903119905minus3

(iv) 119903

119905minus1 119903119905minus2 119903119905minus3 119903119905minus4

(v) 119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5

and (vi)119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5 119903119905minus6

In all cases the output is the logreturn of current oil prices 119903

119905

Each of DWs series plays distinct role in original timeseries and has different effects on the original prices oilseries The selection of dominant DWs as inputs of MLRmodel becomes important and effective on the output dataand has positive effect excessively on modelrsquos ability Themodel becomes exponentially more complex as the numberof subtime series as input variables increases Using a largenumber of input variables should be avoided to save time andcalculation effort Therefore the effectiveness of new seriesobtained by PCA is used as input to theMLRmodelThePCAapproach helps us to reduce the number of original variablesto a set of new variables Generally the objective of PCA is toidentify a new set of variables such that each variable called aprincipal component is a linear combination of the originalvariables The new set of variables accounts for 85minus90 of

The Scientific World Journal 5

01020304050607080

0 1000 2000 3000 4000 5000

A3-a

ppro

xim

atio

n

Daily (January 1 1986ndashSeptember 30 2006)

(a)

0

1

2

3

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D1

Daily (January 1 1986ndashSeptember 30 2006)

(b)

0

2

4

0 1000 2000 3000 4000 5000

minus8

minus6

minus4

minus2D2

Daily (January 1 1986ndashSeptember 30 2006)

(c)

0

1

2

3

4

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D3

Daily (January 1 1986ndashSeptember 30 2006)

(d)

Figure 3 Decomposed wavelet subtime series components (Ds) ofWTI crude oil price data

Decompose input using DWT

Running PCAthe effective of PCs as

input for MLR

Data collection (yt)

Preprocessing for data

rt = lnytytminus1

AM middot middot middot DM

Postprocess results (MLR)yt = ytminus1exp(rt)

D1 D2

Figure 4 The structure of the WMLR model

Table 1 Eigen value and cumulative variance contribution rate ofthe 8 principal components

PC 1 2 3 4 5 6 7 8Eigen value 197 179 159 133 067 041 021 003CumulativeVariance Rate 025 047 067 084 092 097 100 100

the total variation were considered as the number of newvariables

For example taking two previous daily oil prices as arandom variable Every previous daily oil price time seriesare decomposed using DWT into three decomposition levelsrespectively Thus there were 8 subseries considered forthe PCA analysis The result of PCA analysis is shown inTable 1 Table 1 shows that the first four principle componentscan explain 84 variation of the data variation with theeigenvalues greater than 1 to be retained in which all the 4PCs were included in the MLR model Thus the 8 originalvariables can be replaced by 4 new irrelevant variables Fortraining MLR the PSO algorithm solving the recognitionproblem is implemented and the program code includingwavelet toolbox was written in MATLAB language TheWMLRmodel structure developed in present study is shownin Figure 4

The forecasting performances of the MLR and WMLRmodels in terms of the MAE RMSE and 119863stat testing phaseare compared and shown in Table 2 Table 2 shows MLRmodel the M1 with 1 lag obtained the best MAE statisticsof 06948 and the M6 with 6 lags obtained the best RMSEstatistics of 09450 while the M1 with 5 lags obtained the best

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Page 2: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

2 The Scientific World Journal

forecasting For example wavelet-neural network (Jammaziand Aloui [13] Qunli et al [14] and Yousefi et al [15])wavelet-least square support vector machines (LSVM) (Baoet al [16]) and wavelet-fuzzy neural network (Liu et al [17])have been employed recently on some studies in crude oilforecasting They observed that the wavelet transform fairlyimproves forecasting accuracy

A major drawback of wavelet transform for directionprediction is that the input variables lie in a high-dimensionalfeature space depends on the number of sub-time seriescomponents Because the number of sub-time series compo-nents for wavelet is inadvisable to be too many in this studyprincipal component analysis (PCA) is proposed to reducethe dimensions of sub-time series components

The multiple linear regressions (MLR) model that ismuch easier to interpret is considered as an alternative toANN model In this paper a hybrid wavelet multiple linearregression (WMLR) model integrating wavelet and MLR isproposed for short-term daily crude oil price forecastingThestudy applies particle swarm optimization (PSO) to adoptthe optimal parameters to construct the MLR model Forverification purpose the West Texas Intermediate (WTI)crude oil sport price is used to test the effectiveness of theproposedWMLR ensemble learning methodology Finally toevaluate themodel ability the proposedmodel was comparedwith individual ARIMA and GARCHmodels

2 Methodology

21TheARIMAModel Themost comprehensive of all popu-lar and widely known statistical methods used for time seriesforecasting are Box-Jenkins models (Box and Jenkins [18])It has achieved great success in both academic research andindustrial applications during the last three decades Thegeneral form of ARIMA models can be expressed as

119910119905=

119901

sum

119894=1

120601119894119910119905minus119894+

119902

sum

119894=1

120579119894119890119905minus119894+ 119890119905 (1)

where 119901 is the order of the autoregressive 119902 is the order of themoving average and 119890

119905is the random error The Box-Jenkins

methodology is basically divided into four steps identifica-tion estimation diagnostic checking and forecasting

22 The GARCH Model GARCH models have found exten-sive application in the literature and themost popular volatil-itymodel is GARCH (1 1)model proposed by Bollerslev [19]The standard GARCH (1 1) can be described as follows

119903119905= 120583119905+ 120576119905= 120583119905+ ℎ12

119905120578119905120578119905sim 119873 (0 1)

ℎ119905= 120596 + 120572120576

2

119905minus1+ 120573ℎ119905minus1

(2)

where 120583119905denote the conditional mean and ℎ

119905is the con-

ditional variances and 120578119905is a standardized error and 119903

119905=

ln(119909119905119909119905minus1) is log return

23 Multiple Linear Regressions Multiple linear regressions(MLR) model is one of the modelling techniques to investi-gate the relationship between a dependent variable and sev-eral independent variables Let the MLR have 119901 independentvariables with 119899 observations Thus the MLR can be writtenas

119884 = 1199080+ 1199081119909119894+ 11990821199092+ sdot sdot sdot + 119908

119901119909119901+ 120576119905 (3)

where 119908 are regression coefficients 119884 is dependent variable119909119894are independent varaiables and 120576

119905is fitting errors The

method of least squares is generally used to estimate thecoefficients model In many applications the results of a leastsquares fit are often unacceptable when themodel is wrong orwhen the model is misspecified (Bozdogan and Howe [20])

In this study particle swarm optimization (PSO) methodis presented to determine the optimal parameters of theMLRmodel The PSO methods have proven to be very effectivein solving a variety of difficult global optimization problemsin forecasting (Chen and Kao [21] and Alwee et al [22])heat problem (Ma et al [23] and Tyagi and Pandit [24]) anddynamic environments (Liu et al [25])

The classic solution of MLR model involves the mini-mization of the sum of the square errors between the model-predicted value and the corresponding data value

min119891 (119908) =119899

sum

119894=1

(119884119894minus 119894)2

(4)

where 119899 is the number of training data samples 119884119894is the

actual value and 119894is the forecasted value of train data

The same methodology was used to solve this problem usingPSO algorithms The solution with a smaller fitness 119891(119908) ofthe training data set has a better chance of surviving in thesuccessive generations

24 Particle Swarm Optimization Particle swarm optimiza-tion (PSO) is a population-based heuristic method inspiredby the collective motion of biological organisms such asbird flocking and fish schooling to simulate the seekingbehavior to a food source (Bratton and Kennedy [26]) Thepopulation of PSO is called a swarm and each individual inthe population of PSO is called a particle The PSO beginswith a random population and searchers for fitness optimumjust like genetic algorithm (GA) To find the optimumsolution each particle adjusts the direction through the bestexperience which it has found (119901best) and the best experiencethat has been found by all other members (119892best) Thereforethe particles fly around in a multidimensional space towardsthe better area over the search process

Each particle consists of three vectors the positionfor 119894th individual particle can be denoted as 119883

119894=

(119909(1)

119894 119909(2)

119894 119909

(119863)

119894) the best previous position 119901best that the

119894th particle has searched is 119875119894= (119901(1)

119894 119901(2)

119894 119901

(119863)

119894) and

the fly velocity of the 119894th is 119881119894= (V(1)119894 V(2)119894 V(119863)

119894) The

performance of each particle is measured using a fitness

The Scientific World Journal 3

function varying from problem in hand During the iterativeprocedure the 119894th particle at iteration 119905 is updated by

V119889119894(119905 + 1) = 120596

1015840times V119889119894(119905) + 119888

1times 1205931times [119901119889

119894(119905) minus 119909

119889

119894(119905)]

+ 1198882times 1205932times [119901119889

119892(119905) minus 119909

119889

119894(119905)]

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(5)

where120596 is called inertia weight 1198881and 1198882are acceleration con-

stants and 1205931and 120593

2are stochastic value of [0 1] In a PSO

system particles change their positions at each time step untila relatively unchanging position has been encountered or amaximum number of iterations have been met In generalthe performance of each particle is measured according to afitness function which is problemdependent InMLRmodel(4) is the fitness function under consideration Figure 1 showsthe flowchart of the developed PSO algorithm For furtherdetails regarding PSO please refer to Kennedy and Eberhart[27] and Bratton and Kennedy [26]

25 Wavelet Analysis Wavelet transformations provide use-ful decomposition of original time series by capturing use-ful information on various decomposition levels Discretewavelet transformation (DWT) is preferred in most of theforecasting problems because of its simplicity and ability tocompute with less time The DWT can be defined as

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

0119904119898

0

119904119898

0

) (6)

where119898 and 119899 are integers that control the scale and timeThemost common choices for the parameters 119904

0= 2 and 120591

0= 1

120595(119905) called themother wavelet can be defined asintinfinminusinfin120595(119905)119889119905 =

0For a discrete time series119909(119905)where119909(119905) occurs at discrete

time 119905 the DWT becomes

119882119898119899= 2minus1198982

119873minus1

sum

119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (7)

where119882119898119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearity independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby (Mallat [28])

119909 (119905) = 119879 +

119872

sum

119898=1

2119872minus119898minus1

sum

119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (8)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum

119898=1

119863119898(119905) (9)

where 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905) (119898 = 1 2 119872) are detail

subseries which can capture small features of interpretationalvalue in the data

Generate initial population

Fitnessevaluation

Convergencecheck

Updating

Parametersoptimization

Yes

Training MLR

Cross validationNo

Figure 1 Flowchart of PSO algorithm

26 Principal Component Analysis In an MLR one of maintasks is to determine the model input variables that affect theoutput variables significantly The choice of input variablesis generally based on a priori knowledge of causal variablesinspections of time series plots and statistical analysis ofpotential inputs and outputs PCA is a technique widely usedfor reducing the number of input variables when we havehuge volume of information and we want to have a betterinterpretation of variables (Camdevyren et al [29])

The PCA approach introduces a few combinations formodel input in comparison with the trial and error processGiven a set of centred input vectors 119909

1 1199092 119909

119898and

sum119898

119905=1119909119905= 0 usually 119899 lt 119898 Then the covariance matrix of

vector is given by

119862 =1

119897

119897

sum

119905=1

119909119905119909119879

119905 (10)

The principal components (PCs) are computed by solving theeigenvalue problem of covariance matrix 119862

120582119894119906119894= 119862119906119894 119894 = 1 2 119898 (11)

where 120582119894is one of the eigenvalues of 119862 and 119906

119894is the

corresponding eigenvector Based on the estimated 119906119894 the

components of 119911119905(119894) are then calculated as the orthogonal

transforms of 119909119905

119911119905(119894) = 119906

119879

119894119909119905 119894 = 1 2 119898 (12)

The new components 119911119894(119905) are called principal components

By using only the first several eigenvectors sorted in descend-ing order of the eigenvalues the number of principal com-ponents in 119911

119905can be reduced So PCA has the dimensional

reduction characteristic The principal components of PCAhave the following properties 119911

119905(119894) are linear combinations

of the original variables uncorrelated and have sequentially

4 The Scientific World Journal

maximum variances (Jolliffe [30]) The calculation variancecontribution rate is

119881119894=

120582119894

sum119898

119894=1120582119894

times 100 (13)

The cumulative variance contribution rate is

119881 (119901) =

119901

sum

119894=1

119881119894 (14)

The number of the selected principal components is based onthe cumulative variance contribution rate which as a rule isover 85sim90

3 Computer Simulation

31 An Application In this study the West Texas Intermedi-ate (WTI) crude oil price series was chosen as experimentalsampleThemain reason of selecting theWTI crude oil is thatthese crude oil prices are the most famous benchmark priceswhich are widely used as the basis of many crude oil priceformulae The daily data from January 1 1986 to September30 2006 excluding public holidays with a total of 5237 wasemployed as experimental data For convenience of WMLRmodeling the data from January 1 1986 to December 312000 is used for the training set (3800 observations) andthe remainder is used as the testing set (1437 observations)Figure 2 shows the daily crude oil prices from January 1 1986to September 30

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study three main performancecriteria are used to evaluate the accuracy of themodelsThesecriteria are mean absolute error (MAE) root mean squarederror (RMSE) and119863stat TheMAE and RMSE can be definedby

RMSE = radic 1119899

119899

sum

119905=1

(119910119905minus 119910119905)2

MAE = 1119899

119899

sum

119905=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816

(15)

In crude oil price forecasting improved decisions usuallydepend on correct forecasting of directions of actual price119910119905and forecasted price 119910

119905 The ability to predict movement

direction can bemeasured by a directional statistic (119863stat) (Yuet al [1]) which can be expressed as

119863stat =1

119873

119899

sum

119905=1

119886119905times 100

119886119905=

1 if (119910119905+1minus 119910119905) (119910119905+1minus 119910119905) ge 0

0 otherwise

(16)

32 Application and Result At first the MLR model withoutdata preprocessing was used to model daily oil prices In

0102030405060708090

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily (January 1 1986ndashSeptember 30 2006)

Figure 2 Daily crude oil prices from January 1 198 to September30 2006

the next step the preprocessed data which uses subtimeseries components obtained using discrete wavelet transform(DWT) on original data were entered to the MLR model inorder to improve themodel accuracy For theMLRmodel theoriginal log return time series are decomposed into a certainnumber of subtime series components Deciding the optimaldecomposition level of the time series data in wavelet analysisplays an important role in preserving the information andreducing the distortion of the datasets However there is noexisting theory to tell how many decomposition levels areneeded for any time series

In the present study the previous log return of daily oilprice time series is decomposed into various subtime series(DWs) at different decomposition levels by using DWT toestimate current price value Three decomposition levels (24 and 8 months) were considered for this study For theWTIseries data time series of 2-day mode (DW1) 4-day mode(DW2) and 8-day mode (DW3) and approximate mode arepresented in Figure 3

For the WTI series six input combinations based onprevious log return of daily oil prices are evaluated to esti-mate current prices value The input combinations evaluatedin the study are (i) 119903

119905minus1 (ii) 119903

119905minus1 119903119905minus2

(iii) 119903119905minus1 119903119905minus2 119903119905minus3

(iv) 119903

119905minus1 119903119905minus2 119903119905minus3 119903119905minus4

(v) 119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5

and (vi)119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5 119903119905minus6

In all cases the output is the logreturn of current oil prices 119903

119905

Each of DWs series plays distinct role in original timeseries and has different effects on the original prices oilseries The selection of dominant DWs as inputs of MLRmodel becomes important and effective on the output dataand has positive effect excessively on modelrsquos ability Themodel becomes exponentially more complex as the numberof subtime series as input variables increases Using a largenumber of input variables should be avoided to save time andcalculation effort Therefore the effectiveness of new seriesobtained by PCA is used as input to theMLRmodelThePCAapproach helps us to reduce the number of original variablesto a set of new variables Generally the objective of PCA is toidentify a new set of variables such that each variable called aprincipal component is a linear combination of the originalvariables The new set of variables accounts for 85minus90 of

The Scientific World Journal 5

01020304050607080

0 1000 2000 3000 4000 5000

A3-a

ppro

xim

atio

n

Daily (January 1 1986ndashSeptember 30 2006)

(a)

0

1

2

3

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D1

Daily (January 1 1986ndashSeptember 30 2006)

(b)

0

2

4

0 1000 2000 3000 4000 5000

minus8

minus6

minus4

minus2D2

Daily (January 1 1986ndashSeptember 30 2006)

(c)

0

1

2

3

4

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D3

Daily (January 1 1986ndashSeptember 30 2006)

(d)

Figure 3 Decomposed wavelet subtime series components (Ds) ofWTI crude oil price data

Decompose input using DWT

Running PCAthe effective of PCs as

input for MLR

Data collection (yt)

Preprocessing for data

rt = lnytytminus1

AM middot middot middot DM

Postprocess results (MLR)yt = ytminus1exp(rt)

D1 D2

Figure 4 The structure of the WMLR model

Table 1 Eigen value and cumulative variance contribution rate ofthe 8 principal components

PC 1 2 3 4 5 6 7 8Eigen value 197 179 159 133 067 041 021 003CumulativeVariance Rate 025 047 067 084 092 097 100 100

the total variation were considered as the number of newvariables

For example taking two previous daily oil prices as arandom variable Every previous daily oil price time seriesare decomposed using DWT into three decomposition levelsrespectively Thus there were 8 subseries considered forthe PCA analysis The result of PCA analysis is shown inTable 1 Table 1 shows that the first four principle componentscan explain 84 variation of the data variation with theeigenvalues greater than 1 to be retained in which all the 4PCs were included in the MLR model Thus the 8 originalvariables can be replaced by 4 new irrelevant variables Fortraining MLR the PSO algorithm solving the recognitionproblem is implemented and the program code includingwavelet toolbox was written in MATLAB language TheWMLRmodel structure developed in present study is shownin Figure 4

The forecasting performances of the MLR and WMLRmodels in terms of the MAE RMSE and 119863stat testing phaseare compared and shown in Table 2 Table 2 shows MLRmodel the M1 with 1 lag obtained the best MAE statisticsof 06948 and the M6 with 6 lags obtained the best RMSEstatistics of 09450 while the M1 with 5 lags obtained the best

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 3: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

The Scientific World Journal 3

function varying from problem in hand During the iterativeprocedure the 119894th particle at iteration 119905 is updated by

V119889119894(119905 + 1) = 120596

1015840times V119889119894(119905) + 119888

1times 1205931times [119901119889

119894(119905) minus 119909

119889

119894(119905)]

+ 1198882times 1205932times [119901119889

119892(119905) minus 119909

119889

119894(119905)]

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(5)

where120596 is called inertia weight 1198881and 1198882are acceleration con-

stants and 1205931and 120593

2are stochastic value of [0 1] In a PSO

system particles change their positions at each time step untila relatively unchanging position has been encountered or amaximum number of iterations have been met In generalthe performance of each particle is measured according to afitness function which is problemdependent InMLRmodel(4) is the fitness function under consideration Figure 1 showsthe flowchart of the developed PSO algorithm For furtherdetails regarding PSO please refer to Kennedy and Eberhart[27] and Bratton and Kennedy [26]

25 Wavelet Analysis Wavelet transformations provide use-ful decomposition of original time series by capturing use-ful information on various decomposition levels Discretewavelet transformation (DWT) is preferred in most of theforecasting problems because of its simplicity and ability tocompute with less time The DWT can be defined as

120595119898119899(119905 minus 120591

119904) =

1

radic1199041198982

0

120595(119905 minus 119899120591

0119904119898

0

119904119898

0

) (6)

where119898 and 119899 are integers that control the scale and timeThemost common choices for the parameters 119904

0= 2 and 120591

0= 1

120595(119905) called themother wavelet can be defined asintinfinminusinfin120595(119905)119889119905 =

0For a discrete time series119909(119905)where119909(119905) occurs at discrete

time 119905 the DWT becomes

119882119898119899= 2minus1198982

119873minus1

sum

119905=0

120595 (2minus119898119905 minus 119899) 119909 (119905) (7)

where119882119898119899

is the wavelet coefficient for the discrete waveletat scale 119904 = 2119898 and 120591 = 2119898119899 According to Mallatrsquos theorythe original discrete time series 119909(119905) can be decomposed intoa series of linearity independent approximation and detailsignals by using the inverse DWTThe inverse DWT is givenby (Mallat [28])

119909 (119905) = 119879 +

119872

sum

119898=1

2119872minus119898minus1

sum

119905=0

1198821198981198992minus1198982

120595 (2minus119898119905 minus 119899) (8)

or in a simple format as

119909 (119905) = 119860119872(119905) +

119872

sum

119898=1

119863119898(119905) (9)

where 119860119872(119905) is called approximation subseries or residual

term at levels 119872 and 119863119898(119905) (119898 = 1 2 119872) are detail

subseries which can capture small features of interpretationalvalue in the data

Generate initial population

Fitnessevaluation

Convergencecheck

Updating

Parametersoptimization

Yes

Training MLR

Cross validationNo

Figure 1 Flowchart of PSO algorithm

26 Principal Component Analysis In an MLR one of maintasks is to determine the model input variables that affect theoutput variables significantly The choice of input variablesis generally based on a priori knowledge of causal variablesinspections of time series plots and statistical analysis ofpotential inputs and outputs PCA is a technique widely usedfor reducing the number of input variables when we havehuge volume of information and we want to have a betterinterpretation of variables (Camdevyren et al [29])

The PCA approach introduces a few combinations formodel input in comparison with the trial and error processGiven a set of centred input vectors 119909

1 1199092 119909

119898and

sum119898

119905=1119909119905= 0 usually 119899 lt 119898 Then the covariance matrix of

vector is given by

119862 =1

119897

119897

sum

119905=1

119909119905119909119879

119905 (10)

The principal components (PCs) are computed by solving theeigenvalue problem of covariance matrix 119862

120582119894119906119894= 119862119906119894 119894 = 1 2 119898 (11)

where 120582119894is one of the eigenvalues of 119862 and 119906

119894is the

corresponding eigenvector Based on the estimated 119906119894 the

components of 119911119905(119894) are then calculated as the orthogonal

transforms of 119909119905

119911119905(119894) = 119906

119879

119894119909119905 119894 = 1 2 119898 (12)

The new components 119911119894(119905) are called principal components

By using only the first several eigenvectors sorted in descend-ing order of the eigenvalues the number of principal com-ponents in 119911

119905can be reduced So PCA has the dimensional

reduction characteristic The principal components of PCAhave the following properties 119911

119905(119894) are linear combinations

of the original variables uncorrelated and have sequentially

4 The Scientific World Journal

maximum variances (Jolliffe [30]) The calculation variancecontribution rate is

119881119894=

120582119894

sum119898

119894=1120582119894

times 100 (13)

The cumulative variance contribution rate is

119881 (119901) =

119901

sum

119894=1

119881119894 (14)

The number of the selected principal components is based onthe cumulative variance contribution rate which as a rule isover 85sim90

3 Computer Simulation

31 An Application In this study the West Texas Intermedi-ate (WTI) crude oil price series was chosen as experimentalsampleThemain reason of selecting theWTI crude oil is thatthese crude oil prices are the most famous benchmark priceswhich are widely used as the basis of many crude oil priceformulae The daily data from January 1 1986 to September30 2006 excluding public holidays with a total of 5237 wasemployed as experimental data For convenience of WMLRmodeling the data from January 1 1986 to December 312000 is used for the training set (3800 observations) andthe remainder is used as the testing set (1437 observations)Figure 2 shows the daily crude oil prices from January 1 1986to September 30

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study three main performancecriteria are used to evaluate the accuracy of themodelsThesecriteria are mean absolute error (MAE) root mean squarederror (RMSE) and119863stat TheMAE and RMSE can be definedby

RMSE = radic 1119899

119899

sum

119905=1

(119910119905minus 119910119905)2

MAE = 1119899

119899

sum

119905=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816

(15)

In crude oil price forecasting improved decisions usuallydepend on correct forecasting of directions of actual price119910119905and forecasted price 119910

119905 The ability to predict movement

direction can bemeasured by a directional statistic (119863stat) (Yuet al [1]) which can be expressed as

119863stat =1

119873

119899

sum

119905=1

119886119905times 100

119886119905=

1 if (119910119905+1minus 119910119905) (119910119905+1minus 119910119905) ge 0

0 otherwise

(16)

32 Application and Result At first the MLR model withoutdata preprocessing was used to model daily oil prices In

0102030405060708090

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily (January 1 1986ndashSeptember 30 2006)

Figure 2 Daily crude oil prices from January 1 198 to September30 2006

the next step the preprocessed data which uses subtimeseries components obtained using discrete wavelet transform(DWT) on original data were entered to the MLR model inorder to improve themodel accuracy For theMLRmodel theoriginal log return time series are decomposed into a certainnumber of subtime series components Deciding the optimaldecomposition level of the time series data in wavelet analysisplays an important role in preserving the information andreducing the distortion of the datasets However there is noexisting theory to tell how many decomposition levels areneeded for any time series

In the present study the previous log return of daily oilprice time series is decomposed into various subtime series(DWs) at different decomposition levels by using DWT toestimate current price value Three decomposition levels (24 and 8 months) were considered for this study For theWTIseries data time series of 2-day mode (DW1) 4-day mode(DW2) and 8-day mode (DW3) and approximate mode arepresented in Figure 3

For the WTI series six input combinations based onprevious log return of daily oil prices are evaluated to esti-mate current prices value The input combinations evaluatedin the study are (i) 119903

119905minus1 (ii) 119903

119905minus1 119903119905minus2

(iii) 119903119905minus1 119903119905minus2 119903119905minus3

(iv) 119903

119905minus1 119903119905minus2 119903119905minus3 119903119905minus4

(v) 119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5

and (vi)119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5 119903119905minus6

In all cases the output is the logreturn of current oil prices 119903

119905

Each of DWs series plays distinct role in original timeseries and has different effects on the original prices oilseries The selection of dominant DWs as inputs of MLRmodel becomes important and effective on the output dataand has positive effect excessively on modelrsquos ability Themodel becomes exponentially more complex as the numberof subtime series as input variables increases Using a largenumber of input variables should be avoided to save time andcalculation effort Therefore the effectiveness of new seriesobtained by PCA is used as input to theMLRmodelThePCAapproach helps us to reduce the number of original variablesto a set of new variables Generally the objective of PCA is toidentify a new set of variables such that each variable called aprincipal component is a linear combination of the originalvariables The new set of variables accounts for 85minus90 of

The Scientific World Journal 5

01020304050607080

0 1000 2000 3000 4000 5000

A3-a

ppro

xim

atio

n

Daily (January 1 1986ndashSeptember 30 2006)

(a)

0

1

2

3

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D1

Daily (January 1 1986ndashSeptember 30 2006)

(b)

0

2

4

0 1000 2000 3000 4000 5000

minus8

minus6

minus4

minus2D2

Daily (January 1 1986ndashSeptember 30 2006)

(c)

0

1

2

3

4

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D3

Daily (January 1 1986ndashSeptember 30 2006)

(d)

Figure 3 Decomposed wavelet subtime series components (Ds) ofWTI crude oil price data

Decompose input using DWT

Running PCAthe effective of PCs as

input for MLR

Data collection (yt)

Preprocessing for data

rt = lnytytminus1

AM middot middot middot DM

Postprocess results (MLR)yt = ytminus1exp(rt)

D1 D2

Figure 4 The structure of the WMLR model

Table 1 Eigen value and cumulative variance contribution rate ofthe 8 principal components

PC 1 2 3 4 5 6 7 8Eigen value 197 179 159 133 067 041 021 003CumulativeVariance Rate 025 047 067 084 092 097 100 100

the total variation were considered as the number of newvariables

For example taking two previous daily oil prices as arandom variable Every previous daily oil price time seriesare decomposed using DWT into three decomposition levelsrespectively Thus there were 8 subseries considered forthe PCA analysis The result of PCA analysis is shown inTable 1 Table 1 shows that the first four principle componentscan explain 84 variation of the data variation with theeigenvalues greater than 1 to be retained in which all the 4PCs were included in the MLR model Thus the 8 originalvariables can be replaced by 4 new irrelevant variables Fortraining MLR the PSO algorithm solving the recognitionproblem is implemented and the program code includingwavelet toolbox was written in MATLAB language TheWMLRmodel structure developed in present study is shownin Figure 4

The forecasting performances of the MLR and WMLRmodels in terms of the MAE RMSE and 119863stat testing phaseare compared and shown in Table 2 Table 2 shows MLRmodel the M1 with 1 lag obtained the best MAE statisticsof 06948 and the M6 with 6 lags obtained the best RMSEstatistics of 09450 while the M1 with 5 lags obtained the best

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 4: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

4 The Scientific World Journal

maximum variances (Jolliffe [30]) The calculation variancecontribution rate is

119881119894=

120582119894

sum119898

119894=1120582119894

times 100 (13)

The cumulative variance contribution rate is

119881 (119901) =

119901

sum

119894=1

119881119894 (14)

The number of the selected principal components is based onthe cumulative variance contribution rate which as a rule isover 85sim90

3 Computer Simulation

31 An Application In this study the West Texas Intermedi-ate (WTI) crude oil price series was chosen as experimentalsampleThemain reason of selecting theWTI crude oil is thatthese crude oil prices are the most famous benchmark priceswhich are widely used as the basis of many crude oil priceformulae The daily data from January 1 1986 to September30 2006 excluding public holidays with a total of 5237 wasemployed as experimental data For convenience of WMLRmodeling the data from January 1 1986 to December 312000 is used for the training set (3800 observations) andthe remainder is used as the testing set (1437 observations)Figure 2 shows the daily crude oil prices from January 1 1986to September 30

In practice short-term forecasting results aremore usefulas they provide timely information for the correction offorecasting value In this study three main performancecriteria are used to evaluate the accuracy of themodelsThesecriteria are mean absolute error (MAE) root mean squarederror (RMSE) and119863stat TheMAE and RMSE can be definedby

RMSE = radic 1119899

119899

sum

119905=1

(119910119905minus 119910119905)2

MAE = 1119899

119899

sum

119905=1

1003816100381610038161003816119910119905 minus 1199101199051003816100381610038161003816

(15)

In crude oil price forecasting improved decisions usuallydepend on correct forecasting of directions of actual price119910119905and forecasted price 119910

119905 The ability to predict movement

direction can bemeasured by a directional statistic (119863stat) (Yuet al [1]) which can be expressed as

119863stat =1

119873

119899

sum

119905=1

119886119905times 100

119886119905=

1 if (119910119905+1minus 119910119905) (119910119905+1minus 119910119905) ge 0

0 otherwise

(16)

32 Application and Result At first the MLR model withoutdata preprocessing was used to model daily oil prices In

0102030405060708090

0 1000 2000 3000 4000 5000

Crud

e oil

pric

es (U

S do

llar)

Daily (January 1 1986ndashSeptember 30 2006)

Figure 2 Daily crude oil prices from January 1 198 to September30 2006

the next step the preprocessed data which uses subtimeseries components obtained using discrete wavelet transform(DWT) on original data were entered to the MLR model inorder to improve themodel accuracy For theMLRmodel theoriginal log return time series are decomposed into a certainnumber of subtime series components Deciding the optimaldecomposition level of the time series data in wavelet analysisplays an important role in preserving the information andreducing the distortion of the datasets However there is noexisting theory to tell how many decomposition levels areneeded for any time series

In the present study the previous log return of daily oilprice time series is decomposed into various subtime series(DWs) at different decomposition levels by using DWT toestimate current price value Three decomposition levels (24 and 8 months) were considered for this study For theWTIseries data time series of 2-day mode (DW1) 4-day mode(DW2) and 8-day mode (DW3) and approximate mode arepresented in Figure 3

For the WTI series six input combinations based onprevious log return of daily oil prices are evaluated to esti-mate current prices value The input combinations evaluatedin the study are (i) 119903

119905minus1 (ii) 119903

119905minus1 119903119905minus2

(iii) 119903119905minus1 119903119905minus2 119903119905minus3

(iv) 119903

119905minus1 119903119905minus2 119903119905minus3 119903119905minus4

(v) 119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5

and (vi)119903119905minus1 119903119905minus2 119903119905minus3 119903119905minus4 119903119905minus5 119903119905minus6

In all cases the output is the logreturn of current oil prices 119903

119905

Each of DWs series plays distinct role in original timeseries and has different effects on the original prices oilseries The selection of dominant DWs as inputs of MLRmodel becomes important and effective on the output dataand has positive effect excessively on modelrsquos ability Themodel becomes exponentially more complex as the numberof subtime series as input variables increases Using a largenumber of input variables should be avoided to save time andcalculation effort Therefore the effectiveness of new seriesobtained by PCA is used as input to theMLRmodelThePCAapproach helps us to reduce the number of original variablesto a set of new variables Generally the objective of PCA is toidentify a new set of variables such that each variable called aprincipal component is a linear combination of the originalvariables The new set of variables accounts for 85minus90 of

The Scientific World Journal 5

01020304050607080

0 1000 2000 3000 4000 5000

A3-a

ppro

xim

atio

n

Daily (January 1 1986ndashSeptember 30 2006)

(a)

0

1

2

3

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D1

Daily (January 1 1986ndashSeptember 30 2006)

(b)

0

2

4

0 1000 2000 3000 4000 5000

minus8

minus6

minus4

minus2D2

Daily (January 1 1986ndashSeptember 30 2006)

(c)

0

1

2

3

4

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D3

Daily (January 1 1986ndashSeptember 30 2006)

(d)

Figure 3 Decomposed wavelet subtime series components (Ds) ofWTI crude oil price data

Decompose input using DWT

Running PCAthe effective of PCs as

input for MLR

Data collection (yt)

Preprocessing for data

rt = lnytytminus1

AM middot middot middot DM

Postprocess results (MLR)yt = ytminus1exp(rt)

D1 D2

Figure 4 The structure of the WMLR model

Table 1 Eigen value and cumulative variance contribution rate ofthe 8 principal components

PC 1 2 3 4 5 6 7 8Eigen value 197 179 159 133 067 041 021 003CumulativeVariance Rate 025 047 067 084 092 097 100 100

the total variation were considered as the number of newvariables

For example taking two previous daily oil prices as arandom variable Every previous daily oil price time seriesare decomposed using DWT into three decomposition levelsrespectively Thus there were 8 subseries considered forthe PCA analysis The result of PCA analysis is shown inTable 1 Table 1 shows that the first four principle componentscan explain 84 variation of the data variation with theeigenvalues greater than 1 to be retained in which all the 4PCs were included in the MLR model Thus the 8 originalvariables can be replaced by 4 new irrelevant variables Fortraining MLR the PSO algorithm solving the recognitionproblem is implemented and the program code includingwavelet toolbox was written in MATLAB language TheWMLRmodel structure developed in present study is shownin Figure 4

The forecasting performances of the MLR and WMLRmodels in terms of the MAE RMSE and 119863stat testing phaseare compared and shown in Table 2 Table 2 shows MLRmodel the M1 with 1 lag obtained the best MAE statisticsof 06948 and the M6 with 6 lags obtained the best RMSEstatistics of 09450 while the M1 with 5 lags obtained the best

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 5: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

The Scientific World Journal 5

01020304050607080

0 1000 2000 3000 4000 5000

A3-a

ppro

xim

atio

n

Daily (January 1 1986ndashSeptember 30 2006)

(a)

0

1

2

3

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D1

Daily (January 1 1986ndashSeptember 30 2006)

(b)

0

2

4

0 1000 2000 3000 4000 5000

minus8

minus6

minus4

minus2D2

Daily (January 1 1986ndashSeptember 30 2006)

(c)

0

1

2

3

4

0 1000 2000 3000 4000 5000

minus4

minus3

minus2

minus1

D3

Daily (January 1 1986ndashSeptember 30 2006)

(d)

Figure 3 Decomposed wavelet subtime series components (Ds) ofWTI crude oil price data

Decompose input using DWT

Running PCAthe effective of PCs as

input for MLR

Data collection (yt)

Preprocessing for data

rt = lnytytminus1

AM middot middot middot DM

Postprocess results (MLR)yt = ytminus1exp(rt)

D1 D2

Figure 4 The structure of the WMLR model

Table 1 Eigen value and cumulative variance contribution rate ofthe 8 principal components

PC 1 2 3 4 5 6 7 8Eigen value 197 179 159 133 067 041 021 003CumulativeVariance Rate 025 047 067 084 092 097 100 100

the total variation were considered as the number of newvariables

For example taking two previous daily oil prices as arandom variable Every previous daily oil price time seriesare decomposed using DWT into three decomposition levelsrespectively Thus there were 8 subseries considered forthe PCA analysis The result of PCA analysis is shown inTable 1 Table 1 shows that the first four principle componentscan explain 84 variation of the data variation with theeigenvalues greater than 1 to be retained in which all the 4PCs were included in the MLR model Thus the 8 originalvariables can be replaced by 4 new irrelevant variables Fortraining MLR the PSO algorithm solving the recognitionproblem is implemented and the program code includingwavelet toolbox was written in MATLAB language TheWMLRmodel structure developed in present study is shownin Figure 4

The forecasting performances of the MLR and WMLRmodels in terms of the MAE RMSE and 119863stat testing phaseare compared and shown in Table 2 Table 2 shows MLRmodel the M1 with 1 lag obtained the best MAE statisticsof 06948 and the M6 with 6 lags obtained the best RMSEstatistics of 09450 while the M1 with 5 lags obtained the best

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 6: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

6 The Scientific World Journal

GARCHARIMAWMLRMLR

5

0

Erro

r

minus5

minus10

Figure 5 The errors of MLR WMLR ARIMA and GARCHmodels for crude oil price forecasting

Table 2 Forecasting performance indices of MLR and WLR

Model Input Lag MLR WMLRMAE RMSE 119863stat MAE RMSE 119863stat

M1 1 06948 09514 04788 06660 09001 05198M2 1 2 06972 09517 04781 06448 08842 05003M3 1 2 3 06985 09545 04816 05345 07505 05797M4 1 2 3 4 06979 09550 04753 04834 06572 06722M5 1 2 3 4 5 06976 09545 04878 05770 08046 05734M6 1 2 3 4 5 6 06969 09450 04850 05385 07389 06444

119863stat statistics of 04878 For WMLR model M4 with 4 lagsobtained the best MAE RMSE and 119863stat statistics of 0483406572 and 06722 respectively The equations of MLR withsix input variables and WMLR with four input variablesrespectively are

119910119905= minus 0025119910

119905minus1minus 0047119910

119905minus2+ 022119910

119905minus3

minus 0082119910119905minus4minus 0060119910

119905minus5+ 00004119910

119905minus6

119903119905= 0076119911

1(119905) minus 0221119911

2(119905) minus 0213119911

3(119905)

minus 05101199114(119905) + 0416119911

5(119905) minus 0930119911

6(119905)

(17)

where 119911119894(119905) are called principal components and 119910

119905=

119910119905minus1

exp(119903119905)

For further analysis the best performance of the LRWMLR ARIMA and ARIMA-GARCH models was com-pared with the best results of ARIMA and forward neuralnetwork (FNN) studied by Yu et al [1] In Table 3 itshows that WMLR outperform MLR ARIMA GARCH YursquoARIMA and Yursquo FNN models in terms of RMSE statisticsThis results show that the new series (DWT) have significantextremely positive effect on MLR model results

Figure 5 shows the Box-plot for the ARIMA ARIMA-GARCH MLR andWMLR models for testing period It canbe seen that the errors of WMLR model are quite close tothe zero Overall it can be concluded that the WMLR modelprovided more accurate forecasting results than the othermodels for crude oil forecasting

Table 3 The RMSE and MAE comparisons for different models

Model RMSE MAEARIMA (2 1 5) 13835 10207GARCH (1 1) 09513 06947MLR 09450 06969WMLR 06572 04834Yursquo ARIMA (Yu et al [1]) 20350 mdashYursquo FNN (Yu et al [1]) 08410 mdash

4 Conclusions

The accuracy of the wavelet multiple linear regression(WMLR) technique in the forecasting daily crude oil has beeninvestigated in this study The PCA is used to choose theprinciple component scores of the selected inputs which wereused as independent variables in theMLRmodel and the par-ticle swarm optimization (PSO) is used to adopt the optimalparameters of the MLR model The performance of the pro-posed WMLR model was compared to regular LR ARIMAand GARCH model for crude oil forecasting Comparisonresults indicated that the WMLR model was substantiallymore accurate than the other models The study concludesthat the forecasting abilities of the MLR model are found tobe improved when the wavelet transformation technique isadopted for the data preprocessingThedecomposed periodiccomponents obtained from the DWT technique are found tobe most effective in yielding accurate forecast when used asinputs in the MLR model The accurate forecasting results

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 7: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

The Scientific World Journal 7

indicate that WMLR model provides a superior alternativeto other models and a potentially very useful new methodfor crude oil forecasting The WMLR model presented inthis study is a simple explicit mathematical formulation TheWMLR model is much simpler in contrast to ANN modeland can be successfully used in modeling short-term crudeoil price In the present study three resolution levels wereemployed for decomposing crude oil time series If moreresolution levels were used the results from WMLR modelmay turn out better This may be a subject of another study

Conflict of Interests

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

Acknowledgments

The authors thankfully acknowledged the financial supportthat was afforded by Universiti Teknologi Malaysia underGUPGrant (VOT 06J13) Besides that the authors would liketo thank theDepartment of Irrigation andDrainageMinistryof natural Resources and Environment Malaysia in helpingus to provide the data

References

[1] 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

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

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

[4] 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

[5] Y Wei Y Wang and D Huang ldquoForecasting crude oil marketvolatility further evidence usingGARCH-classmodelsrdquoEnergyEconomics vol 32 no 6 pp 1477ndash1484 2010

[6] L Liu and J Wan ldquoA Study of Shangai fuel oil futures pricevolatility based onhigh frequency data long-range dependencemodeling and forecastingrdquo Economic Modelling vol 29 pp2245ndash2253 2012

[7] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[8] MAKaboudan ldquoCompumetric forecasting of crude oil pricesrdquoinThe Proceedings of IEEE Congress on Evolutionary Computa-tion pp 283ndash287

[9] 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

[10] 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

[11] L Yu S Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreign

exchange ratesrdquoComputers andOperations Research vol 32 no10 pp 2523ndash2541 2005

[12] W Xie L Yu S Y Xu and S YWang ldquoA newmethod for crudeoil price forecasting based on support vector machinesrdquo LectureNotes in Computer Science vol 3994 pp 441ndash451 2006

[13] 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

[14] W Qunli H Ge and C Xiaodong ldquoCrude oil price forecastingwith an improved model based on wavelet transform and RBFneural networkrdquo in Proceedings of the International Forum onInformation Technology and Applications (IFITA rsquo09) pp 231ndash234 May 2009

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

[16] Y Bao X Zhang L Yu K K Lai and S Wang ldquoHybridizingwavelet and least squares support vector machines for crudeoil price forecastingrdquo in Proceedings of the 2nd InternationalWorkshop on Intelligent Finance Chengdu China 2007

[17] J Liu Y Bai and B Li ldquoA new approach to forecast crudeoil price based on fuzzy neural networkrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) pp 273ndash277 August 2007

[18] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[19] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[20] H Bozdogan and J A Howe ldquoMisspecifiedmultivariate regres-sion models using the genetic algorithm and informationcomplexity as the fitness functionrdquo European Journal of Pureand Applied Mathematics vol 5 no 2 pp 211ndash249 2012

[21] S M Chen and P Y Kao ldquoTAEIX forecasting based onfuzzy time series partical swarm optimization techniques andsupport vectormachinesrdquo Information Sciences vol 247 pp 62ndash71 2013

[22] R Alwee S M Shamsuddin and R Sallehuddin ldquoHybridsupport vector regression and autoregressive integratedmovingaverage models improved by particle swarm optimization forproperty crime rates forecasting with economic indicatorsrdquoTheScientific World Journal vol 2013 Article ID 951475 11 pages2013

[23] R J Ma N Y Yu and J Y Hu ldquoApplication of particleswarm optimization algorithm in the heating system planningproblemrdquo The Scientific World Journal vol 2013 Article ID718345 11 pages 2013

[24] G Tyagi and M Pandit ldquoCombined heat and power dispatchusing time varying acceleration coefficient particle swarm opti-mazationrdquo International Journal of Engineering and InnovativeTechnology vol 1 no 4 pp 234ndash240 2012

[25] L Liu S Yang and D Wang ldquoParticle swarm optimizationwith composite particles in dynamic environmentsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 40 no 6 pp 1634ndash1648 2010

[26] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intel-ligence Symposium (SIS rsquo07) pp 120ndash127 IEEE Service CenterPiscataway NJ USA April 2007

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Page 8: Research Article Crude Oil Price Forecasting Based on ...downloads.hindawi.com/journals/tswj/2014/854520.pdf · K =L 1, if ; +1

8 The Scientific World Journal

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] 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

[29] H Camdevyren N Demyr A Kanik and S Keskyn ldquoUseof principal component scores in multiple linear regressionmodels for prediction of Chlorophyll-a in reservoirsrdquo EcologicalModelling vol 181 no 4 pp 581ndash589 2005

[30] I T Jolliffe Principal Component Analysis Springer New YorkNY USA 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

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Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014