DAILY CRUDE OIL PRICE FORECASTING MODEL USING ARIMA, GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTIC AND SUPPORT VECTOR MACHINES

Transcription

1 Amercan Journal of Appled Scences (3): 45-43, 04 ISS: Scence Publcaton do:0.3844/ajassp Publshed Onlne (3) 04 ( DAILY CRUDE OIL PRICE FORECASTIG MODEL USIG ARIMA, GEERALIZED AUTOREGRESSIVE CODITIOAL HETEROSCEDASTIC AD SUPPORT VECTOR MACHIES Rana Abdullah Ahmed and An Bn Shabr, Department of Mathematcal Scences, Unverst Teknolog Malaysa, Skuda, Johor, 830, Malaysa Department of Mathematcs, College of Basc Educaton, Unversty of Mousl, Mousl, Iraq Receved ; Revsed 03--9; Accepted ABSTRACT Crude ol prce forecastng s ganng ncreased nterest globally. Ths nterest s due manly to the economc value attached to the product. For ths reason, new forecastng methods are proposed n the lterature. Ths paper proposes a novel technque for forecastng crude ol prce based on Support Vector Machnes (SVM). The study adopts the data on crude ol prce of West Texas Intermedate (WTI) for ts expermental purposes. Ths s because many studes have prevously used ths same data and t wll afford a common bass for assessment. To evaluate the performance of the model, the study employs two measures, RMSE and MAE. These are used to compare the performance of the proposed technque and that of ARIMA and GARCH methods for the most effcent n crude ol prce forecastng. The results reveal that the proposed method outperforms the other two n terms of forecast accuracy whle t acheved a forecast error of that of ARIMA and GARCH were and.034 respectvely judgng by ther RMSE. Keywords: Support Vector Machne, Forecastng, GARCH, Ol Prce and ARIMA. ITRODUCTIO Forecastng crude ol prces s mportant as t affects other key sectors of the economy ncludng the stock market. One of the mportant areas n economc research s forecastng the trend of prce change of nternatonal crude ol. It s also a ponter n numerous ndustres for quck management nterventon due to the recent extreme fluctuatons n the prce of nternatonal crude ol. Ths makes t crucal to develop relable models that would assst adequately n forecastng the fluctuaton of nternatonal crude ol prce. Ths s amed at facltatng the partes nvolved n takng approprate acton to avod assocated rsk. The ncrease n prce of nternatonal crude ol and ts daly changes does not only affect the fnancal markets and economes, but t also affects ndvduals too. Ths s because of prce ncrease of crude ol mpacts greatly on the prce of petrol whch has ts country and by extenson the gross domestc product (GDP). GDP has been defned by (Chrystal & Lpsey) as the total goods and servces produced n a country wthn a gven year. The aforementoned reasons makes the predcton of crude ol prces a very mperatve task to decrease the mpact of prce fluctuatons and assst polcy makers and ndvduals to take nformed decsons that would help n copng wth prce fluctuatons arsng from the energy markets. However, because of the mentoned reasons predctng crude ol s not a smple task. These have all made the predcton of the crude ol prce a wdely researched nto area n the energy market. The models found n the lterature on crude ol prce forecastng nclude the popular Box-Jenkns method as n (Lu, 99; Chnn et al., 005; Agnolucc, 009). Othermodels explored nclude GARCH-type models as n (Ahmed and Shabr, 03; Hou and Saudy, 0; Sadorsky, 006). In ths study, we attempt to extend the models used n the study of crude ol prce to attendant effect on goods and servces produced n the Correspondng Author: An Bn Shabrl, Department of Mathematcal Scences, Unverst, Teknolog Malaysa, Skuda, Johor, 830, Malaysa 45

2 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 the realm of the artfcal ntellgence partcularly fttng a Support Vector Machne (SVM) to the forecast such data of hgh volatlty. Secton that follows ths ntroducton dscusses revews of related works, Secton 3 dscusses the methodology n whch we concentrate on the mathematcal formulatons of the three methods n ths study and Secton 4 dscusses the results of the fndngs, whle Secton 5 gves the concluson of the paper. Secton 6 s devoted to acknowledgement n whch we apprecated the assstance receved from corporate bodes and ndvduals towards makng ths research come to lght.. REVIEW OF RELATED LITERATURE Lu (99) employed Box-Jenkns technque to study the dynamc relatonshps between US crude ol prces, gasolne prces and the stock of gasolne wth transferrng functon models US, whle Kumar (99) used tme seres models to nvestgate and compare the forecast accuracy of future prces of crude ol. The study ft an ARMA (,) model as the best ft model and compared wth future crude ol prces wth. The merts of ARIMA models are twofold (Wang et al., 005). Intally, ARIMA models are a set of typcal lnear models whch are proposed for the lnear tme seres and captured lnear characterstcs n the tme seres. Subsequently, the theoretcal base of ARIMA models s deal. Chnn et al. (005) studed the predctve content of energy futures. They examned the relatonshp between spot and futures prces for energy commodtes. An ARIMA (,,) was used for crude ol prces forecast. Sadorsky (006) showed that the out-of-sample forecasts of a sngle equaton GARCH model are best for those of Vector Auto-regresson, state space and bvarate GARCH models, are more superor n forecastng the futures prces of petroleum. Agnolucc (009) utlzed dverse knds of GARCH models and mentoned nstablty models to predct daly WTI future prce nstablty, but the emprcal results exposed that ther performances were ncompatble wth regard to dverse measures and statstcal tests. Marzo and Zagagla (00) appled numerous GARCH models to predct the nstablty of daly futures prces of crude ol traded on YMEX. The authors concluded they have not found a contnuously greater model based on dverse statstcal tests such as DM test, drecton accuracy test and performance 46 measures as Success Rato, Heteroscedastcty adjusted MSE, MAE and MSE. Hou and Saudy (0) an alternatve approach nvolvng nonparametrc method to model and forecast ol prce return volatlty, the results demonstrate that the out-of-sample nstablty predcton of the nonparametrc GARCH model defers greater performance qualfed for a broad class of parametrc GARCH models. Ahmed and Shabr (03) appled ftted GARCH model to crude ol spot prces. Ths was done n order to llustrate the advantages of nonlnear models. In the study the authors ft three GARCH models namely; GARCH-, GARCH-t and GARCH G to crude ol spot prces. The results revealed that GARCH- model s the best model for forecastng for Brent whle GARCH-G model s the best for forecastng of WTI crude ol spot prces. Morana (00) showed how the ol prce allocaton can be predcted by usng the GARCH propertes of ol prce changes over short-term horzons. He used a sem-parametrc approach to ol prce forecastng and t was based on bootstrap approach. Accordng to Marmoutou et al. (009), the GARCH (,) -model may provde equally good results when compared to a combned GARCH and Extreme Value Theory (GARCH-EVT). Sadorsky (999) showed that ol prce nstablty alarms have dssymmetrcal effects on the fnancal system. The fluctuatons n ol prces nfluence fnancal actvty, but modfcaton n fnancal actvty has lttle mpact on ol prces. Most recently, Support Vector Machne (SVM) a novel neural network algorthm, was developed by Vapnk (995) has ganed sgnfcant nroad n the feld of forecastng. Amongst the unque propertes of SVM s that t s opposed to the over-fttng dffculty and can draw model nonlnear relatons n a stable and effcent way. Addtonally, SVM s nstructed as a curved optmzaton problem resultng n the global explanaton that n many cases defers exclusve explanatons. Intally, SVMs have been expanded for categorzaton tasks (Burges, 998). SVMs have been expanded to resolve tme seres predcton and nonlnear regresson problems, wth the ntroducton of Vapnk s ε-nsenstve loss functon and they show excellent performance (Huang et al., 005; Muller et al., 997). Derved from ths standard, SVMs wll ultmately produce better smplfcaton performance n comparson wth other neural networks. Due to such

3 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 benefts, SVM method has been used n the area of economc tme seres forecastng (Tay and Cao, 00; 00; Km, 003; Huang et al., 005). Whereas, n comparng wth customary neural networks, the outstandng applcaton of SVMs s derved from the state that the modeled data should have defnte consstency. Accordngly, for the tme seres data wth changng dynamcs, a partcular SVM model could not acheve well n capturng such dynamc and unstructured nput-output relatonshp ntrnsc n the economc data. Khashman and wulu (0a) showed an ntellgent system that forecasts the crude ol prce. Ths ntellgent system s derved from SVM, the outcomes ganed were very hopeful as t establshed that SVM could be utlzed wth a hgh accuracy n forecastng the prce of crude ol. Xao-Ln and Ha-We (0) adopt three basc kernel functons of SVM to buld the predcton model of the crude ol prce, t used a partcle swarm algorthm to optmze the parameters. The result show the predcton model whose parameters have been optmzed by a genetc algorthm. Khashman and wulu (0b) nvestgated and compared the applyng of a back propagaton neural network and an SVM to the task of forecastng ol prces and the outcomes propose the neural networks can be competently appled to forecast future ol prces wth mnmal computatonal expendture. 3. THE METHODOLOGY In ths Secton, the paper dscusses the three technques that feature promnently n ths study. These are ARIMA, GARCH and SVM. Emphass s lad on how the proposed SVM technque would be mplemented n crude ol prce forecastng. 3.. ARIMA Modelng Box and Jenkns (976) ntroduced the ARIMA model and ever snce then the method has turned out to be one of the most famous approaches to predctng. The future value of a varable n an ARIMA model s presumed to be a lnear combnaton of past errors and past values, stated as follows: y = θ + φ y + φ y φ y t 0 t t p t p +ε θ ε θ ε... θ ε t t t q t q () 47 where, y t s the actual value and φ and θ j are the coeffcents, p and q are ntegers that are frequently submtted to as autoregressve, ε t s the random error at tme t and movng average polynomals, n that order. Fundamentally, ths method has three stages: Model classfcaton, parameter evaluaton and dagnostc examnaton. For nstance, the ARIMA (,0,) model can be characterzed as follows Equaton (): yt = θ 0 + φ yt + εt θε t () Equaton () demands some sgnfcant partcular cases of the ARIMA famly of models. If q = 0, then () becomes an AR model of order p. When p = 0, the model decreases to a MA model of order q. One essental task of the ARIMA model buldng s to conclude the sutable model order (p, q). Accordng the prevous work, Box and Jenkns (976) developed a practcal approach to buldng ARIMA models, whch has the fundamental mpact on the tme seres analyss and forecastng applcatons. Box and Jenkns recommended to apply the Partal Autocorrelaton Functon (PACF) and the Autocorrelaton Functon (ACF) of the sample data as the fundamental tools to recognze the order of the ARIMA model. In the classfcaton step, data transformaton s frequently necessary to make the tme seres statonary. Statonarty s an essental stage n creatng an ARMA model appled for predctng. A statonary tme seres s descrbed by statstcal characterstcs for nstance the mean and the autocorrelaton structure beng stable ultmately. Whle the expermental tme seres shows heteroscedastcty and trend, power transformaton and dfferencng are used to the data to elmnate the trend and to become constant the varance before can be ftted an ARIMA model. 3.. GARCH Modelng The ARIMA (p, d, q) model cannot capture the heteroscedastc outcomes of a tme seres procedure, characterstcally examned n the shape of hgh kurtoss, or gatherng of volatltes and the nfluence effect. Engle (98) ntated the Autoregressve Condtonal Heteroscedastc (ARCH) model, afterward Bollerslev (986) generalzed t thus the name

4 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 Generalzed Autoregressve Condtonal Heteroscedastc model (GARCH). The term condtonal mples the level of assocaton on the past sequence of observatons and the autoregressve descrbes the feedback mechansm that ncorporates past observatons nto the present (Laux et al., 0). The varance equaton of the GARCH (p, q) model can be expressed as Equaton (3 and 4): ε = Z σ t t t Z ~ Ψ(0,) p p t t t j = j= σ = ω + α ε + β σ = ω + α(b) ε + β(b) σ t t j (3) (4) where, Ψ t (0, ) s the lkelhood densty functon of the resduals or nnovatons wth unt and zero mean varance. Intentonally, τ are extra dstrbutonal parameters to explan the shape and the skew of the dstrbuton. The GARCH model can be reduced to the ARCH model f all the coeffcentsβ are zero. Smlar to ARMA models a GARCH requrement frequently gudes to a more economcal representaton of the chronologcal dependences and therefore presents a comparable addtonal flexblty over the lnear ARCH model when parameterzng the condtonal varance. Bollerslev (986) has demonstrated that the GARCH (p, q) procedure s wde-sense statonary f the followng condtons hold: E(ε t ) = 0 ω var( ε t ) = ( α() β()) cov( εt, εs ),t sf andonlyf () + () < The smple GARCH (, ) model has been establshed to offer a good demonstraton of an extensve dversty of volatlty procedures n most applcatons, (Bollerslev et al., 99) Support Vector Machnes Vapnk (995) proposed the Support Vector Machnes (SVMs). Accordng to the Structured Rsk Mnmzaton (SRM) prncple, SVMs look for reducng an upper bound of the generalzaton error rather than the emprcal error as n other neural 48 networks. Furthermore, the SVMs models create the revert functon by concernng a set of hgh dmensonal lnear functons. The SVM regresson functon s formulated as follows Equaton (5): y(x) = w φ (x) + b (5) where, Φ(x) s named the feature, whch s nonlnear planed from the nput space x. The coeffcents w and b are evaluated by mnmzng Equaton (6 and 7): R(C) = C L ε (d, y ) + w (6) = d y ε d y ε L ε(d, y ) = (7) 0 others where, both C and ε are prescrbed parameters. The frst term L s (d, y ) s named the ε-ntensve loss functon. The d s the actual stock prce durng the th perod. Ths functon shows that errors below are not penalzed. Also the term ( C( ) L (d, y ) measures = z the emprcal error. The next term, w s the flatness of the functon. C assesses the trade-off between the flatness of the model and the emprcal rsk. ξ and ξ were ntroduced as the postve slack varables, whch sgnfy the dstance from the actual values to the correspondng boundary values of ε-tube. Equaton (4) s converted to the followng constraned formaton: Mnmze: ( ) T R(w, ξ, ξ ) = ww + C ( ξ + ξ ) = Subjected to Equaton (9 to ): (8) w φ (x ) + b d ε + ξ (9) d w φ(x ) b ε + ξ (0) where ξ ξ 0, =,,..., () Fnally, ntroducng Lagrange multplers and maxmzng the dual functon of Equaton (8) we have:

5 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 = = R( α α ) = d ( α α ) ε ( α α ) ( α α j) ( α j α j)k(x,x j ) = j= Wth the constrants Equaton (3 to 5): () ( α α ) = 0 (3) = 0 α C (4) 0 α C =,,..., (5) In Equaton (), α and α are called Lagrangan multplers. They satsfy the equaltes Equaton (6): scenaro, testng and valdatng data requre data collecton. Although, the data collected from a varety of sources must be chosen along wth the equvalent norms. Sample preprocessng s the second phase that comprses of two steps: Frst step nvolves data normalzaton and second step s data dvson. In the process of developng any model, famlarty wth the accessble data s one of the greatest sgnfcance. SVM s no excepton to ths rule, as well; data normalzaton n can nfluence model performance sgnfcantly. Subsequently, data collecton should be dvded nto two sub-sets: Frst n-sample data and second out-of-sample data whch are appled for model development and model evaluaton n that order. SVM tranng and learnng s the thrd phase. Ths phrase comprses three major tasks: Determnaton of SVM archtecture, sample tranng and sample valdaton, whch s the center procedure of the SVM model. α α = 0 l α α + = f (x,a,a) ( )K(x,x ) b (6) Here, K(x, x ) s named the kernel functon. The amount of the kernel s equvalent to the nner product of two vectors x and x j n the feature space φ(x ) and φ(x j ), such that K(x, x ) = φ(x )φ(x j ). Any functon that fulfllng Mercer s condton Vapnk (995) can be appled as the kernel functon. The Gaussan kernel functon: ( ) K(x,x ) = exp x x / σ j j Is specfc n ths study. The SVMs were used to evaluate the nonlnear behavor of the predctng data set because Gaussan kernels am to present good performance under common effcency assumptons Proposed SVM Implementaton The flowchart of the proposed SVM technque s shown n Fg.. Ths gves a vvd llustraton of the procedure for expandng an SVM for tme seres predctng. The flowchart n Fg. can be splt nto four phases. The frst phase s data samplng. To expand an SVM model for a predctng tranng, Fg.. A flow chart of SVM-based forecastng system 49

6 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 SVM-based crude ol prce forecastng nvolves four steps: Data samplng. For ths research varous data can be collected, for example YMEX, WTI. Data collected can be classfed nto dverse tme scales: Daly, weekly and monthly. For daly data, there are a varety of mssng ponts and nconsstences for the marketplace has been blocked or stopped because of unexpected events or weekends. Consequently, weekly data and monthly data should be approved as alternatves Data preprocessng. It may requre to be transformed the collected ol prce data nto a defnte sutable range for network learnng va logarthm transformaton, varaton or other methods. After that the data should be splt nto out-of-sample data and n-sample data Tranng and learnng. In ths step the tranng results determne the SVM archtecture and parameters. There are no norms n choosng the parameters other than a tral-and-error bass. In ths study, the RBF kernel s appled because the RBF kernel tends to provde good performance under common softness assumptons As a result, t s partcularly constructve f no extra nformaton of the data s accessble. In concluson, an acceptable SVM-based model for ol prce predctng s attaned. Future prce forecastng Selectng correspondng SVM parameters s the modelng: Kernel functon and penalty factor c, whch nfluence sgnfcantly on the predctng outcomes. Statstcal software was used to create evaluaton among sgmod kernel functon, radal bass functon, kernel functon and polynomal kernel functon. In concluson, the radal bass functon was selected for the hgh predcton accurateness and concurrently through many trals of the parameter computaton Data For ths study, the West Texas Intermedate (WTI) crude ol spot prce was adopted for expermental purposes. The reason of choosng the ol prce sgns s that, the crude ol prces are the most well-known standard prces, whch are extensvely appled at the 430 foundaton of many crude ol prce codes. The crude ol prce data utlzed n ths study are daly data and are generously avalable from the energy nformaton admnstraton (EIA).The data covers the perod January, 986 to September 30, 006, thereby gvng a total of 537 observatons. The data s presented n Fg.. A dfference n unt can result n a dfference n data magntude, whch tends to affect predcton accuracy n the long run. The normalzaton processng can resolve ths ssue. In ths study, the data was normalzed to a scalable range of [0,] n the tranng set and predcton set, usng the normalzaton equaton: ( ) ( ) X = X X / X X n mn max mn The normalzed data s shown n Fg Evaluaton of Volatlty Forecasts Ths study adopted two very popular measures for evaluatng the forecast accuracy of the seres and these are: Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). These measures are evaluated by assessng ther returns. The one wth the lowest error measure s judged the best. These measures are defned as follows. Mean Absolute Error (MAE) s gven by: MAE = X X p ˆ t = ( t ) Fg.. The tme seres for WTI daly t

7 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 Fg. 3. The normalzed data for WTI Table. The evaluaton of forecastng results for WTI crude ol prce Methodology RMSE MAE ARIMA GARCH SVM and Root Mean Squared Error (RMSE) s gven by: (( ) ) / ˆ t t k RMSA = x x p t = Where: X t : The return of the horzon before the current tme t X : The average return ˆp t : Is the forecast value of the condtonal varance over n steps ahead horzon of the current tme t 3.7. Results and Analyss The results are shown n Table. From Table t can be seen that the proposed method SVM outperforms GARCH and ARIMA technques. From the pont of vew of the RMSE t returned a forecast error of followed by ARIMA wth a forecast error of and then GARCH wth.034. For the MAE SVM stll post the best result wth forecast accuracy of ; ARIMA and GARCH COCLUSIO In ths study a novel approach based on artfcal ntellgence for crude ol prce modelng s proposed. The proposed technque forecast accuracy performance was evaluated usng two measures of error RMSE and MAE and compare wth some other well-known technques n crude ol spot prce forecastng lke ARIMA and GARCH. The results revewed that the proposed SVM method outperforms the others. The study therefore recommends that the proposed SVM method be employed n future for crude ol prce forecastng. 5. REFERECES Agnolucc, P., 009. Volatlty n crude ol futures: A comparson of the predctve ablty of GARCH and mpled volatlty models. Energy Econ., 3: DOI: 0.06/j.eneco Ahmed, R.A. and A.B. Shabr, 03. Fttng GARCH models to crude ol spot prce data. Lfe Sc. J., 0: Bollerslev, T., 986. Generalzed autoregressve condtonal heteroskedastcty. J. Econometr., 3: DOI: 0.06/ (86) Bollerslev, T., R.Y. Chou and K.F. Kroner, 99. ARCH modelng n fnance : A revew of the theory and emprcal evdence. J. Econ., 5: DOI: 0.06/ (9)90064-X

8 Rana Abdullah Ahmed and An Bn Shabr / Amercan Journal of Appled Scences (3): 45-43, 04 Box, G.E.P. and G. Jenkns, 976. Tme Seres Analyss: Forecastng and Control. st Edn., Holden-Day, San Francsco, ISB-0: , pp: 575. Burges, C.J.C., 998. A tutoral on support vector machnes for pattern recognton. Data Mnng Knowl. Dscovery, : -67. DOI: 0.03/A: Chnn, M.D., M.R. LeBlanc and O. Cobon, 005. The Predctve Content of Energy Futures: An Update on Petroleum, atural Gas, Heatng Ol and Gasolne. st Edn., atonal Bureau of Economc Research, pp: 7. Engle, R.F., 98. Autoregressve condtonal heteroscedastcty wth estmates of the varance of unted kngdom nflaton. Econometrca, 50: Hou, A. and S. Suard, 0. A nonparametrc GARCH model of crude ol prce return volatlty. Energy Econ., 34: DOI: 0.06/j.eneco Huang, W., Y. akamora and S. Wang, 005. Forecastng stock market movement drecton wth support vector machne. Comput. Operat. Res., 3: DOI: 0.06/j.cor Khashman, A. and.i. wulu, 0a. Intellgent predcton of crude ol prce usng support vector machnes. Proceedngs of the IEEE 9th Internatonal Symposum on Appled Machne Intellgence and Informatcs, Jan. 7-9, IEEE Xplore Press, Smolence, pp: DOI: 0.09/SAMI Khashman, A. and.i. wulu, 0b. Support vector machnes versus back propagaton algorthm for ol prce predcton. Proceedngs of the 8th Internatonal Conference on Advances n eural etworks, May 9-Jun. 0, Sprnger Verlag Berln, Guln, Chna, pp: DOI: 0.007/ _60 Km, K.J., 003. Fnancal tme seres forecastng usng support vector machnes. eurocomputng, 55: DOI: 0.06/S095-3(03)0037- Kumar, M.S., 99. The forecastng accuracy of crude ol futures prces. Int. Monetary Fund, 39: DOI: 0.307/ Laux, P., S. Vogl, W. Qu, H.R. Knoche and H. Kunstmann, 0. Copula-based statstcal refnement of precptaton n RCM smulatons over complex terran. Hydrol. Earth Syst. Sc., 5: DOI: 0.594/hess Lu, L.M., 99. Dynamc relatonshp analyss of us gasolne and crude ol prces. J. Forecast., 0: DOI: 0.00/for Marmoutou, V., B. Raggad and A. Trabels, 009. Extreme value theory and value at rsk: Applcaton to ol market. Energy Econ., 3: DOI: 0.06/j.eneco Marzo, M. and P. Zagagla, 00. Volatlty forecastng for crude ol futures. Appled Econ. Lett., 7: DOI: 0.080/ Morana, C., 00. A semparametrc approach to shortterm ol prce forecastng. Energy Econ., 3: DOI: 0.06/S (00)00075-X Muller, K.R., J.A. Smola and B. Scholkopf, 997. Predctng tme seres wth support vector machnes. Proceedngs of the Internatonal Conference on Artfcal eural etworks, Oct. 8-0, Swtzerland, pp: DOI: 0.007/BFb00083 Sadorsky, P., 999. Ol prce shocks and stock market actvty. Energy Econ., : DOI: 0.06/S (99)0000- Sadorsky, P., 006. Modelng and forecastng petroleum futures volatlty. Energy Econ., 8: DOI: 0.06/j.eneco Tay, E.H. and L.J. Cao, 00. Applcaton of support vector machnes n fnancal tme seres forecastng. Omega, 9: DOI: 0.06/S (0) Tay, F.E. and L.J. Cao, 00. Modfed support vector machnes n fnancal tme seres forecastng. eurocomputng, 48: DOI: 0.06/S095-3(0) Vapnk, V.., 995. The ature of Statstcal Learnng Theory. st Edn., Sprnger, ew York, ISB-0: , pp: 34. Wang, S.Y., L.A. Yu and K.K. La, 005. Crude ol prce forecastng wth TEI@I methodology. J. Syst. Sc. Complexty 8: Xao-Ln, Z. and W. Ha-we, 0. Crude ol prces predctve model based on support vector machne and partcle swarm optmzaton. Proceedngs of the Internatonal Conference on Software Engneerng, Knowledge Engneerng and Informaton Engneerng, (SEKEIE ), pp: DOI: 0.007/ _89