Comparison of support-vector machines and back propagation neural networks in forecasting the six major Asian stock markets

Transcription

1 Int. J. Electronc Fnance, Vol., No., Comparson of support-vector machnes and back propagaton neural networks n forecastng the sx major Asan stock markets Wun-Hua Chen and Jen-Yng Shh Graduate Insttute of Busness Admnstraton, Natonal Tawan Unversty, No., Sec. 4, Roosevelt Road, Tape 06, Tawan, ROC Fax: E-mal: andychen@ntu.edu.tw E-mal: d @ntu.edu.tw Soushan Wu* College of Management, Chang-Gung Unversty, 59, Wen-Hwa st Road, Taoyuang, Tawan, ROC E-mal: swu@mal.cgu.edu.tw *Correspondng author Abstract: Recently, applyng the novel data mnng technques for fnancal tme-seres forecastng has receved much research attenton. However, most researches are for the US and European markets, wth only a few for Asan markets. Ths research apples Support-Vector Machnes (SVMs) and Back Propagaton (BP) neural networks for sx Asan stock markets and our expermental results showed the superorty of both models, compared to the early researches. Keywords: fnancal forecastng; Support-Vector Machnes (SVMs); Back Propagaton (BP) neural networks; Asan stock markets. Reference to ths paper should be made as follows: Chen, W-H., Shh, J-Y. and Wu, S. (006) Comparson of support-vector machnes and back propagaton neural networks n forecastng the sx major Asan stock markets, Int. J. Electronc Fnance, Vol., No., pp Bographcal notes: Wun-Hwa Chen receved hs PhD n Management Scence from the State Unversty of New York at Buffalo n 989. He s currently a Professor of Operatons Management n the Department of Busness Admnstraton at Natonal Tawan Unversty. Hs research and teachng nterests le n producton plannng and schedulng, AI/OR algorthmc desgn, data mnng models for customer relatonshp management and mathematcal fnancal modellng. He has publshed papers n numerous journals ncludng Annals of Operatonal Research, IEEE Transactons on Robotcs and Automaton, European Journal of Operatonal Research and Computers and Operatons Research. In addton, he has held the Charmanshp n Informaton Technology Advsory Commttees of several government agences n Tawan and has extensve consultng experences wth many major corporatons n Tawan. Copyrght 006 Inderscence Enterprses Ltd.

2 50 W-H. Chen, J-Y. Shh and S. Wu Jen-Yng Shh s a doctoral student n the Graduate Insttute of Busness Admnstraton at Natonal Tawan Unversty. Her current research nterests nclude data mnng and text mnng for fnancal applcatons and management ssues. She receved her BS n Fnance from Natonal Tawan Unversty n 995 and an MBA n Busness Admnstraton from Natonal Tawan Unversty n 997. Soushan Wu receved hs PhD n Fnance from the Unversty of Florda n 984. He s a Char Professor and Dean of the College of Management, Chang- Gung Unversty, Tawan. He s also a vstng scholar at Clemson Unversty, Hong Kong Polytechnc Unversty. Hs research nterests nclude management scence, nvestment scence, captal markets and nformaton systems. He has publshed over 70 papers n Research n Fnance, Fnancal Management, Asa-Pacfc Journal of Fnance, Internatonal Journal of Accountng and Informaton Systems, etc. He has also served as an ad hoc revewer for academc journals n the felds of Accountng, Fnance and Informaton Management. He serves as a Chef Edtor of the Journal of Management (n Tawan). Introducton In the busness and economc envronment, t s very mportant to accurately predct varous knds of fnancal varables to develop proper strateges and avod the rsk of potentally large losses. Few lterature works have been devoted to the predcton of these varables, such as stock market return and rsk predcton (Tay and Cao, 00; Thomason, 999b), sales forecastng (Stener and Wttkemper, 995) and monetary polcy. However, stock markets are often affected by many factors n fundamental and techncal analyss. In ths stuaton, data mnng technques may be one of the ways to satsfy the demand for the accurate predcton of fnancal varables. In the past, most predcton models were based on conventonal statstcal methods, such as tme-seres and multvarate analyss. In recent years, however, Artfcal Intellgence (AI) methodologes, ncludng the Artfcal Neural Networks (ANNs), Genetc Algorthms (GA) and fuzzy technologes, have become popular research models. They have been appled to many fnancal realms, ncludng economc ndces predcton (Gestel et al., 00; Legh et al., 00; Texera and Rodrgues, 997; Zhang et al., 00), stock/futures prce predcton (Kamjo and Tangawa, 990; Tay and Cao, 00), currency exchange rate predcton (Q and Zhang, 00; Shazly and Shazly, 999), opton tradng (Baley et al., 998), bond ratng (Km and Han, 00; Shn and Han, 00), bankruptcy predcton (Lee et al., 996; Zhang et al., 00) and portfolo optmsaton (Kosaka et al., 99). These AI technques were developed to meet the ncreasng demand for tools and methods that can predct, detect, classfy and summarse the structure of varables and defne the relatonshps between them wthout relyng too much on specfc assumptons, such as lnearty; or on error dstrbutons, such as normalty. Many AI applcatons n the fnancal analyss have addressed ths demand by developng the parametrc non-lnear models. In captal market research, stock or ndex prces are notorously dffcult to predct usng the tradtonal forecastng methods, such as least squares regresson, because of ther chaotc nature. Consequently, several AI-based methodologes have been proposed that clam better predcton capabltes than tradtonal models. Most researches,

3 Comparson of SVMs and BP neural networks 5 however, concentrated on the US and European captal markets; a very few studes have focused on the Asan captal markets, whch are famous for ther thn-dsh-lke characterstcs, that s, they overreact to good or bad events, and are very volatle. In ths paper, we apply two AI technques, Back Propagaton (BP) neural networks and Support Vector-Machnes (SVMs), to construct the predcton models for forecastng the sx major Asan stock market ndces and compare the results wth a benchmark developed by an autoregressve (AR) tme-seres model. ANNs are a feld of study wthn the AI area, where researchers employ a bologcally nspred method to process nformaton. They are good for solvng some real-world problems especally n the areas of forecastng and classfcaton decsons. However, most of the research has learnng pattern lmtatons because of the tremendous nose and complexty of the nput data. Some research ndcates that the BPs and SVMs can perform very well n the fnancal tme-seres forecastng, for example, BP algorthms have been appled as alternatves to statstcal technques because of ther superor accuracy. On the other hand, SVMs deal wth the non-lnear functon fttng problem very effcently because they seek to mnmse an upper bound of the generalsaton error. Furthermore, they are unque, optmal and free from local mnma. We therefore explore the applcablty of BP and SVMs to the fnancal predcton problem. The rest of ths paper s organsed as follows. In Secton, prevous research on fnancal predcton s revewed. The two predcton methods used n ths paper are dscussed n Sectons 3 and 4. Our research data and modellng are presented n Secton 5 and our emprcal results are gven n Secton 6. Fnally, n Secton 7, we present our conclusons. Related works. Revew of conventonal structural models Many models have been developed to predct stock market behavour, for example, one-or mult-step ahead prce predcton, prce change drecton, returns and rsks, portfolo asset allocaton and tradng strategy decsons. Structural models are the major approach n the stock market predcton. Two renowned models are: the Captal Asset Prcng Model (CAPM) (Sharpe, 964), whch descrbes the relatonshp between an ndvdual stock return ncludng ts rsk and the market return; and the Arbtrage Prcng Theory (APT) (Roll and Ross, 980), whch descrbes the relatonshp between an ndvdual stock return and some macroeconomc varables. In addton, Brock et al. (99) found non-lneartes n market prces and showed that the use of techncal analyss ndcators, under certan assumptons, may generate effcent tradng rules (Brock et al., 99). The prces of other fnancal commodtes also have ths non-lnear dynamc property. Savt (989), for example, suggested a non-lnear dynamc model for opton prces. In the last decade, statstcal models, especally tme-seres models and multvarate models, have become very popular n the constructon of varous knds of stock market behavour predcton models, as well as the economc ndcator models. Wth regard to tme-seres models, Granger and Sn (000) proposed AR- and ARCH-lke models for explorng the predctablty of an alternatve measure of rsk. Also, Östgermark and

4 5 W-H. Chen, J-Y. Shh and S. Wu Hernesnem (995) measured the nfluence of the freshness of nformaton on the ablty of an nvestor to forecast a stock ndex and the stock ndex future by applyng vector models, known as VARMAX-models. Ths study used data from the Fnnsh FOX ndex, depctng prce movements of the Helsnk Stock Exchange and the FOX futures ndex (Östgermark and Hernesnem, 995). Most multvarate analytcal predcton models choose some crtcal determnants as nput varables to predct stock market behavour and try to detect the relatonshps between the nput and output varables, such as stock return, rsk, etc. The choce of the nput varables s somewhat arbtrary, as there s no formal theory that supports stock returns as a functon of mcro- and macrondces. The former ncludes some techncal analytcal ndcators, whle the latter ncludes nterest rates, Gross Domestc Producton (GDP), etc. The representatve models of multvarate analytcal predcton models are lnear regresson models and multvarate dscrmnant analytcal models (Baestaens and van de Bergh, 995; Chen et al., 986; Ferson and Harvey, 99).. AI models Conventonal statstcal models have had lmted success, prmarly because the structural relatonshp between an asset prce and ts determnants changes over tme; hence, AI has become more popular n the captal market research n the recent years. A number of models use AI to forecast the prce behavour of the fnancal commodtes and compare the results wth the tradtonal statstcal methodologes or the prce behavour of dervatve fnancal commodtes. Refenes et al. (995), for example, modelled stock returns wth ANNs n the framework of APT and compared ther study wth regresson models. They showed that even the smple neural learnng procedures, such as BP algorthms easly outperform the best practce of regresson models n a typcal stock rankng applcaton wthn the framework of APT. Tsbours and Zedenberg (995) appled two models, a BP model and a temporal dfference model, to sx stock returns to test the weak form of Effcent Market Hypothess (EMH). They found some evdence to queston the null hypothess that the stock market s weakly effcent. Stener and Wttkemper (995) nvestgated the performance of several ANN models that forecast the return of a sngle stock and showed that ANNs outperform comparable statstcal technques n ths area. They reported that the relatonshp between the return of a stock and factors of the captal market s usually non-lnear and may contan dependences n tme. There may also be multple connectons between several factors (Stener and Wttkemper, 995). Shazly and Shazly (999) suggested a hybrd system combnng neural networks and genetc tranng to forecast the three-month spot rate of exchange for four currences: the Brtsh pound, the German mark, the Japanese yen and the Swss franc. The performance of the model seems to be better than the predctons made by the forward and futures rates (Shazly and Shazly, 999). Tay and Cao (00) appled SVMs to the prce predcton of fve real futures contracts from the Chcago Mercantle Market. Ther objectve was to examne the feasblty of usng SVMs n fnancal tme-seres forecastng by comparng them wth a multlayer BP neural network. On the bass of some devaton and drecton performance crtera, the results show that the SVM outperforms the BP neural network (Tay and Cao, 00). Gestel et al. (00) appled the Bayesan evdence framework, already used successfully n the desgn of multlayer perceptons, for Least Squares SVM (LS-SVM) regresson to nfer non-lnear models for

5 Comparson of SVMs and BP neural networks 53 predctng a tme-seres and the related volatlty. They appled the model to the predcton of the US short-term nterest rate and the DAX 30 ndex, and compared the results wth the tme-seres models, such as AR and GARCH (,) and buy and hold strateges. LS-SVM outperformed the other models n terms of the percentage of correct sgn predctons by 5 8% (Gestel et al., 00). Wttkemper and Stener (996) used fnancal statements from 67 German corporatons for the perod to predct a stock s systematc rsk by dfferent methods, ncludng the tradtonal methods such as the Blume and Fund models, neural network models and GA models. They showed that the neural networks, whose topologes had been optmsed by a genetc algorthm, gave the most precse forecasts. Although the superorty of the AI models has been shown n the US and European fnancal markets, few researches have been carred out n Asan markets (see Table ). As t s not possble to generalse the superorty of a sngle AI model for all markets, we desgned two AI models to examne tme-seres forecastng n the sx major Asan stock markets as lsted earler, some of whch are renowned for ther thn-dsh-lke characterstcs. Table Research nto the superorty of AI models Research Algorthm Commodty Market Refenes et al. (995) BP neural network Stock UK Tsbours and Zedenberg (995) Stener and Wttkemper (995) Shazly and Shazly (999) BP model and a temporal dfference model Stock US ANN models Stock ndex German A hybrd system combnng neural networks and genetc tranng Currency Tay and Cao (00) SVMs Futures US Gestel et al. (00) LS-SVM T-bll rate and stock ndex Wttkemper and Stener (996) Neural networks whose topologes are optmsed by a genetc algorthm Stock UK, German, Japan and Swss US and German German 3 SVMs for tme-seres forecastng 3. Introducton to SVMs The SVM, whch orgnated as an mplementaton of Vapnk s Structural Rsk Mnmzaton (SRM) prncple (Vapnk, 995), s now beng used to solve a varety of learnng, classfcaton and predcton problems. In many ways, an SVM performs the same functon as an ANN. For example, when both the nput and output data are avalable (supervsed learnng n ANN), the SVM can perform classfcaton and regresson; but when only the nput data are avalable, t can perform clusterng,

6 54 W-H. Chen, J-Y. Shh and S. Wu densty estmaton and prncple component analyss. The SVM s more than just another algorthm. It has the followng advantages over an ANN: t can obtan the global optmum the overfttng problem can be easly controlled and 3 emprcal testng has shown that the performance of SVMs s better than ANNs n classfcaton (Ca and Ln, 00; Morrs, and Autret, 00) and n regresson (Tay and Cao, 00). The SVM deals wth classfcaton and regresson problems by mappng the nput data nto hgh-dmensonal feature spaces. Its central feature s that the regresson surface can be determned by a subset of ponts or Support-Vectors (SV); all other ponts are not mportant n determnng the surface of the regresson. Vapnk ntroduced a ε-nsenstve zone n the error loss functon (Fgure ). Tranng vectors that le wthn ths zone are deemed correct, whereas those that le outsde the zone are deemed ncorrect and contrbute to the error loss functon. As wth classfcaton, these ncorrect vectors also become the support-vector set. Vectors lyng on the dotted lne are SV, whereas those wthn the ε-nsenstve zone are not mportant n terms of the regresson functon. Fgure Approxmaton functon (sold lne) of the SV regresson usng a ε-nsenstve zone (the area between the dotted lnes) 3. Support-vector regresson Here, we ntroduce the most mportant features of Support-Vector Regresson (SVR). Frstly, we dscuss the smplest model of SVR, a lnear regresson n the feature space. Though t s the easest algorthm but usually not very useful n real-world applcatons, t s a key for understandng complex SVR models. Ths s followed by a dscusson of the kernel functons that deal wth non-lnear decson surfaces. A dscusson of SVMs s gven by Burges (998), Crstann and Taylor (000) and Smola and Scholkopf (998). SVR s based on a smlar concept to SVM classfers. By generalsng the SVM algorthm for regresson estmaton, the SVR algorthm tres to construct a lnear functon such that the tranng ponts le wthn a dstance ε (Fgure ). As wth SVM classfers, we can also derve a quadratc programmng problem to generalse non-lnear regresson by usng the kernel functons.

7 Comparson of SVMs and BP neural networks 55 Gven a set of tranng data {(, ),..., ( l, l) } x y x y X R, where X denotes the space of the nput patterns, the goal of SVR s to fnd a functon f( x) that has at most ε devaton from the targets y for all of the tranng data and, at the same tme, s as flat as possble. Let the lnear functon f takes the form: T f( x) w x b wth w X, b R = + () Flatness n () means a smaller w. The problem can be formulated as mn w s.t. T y w x b ε T wx + b y ε () However, not all the problems are feasble n ths hard margn. As wth the classfcaton * problem, non-negatve slack varables, ξ, ξ, can be ntroduced to deal wth the otherwse nfeasble constrants of the optmsaton problem (). Thus, the formaton s stated as mn w l ( ξ ξ ) * + C + = T ε ξ y w x b + s.t. wx+ b y ε + ξ * ξ, ξ 0 T * (3) The constant C > 0 determnes the error margn trade-off between the flatness of f and the amount of devaton larger than ε that s tolerated. The formulaton corresponds to a ε -nsenstve loss functon ξ, descrbed by ε ξ ε 0 f ξ ε = ξ ε otherwse (4) The Lagrange relaxaton of (3) s as follows: L = w + C ( + ) ( + y + w x + b) l l * T ξ ξ α ε ξ = = l l * * T * * α ε + ξ + y w x b ηξ + ηξ = = ( ) ( ) (5)

8 56 W-H. Chen, J-Y. Shh and S. Wu * * where α, α, η andη are Lagrange multplers. Thus, the dual optmsaton problem of (5) s as follows: max ( )( ) + + s.t. = α α l l l * * T * * α α αj αj x xj ε ( α α ) y( α α ) j, = = = l * ( α α ) = 0 *, [0, C] (6) Equaton () can now be rewrtten as follows: l ( * l α ) and therefore ( ) ( * ) T α α α (7) w= x f x = x x+ b = = w s determned by tranng patterns x, whch are SVs. In a sense, the complexty of the SVR s ndependent of the dmensons of the nput space because t only depends on the number of SV. To enable the SVR to predct a non-lnear stuaton, we map the nput data nto a feature space. The mappng to the feature space F s denoted by n Φ: R F x Φ( x) The optmsaton problem (6) can be rewrtten as follows: max ( )( ) Φ( ) Φ( ) + + s.t. = α α l l l * * T * * α α αj αj x xj ε ( α α ) y( α α ) j, = = = l * ( α α ) = 0 *, [0, C] (8) T The decson functon can be computed by the nner products of Φ( x) Φ( x ) wthout explctly mappng to a hgher dmenson, whch s a tme-consumng task. Hence, the kernel functon s as follows: Kxz x z T (,) Φ() Φ() By usng a kernel functon, t s possble to compute the SVR wthout explctly mappng n the feature space. The condton for choosng kernel functons should conform to Mercers condton, whch allows the kernel substtutons to represent dot products n some Hlbert space.

9 Comparson of SVMs and BP neural networks 57 4 ANNs for tme-seres forecastng 4. Introducton to ANNs An ANN s a bologcally nspred form of dstrbuted computaton. It smulates the functons of the bologcal nervous system by a composton of nterconnected smple elements (artfcal neurons) operatng n parallel, as shown n Fgure. An element s a smple structure that performs three basc functons: nput, processng and output. ANNs can be organsed nto several dfferent connecton topologes and learnng algorthms (Lppmann, 987). The number of nputs to the network s constraned by the problem type, whereas the number of neurons n the output layer s constraned by the number of outputs requred by the problem type. However, the number of hdden layers and the szes of the layers are decded by the desgner. Fgure A neural network wth one hdden layer ANNs have become popular because they can be generalsed and do not make any assumpton about the underlyng dstrbuton of data. In addton, they have the ablty to learn any functon, for example, the two-layer sgmod/lnear network can represent any functonal relatonshp between the nput and output f the sgmod layer has enough neurons. ANNs have been successfully appled n dfferent areas, ncludng marketng, retal, bankng and fnance, nsurance telecommuncatons and operatons management (Smth and Gupta, 000). ANNs apply many learnng rules, of whch BP s one of the most commonly used algorthms n fnancal research. We therefore use the BP to develop ANN models n ths paper. In the next secton, we dscuss the BP neural networks. 4. Back Propagaton BP trans multlayer feedforward networks wth dfferentable transfer functons to perform functon approxmaton, pattern assocaton and pattern classfcaton. It s the process by whch the dervatves of network error, wth respect to network weghts and bases, are computed to perform computatons backwards through the network. Computatons are derved usng the chan rule of calculus. There are several dfferent BP tranng algorthms wth a varety of dfferent computaton and storage requrements. No sngle algorthm s best suted to all the problems. All the algorthms use the gradent of the performance functon to determne how to adjust the weghts to mnmse the performance. For example, the performance functon of feed forward networks s the Mean Square Error (MSE).

10 58 W-H. Chen, J-Y. Shh and S. Wu The basc BP algorthm adjusts the weghts n the steepest descent drecton (negatve of the gradent); that s, the drecton n whch the performance functon decreases most rapdly. The tranng process requres a set of examples of proper network nputs and target outputs. Durng the tranng, the weghts and bases of the network are teratvely adjusted to mnmse the network performance functon. Fgure 3 represents the pseudocode of a BP tranng algorthm. Fgure 3 Gradent-descent algorthm For the functon approxmaton problems n networks that contan up to a few hundred weghts, the Levenberg Maruardt algorthm generally has the fastest convergence. Ths advantage s especally useful f a very accurate tranng s requred. One method for mprovng generalsaton s early stoppng, whch dvdes the avalable tranng samples nto two subsets: the tranng set, used for computng the gradent and updatng the network weghts and bases the valdaton set, the error of the valdaton set, whch s montored durng the tranng process, wll normally decrease durng the ntal phase of tranng, as does the tranng set error. However, t typcally ncreases when the network begns to overft the data. After t has ncreased for a specfed number of teratons, the tranng s stopped, and the weghts and bases at the mnmum of the valdaton error are returned. 5 Results of emprcal testng 5. Data sets We examned the sx major Asan stock market ndces n our experment, namely the Nkke 5 (NK), the All Ordnares (AU), the Hang Seng (HS), the Strats Tmes (ST), the Tawan Weghted (TW) and the KOSPI (KO). Data were collected from the Yahoo Fnance, the Tawan Stock Exchange and the Korean Stock Exchange. The tme perods

11 Comparson of SVMs and BP neural networks 59 used and the ndces statstcal data are lsted n Table, and the daly closng prces used as the data sets are plotted n Fgure 4. The tme perods cover many mportant economc events, whch we beleve are suffcent for the tranng models. Table Descrpton of data sets Index Symbol Tme perod Hgh Low Mean SD Nkke 5 NK 984//4 00/5/3 38, , All Ordnares AU 984/8/3 00/5/ Hang Seng HS 986//3 00/5/3 8, Strats Tmes ST 987//8 00/5/ TAIEX TW 97//5 00/5/3, KOSPI KO 980//4 00/5/ Fgure 4 Indces of the sx Asan stock markets We further transformed the orgnal closng prce nto a fve-day Relatve Dfference n Percentage of prce (RDP). Accordng to Thomason (999a,b) and Tay and Cao (00), ths transformaton has many advantages. The most mportant s that the dstrbuton of the transformed data becomes more symmetrcal so that t follows a normal dstrbuton

12 60 W-H. Chen, J-Y. Shh and S. Wu more closely. Ths mproves the predctve power of the neural network. Followng the predcton model used n the prevous research (Tay and Cao, 00; Thomason, 999a,b), the nput varables n ths paper are determned from four-lagged RDP values based on fve-day perods (RDP-5, RDP-0, RDP-5 and RDP-0); and one transformed closng ndex (EMA5), obtaned by subtractng a fve-day exponental movng average from the closng ndces (Table 3). Applyng the RDP transform to the closng ndces over the perod of the data removes the trend n prces. EMA5 s used to retan as much nformaton as possble from the orgnal closng ndex because the applcaton of the RDP transform to the orgnal closng ndex may remove some useful nformaton. The fve-day-based nput varables correspond to techncal analyss concepts. The output varable RDP+5 s obtaned by frst smoothng the closng ndex wth a three-day exponental movng average. The applcaton of a smoothng transform to the dependent varable generally enhances the predcton performance of the neural network (Thomason, 999b). Table 3 Input and output varables Indcator Calculaton Input varables EMA5 p () EMA5 () RDP-5 ( p ( ) p ( 5))/ p ( 5) 00 RDP-0 ( p ( ) p ( 0))/ p ( 0) 00 RDP-5 ( p ( ) p ( 5)) / p ( 5) 00 RDP-0 ( p ( ) p ( 0)) / p ( 0) 00 Output varables RDP+5 ( p ( + 5) p ( ))/ p ( ) 00 p () = EMA() 3 5. Preprocessng data and desgn of the models Frstly, RDP values beyond the ± Standard Devatons (SD) are selected as outlers and replaced wth the closest margnal values. Next, the preprocessng technque for data scalng s performed. For the sake of comparson across data sets, all data ponts are scaled n the range of to, ncludng the output data. Each of the sx data sets s parttoned nto three subsets accordng to the tme sequence n the rato 80:0:0. The frst part s used for tranng; the second s a valdatng set that selects optmal parameters for the SVR and prevents the overfttng found n the BP neural networks and the last part s used for the testng. 5.3 Performance crtera Although the MSE s a perfectly acceptable measure of performance, n practce the ultmate goal of any testng strategy s to confrm that the results of models are robust and capable of measurng the proftablty of a system. It s mportant, therefore, to

13 Comparson of SVMs and BP neural networks 6 desgn a test from the outset. Ths s not always carred out wth the level of rgor that t merts, partly because of unfamlarty wth the establshed methods or practcal dffcultes ntrnsc to non-lnear systems. Consequently, we desgned test sets to evaluate the effects of the models. Accordng to Tay and Cao (00) and Thomason (999a,b), the predcton performance s evaluated usng the followng statstcs: MSE, Normalsed Mean Squared Error (NMSE), Mean Absolute Error (MAE), Drectonal Symmetry (DS) and Weghted Drectonal Symmetry (WDS). These crtera are defned n Table 4. MSE, NMSE and MAE measure the correctness of a predcton n terms of levels and the devaton between the actual and predcted values. The smaller the values, the closer the predcted tme-seres values wll be to the actual values. Although predctng the levels of prce changes (or frst dfferences) s desrable, n many cases the sgn of the change s equally mportant. Most nvestment analysts are usually far more accurate at predctng drectonal changes n an asset prce than predctng the actual level. DS provdes an ndcaton of the correctness of the predcted drecton of RDP+5 gven n the form of percentages (a large value suggests a better predctor). WDS measures both the magntude of the predcton error and ts drecton. It penalses ncorrectly predcted drectons and rewards drectons that are correctly predcted. The smaller the value of WDS, the better the forecastng performance wll be n terms of both the magntude and drecton. Table 4 Performance metrcs and ther calculaton Metrcs MSE NMSE Calculaton n MSE = ( a p) n = δ n = n NMSE = ( a p) δ n = ( a a) n = MAE DS n MAE = a p n = n 00 DS = d n = ( a a )( p p ) 0 d = 0 otherwse WDS n n ' d a p d a p = = WDS = 0 ( a a )( p p ) 0 d = otherwse ' ( a a )( p p ) 0 d = 0 otherwse

14 6 W-H. Chen, J-Y. Shh and S. Wu 5.4 SVM mplementaton We appled Vapnk s SVM for regresson by usng LIBSVM ( ntu.edu.tw/~cjln/lbsvm/). The typcal kernel functons used n SVRs are the polynomal kernel kx (, y) = ( x y+ ) d and the Gaussan kernel kxy (,) = exp( / δ ( x y) ), where d s the degree of the polynomal kernel and δ s the bandwdth of the Gaussan kernel. We chose the Gaussan kernel as our kernel functon because t performs well under general smoothness assumptons. The parameters to be determned are kernel bandwdth δ, C and ε. Accordng to Thomason (999a,b) and Tay and Cao (00), t showed that SVRs are nsenstve to ε, as long as t s a reasonable value. Thus, we chose 0.00 for ε. For δ and C, we used tenfold cross valdaton to choose sutable values. The values of the canddates were δ {,, 0.5, 0., 0.0, 0.00, 0.000} and C {500,50,00,50,0,}, respectvely. Table 5 gves the parameters obtaned from the trals of each market scenaro. Table 5 Parameter set of each ndex Country δ C NK AU 0 HS ST TW KO BP mplementaton The BP models used n ths experment are mplemented usng the Matlab 6. s ANN toolbox. Several learnng technques, such as the quas-newton method, Levenberg-Marquardt algorthm and conjugate gradent methods could also be used. For effcency, however, we use the Levenberg Marquardt algorthm. The archtecture of the BP model s as follows: the number of hdden nodes vares from 8 to 0 and the optmum number of hdden nodes (0) that mnmses the error rate on the valdaton set s determned. The actvaton functon of the hdden layer s sgmod and the output node uses the lnear transfer functon. 6 Results and dscusson For each market, we also desgned an AR () tme-seres model as the benchmark for comparson. The forecastng results of the SVM, BP and AR () for the test set are collated n Table 6. In general, SVMs and BP models outperformed the AR () model n the devaton performance crtera. However, n the drecton performance crtera, the AR () model proved superor, possbly because SVMs and BPs are traned n terms of

15 Comparson of SVMs and BP neural networks 63 devaton performance; the former to fnd the error bound and the latter to mnmse MSE. Thus, n the devaton performance crtera, they perform better than the benchmark model AR (). Wth regard to the drecton crtera, the tme-seres model has an ntrnsc mert because t emphasses the trend of the seres. Therefore, the two AI models cannot outperform AR n the drecton crtera n the sx Asan markets under study. Table 6 Comparson of the results of SVM, BP and AR () model on the test set MSE NMSE MAE DS WDS NK SVM BP AR () AU SVM BP AR () HS SVM BP AR () ST SVM BP AR () TW SVM BP AR () KO SVM BP AR () The performances of the SVM and BP models are farly good compared to Tay and Cao (00). The devaton crtera, such as NMSEs are less than, except for the AU ndex n the SVMs. The NMSEs of BP are less than, except for the AU and KO ndces. As to the correctness of predcted drecton, all the values of DS are more than 50%. We also obtan smaller WDS values. Fgure 5 shows the predcted effects for the perod September October 00, whch ncluded the 9/ terrorst attacks. The ftted lnes of the BP and SVM models seem close to each other and both are closer to the real value lne than the AR () lne because of the smaller values of the devaton measurement. However, ther drecton performance s not as good as the AR () lne for the smaller value of the drecton measurement n the SVM and BP models.

16 64 W-H. Chen, J-Y. Shh and S. Wu Fgure 5 RDP+5 values of ndces ftted by AR (), BP and SVM models and ther real values. (a) Nkke 5, (b) All Ordnares, (c) Hang Seng, (d) Strats Tmes, (e) TAIEX and (f) KOSPI Table 7 gves the comparson of the results of the BP and SVM models. There s no dfference n the AU ndex for the three devaton performance crtera (MSE, NMSE and MAE). In the HS, TW and KO data sets, the SVM models perform slghtly better than the BP models. On the other hand, the BP method performs only a lttle better than SVMs n the NK and ST data sets. In total, there s no sgnfcant dfference between the performances of the two models n the devaton measurements. Takng the AR () model as a benchmark to evaluate the two AI models, both seem to perform better than AR () for all sx markets n terms of the devaton measurements. For the other two predcted drecton crtera (DS and WDS), SVMs work better than BPs n the AU, ST and TW ndces; however, BPs work better than SVMs n the NK, HS and KO ndces. In all sx markets, we fnd that the AR () model outperforms the SVM and BP models n terms of the predcted drecton crtera.

17 Comparson of SVMs and BP neural networks 65 Because there s no consstency n the superorty of the two AI models, our result dffers from that of Tay and Cao (00). The performance of SVMs s not always better than the BPs n these performance crtera. Table 7 summarses the comparson between the two models. Generally speakng, based on the fve performance crtera, SVMs ft the AU, HS, TW and KO ndces well, whereas BP fts NK and ST ndces. Table 7 Comparson of BP and SVMs on the test set MSE NMSE MAE DS WDS Summary NK AU HS ST + + TW KO represents no dfference between SVM and ANN, + represents SVMs that perform well and represents ANNs that perform well. 7 Concluson In ths paper, we have examned the feasblty of applyng two AI models, SVM and BP, to fnancal tme-seres forecastng for the Asan stock markets, some of whch are famous for ther thn-dsh-lke characterstcs. Our experments demonstrate that both models perform better than the benchmark AR () model n the devaton measurement crtera. Although prevous research clams that the SVM method s superor to the BP, there are some exceptons, for example, the NK and the ST ndces. In general, the SVM and the BP models perform well n the predcton of ndces behavour n terms of the devaton crtera. Our experments also show that the AI technques can assst the stock market tradng and the development of the fnancal decson support systems. Future research should apply more complex SVM methods and other ANN algorthms to the forecastng of Asan stock markets. Drecton predcton crtera are mportant sgnals n the tradng strateges of nvestors. Improvng these crtera n AI models s, therefore, another mportant ssue n the fnancal research. In our current models, we only select the return of ndex prce nformaton as nput varables. Enhancng the performance of predcton models by ncludng other effcent nput varables (such as some macroeconomc varables), the relatonshps between dfferent markets and other tradng nformaton about the markets should also be the subject of future research. References Anthony, M. and Bggs, N.L. (995) A computatonal learnng theory vew of economc forecastng wth neural nets, Neural Networks n the Captal Markets, pp Baestaens, D.E. and van den Bergh, W.M. (995) Trackng the Amsterdam stock ndex usng neural networks, Neural Networks n the Captal Markets, pp.49 6.

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