Hybridizing Exponential Smoothing and Neural Network for Financial Time Series Predication

Transcription

1 Hybridizing Exponenial Smoohing and Neural Nework for Financial Time Series Predicaion Kin Keung Lai 1, 2, Lean Yu 2, 3, Shouyang Wang 1, 3, and Wei Huang 4 1 College of Business Adminisraion, Hunan Universiy, Changsha , China 2 Deparmen of Managemen Sciences, Ciy Universiy of Hong Kong, Ta Chee Avenue, Kowloon, Hong Kong {mskklai, msyulean}@ciyu.edu.hk 3 Insiue of Sysems Science, Academy of Mahemaics and Sysems Science, Chinese Academy of Sciences, Beijing , China {yulean, sywang}@amss.ac.cn 4 School of Managemen, Huazhong Universiy of Science and Technology, 1037 Luoyu Road, Wuhan , China Absrac. In his sudy, a hybrid synergy model inegraing exponenial smoohing and neural nework is proposed for financial ime series predicion. The proposed model aemps o incorporae he linear characerisics of an exponenial smoohing model and nonlinear paerns of neural nework o creae a synergeic model via he linear programming echnique. For verificaion, wo real-world financial ime series are used for esing purpose. 1 Inroducion A challenging ask in financial markes such as sock marke and foreign exchange marke is o predic he movemen direcion of financial markes so as o provide valuable decision informaion for invesors. Thus, various kinds of forecasing mehods have been developed by many researchers and business praciioners. Of he various forecasing models, he exponenial smoohing model has been found o be one of he effecive forecasing mehods. Since Brown [1] began o use simple exponenial smoohing o predic invenory demand, he exponenial smoohing models have been widely used in business and finance [2-3]. For example, Gardner [2] inroduced exponenial smoohing mehods ino supply chain managemen for predicing demand, and achieved saisfacory resuls. Leung e al. [3] used an adapive exponenial smoohing model o predic Nikkei 225 indices, and achieved good resuls. However, he exponenial smoohing mehod is only a class of linear model and hus i can only capure linear feaure of financial ime series. Bu financial ime series are ofen full of nonlineariy and irregulariy. Furhermore, as he smoohing consan decreases exponenially, he disadvanage of he exponenial smoohing model is ha i gives simplisic models ha only use several previous values o forecas he fuure. The exponenial smoohing model is, herefore, unable o find suble nonlinear paerns in he financial ime series daa. Obviously, he approximaion of linear models o complex real-world problems is no always sufficien. Hence, i is necessary o consider oher nonlinear mehods o complemen he exponenial smoohing model. V.N. Alexandrov e al. (Eds.): ICCS 2006, Par IV, LNCS 3994, pp , Springer-Verlag Berlin Heidelberg 2006

2 494 K.K. Lai e al. Recenly, arificial neural nework (ANN) models have shown heir nonlinear modeling capabiliy in financial ime series forecasing. Since Lapedes and Farker [4] used ANN o predic he chaoic ime series, he ANN models are widely used in he ime series forecasing. For example, Refenes e al. [5] applied mulilayer forward nework models o forecas foreign exchange prices and obained good resuls. Alhough he ANN models achieve success in financial ime series forecasing, hey have some disadvanages. Since he real world is highly complex, here exis some linear and nonlinear paerns in he financial ime series simulaneously. I is no sufficien o use only a nonlinear model for ime series because he nonlinear model migh miss some linear feaures of ime series daa. Furhermore, previous sudies [6-7] are shown ha using ANN o model linear problems may produce mixed resuls. In such siuaions, i is necessary o hybridize he linear model and nonlinear model for financial ime series forecasing. This is because he ANN model and exponenial smoohing model are complemenary. On one hand, he ANN model can find suble nonlinear feaures hidden in he ime series daa, bu may miss some linear paerns when forecasing. On he oher hand, he exponenial smoohing model can give good resuls in he linear paerns of ime series, bu canno capure he nonlinear paerns, which migh resul in inaccurae forecass. Moivaed by he previous findings, his sudy proposes a hybrid synergy model o financial ime series predicion inegraing exponenial smoohing and ANN via linear programming echnique. The exponenial smoohing model raher han oher linear models such as ARIMA is chosen as neural nework model s complemen for several reasons. Firs of all, he major advanage of exponenial smoohing mehods is ha hey are simple, inuiive, and easily undersood. These mehods have been found o be quie useful for shorerm forecasing of large numbers of ime series. A he same ime, exponenial smoohing echniques have also been found o be appropriae in such applicaions because of heir simpliciy. Second, he exponenial smoohing model has less echnical modeling complexiy han he ARIMA model and hus makes i more popular in pracice. As Lilien and Koler [8] repored, exponenial smoohing models have been widely used by approximaely 13% of indusry. Third, Mills [9] found lile difference in forecas accuracy beween exponenial smoohing echniques and ARIMA models. In some examples, exponenial smoohing models can even obain beer resuls han neural nework model. Foser e al. [10] once argued ha he exponenial smoohing is superior o neural neworks in forecasing yearly daa. Generally, he exponenial smoohing model is regarded as an inexpensive echnique ha gives forecass ha is good enough in a wide variey of applicaions. The remainder of he sudy is organized as follows. Secions 2 and 3 provide some basic backgrounds abou he exponenial smoohing and neural nework mehods for financial ime series forecasing. In Secion 4, he hybrid mehodology combining he exponenial smoohing and neural nework model via linear programming is inroduced. For verificaion, wo experimens are performed in Secion 5. Finally, Secion 6 concludes he paper. 2 The Exponenial Smoohing Forecasing Model In he applicaion of he exponenial smoohing model, here are hree ypes of models ha are widely used in differen ime series. Simple exponenial smoohing (Type I) is

3 Hybridizing Exponenial Smoohing and Neural Nework 495 used when he ime series has no rend. Double exponenial smoohing (Type II) is an exponenial smoohing mehod for handling a ime series ha displays a slowly changing linear rend. Two approaches are covered: one-parameer double exponenial smoohing, which employs a single smoohing consan; and Hol-Winers woparameer double exponenial smoohing, which employs wo smoohing consans. The hird is Winers mehod (Type III), which is an exponenial smoohing approach o predicing seasonal daa. This mehod also conains wo approaches: muliplicaive Winers mehod, which is appropriae for increasing seasonal variaion; and addiive Winers mehod, which is appropriae for consan seasonal variaion [1-2]. In financial ime series, here is irregulariy, randomiciy and no rend. These feaures show ha he simple exponenial smoohing mehod is suiable for financial ime series forecasing for he specified ime period. Therefore, only he ype I of he exponenial smoohing model is described in deail. For he ype II & III, ineresed readers can be referred o [1-3] for more deails. Suppose ha he ime series y 1, y 2,, y n is described by he model y = β 0 + ε (1) where β 0 is he average of he ime series and ε random error. Then he esimae S of β0 made in ime is given by he smoohing equaion S = y + ( 1 α ) S 1 α (2) where α is a smoohing consan beween 0 and 1 and S -1 is he esimae of β0 in -1. Thus a poin forecas made in ime for y +1 is From Equaion (2), we have yˆ 1 = S = αy + (1 α ) yˆ (3) + S 1 = α y 1 + ( 1 α) S (4) 2 Subsiuing Equaion (4) o Equaion (2), hen 2 S = α y + ( 1 α)( αy 1 + (1 α) S 2 ) = αy + α(1 α) y 1 + (1 α) S (5) 2 Similarly, subsiuing recursively for S -2, S -3,, S 1 and S 0, we obain S = yˆ + L S (6) = αy + α(1 α) y 1 + α(1 α) y α(1 α) y1 + (1 α) Here we see ha S, he esimae made in ime of he average β0 of he ime series, can be expressed in erm of observaions y, y -1,, y 1 and he iniial esimae S 0. The coefficiens measuring he conribuions of he observaions y, y -1,, y 1 ha is, 1 α, α (1 α ), L, α (1 α ) decrease exponenially wih ime. For his reason his mehod is referred as simple exponenial smoohing. In order o use Equaion (6) o predic he ime series, we need o deermine he value of he smoohing consan α and he iniial esimae S 0. For he smoohing consan, he ordinary leas square (OLS) can be used o deermine α ; while for he 0

4 496 K.K. Lai e al. iniial value S 0, we can le S 0 be equal o y 1, i.e. S 0 = y 1, or le S 0 be equal o he simple arihmeic average of a few previous observaions. For example, S 0 = (y 1 +y 2 +y 3 )/3. Once deermining α and S 0, exponenial smoohing model can be used for predicion. The advanage of he exponenial smoohing mehod is ha i is capable of fiing he linear paerns of he ime series well and easy o use. Bu he financial ime series is ofen irregular and nonlinear, i is no sufficien o use exponenial smoohing for financial ime series modeling. 3 The Neural Nework Forecasing Model In his sudy, one of he widely used ANN models, he back-propagaion neural nework (BPNN) [11], is used for ime series forecasing. The main reason is ha some sudies (e.g. [11-12]) have shown ha he BPNN wih an ideniy ransfer funcion in he oupu uni and logisic funcions in he middle-layer unis can approximae any coninuous funcion arbirarily well given a sufficien amoun of middle-layer unis [12]. Yu e al. [13] have also found ha BPNN has been one popular model ha can approximae various nonlineariies in he daa series. Generally, he BPNN can be rained by he hisorical daa of a ime series in order o capure he non-linear characerisics of he specific ime series. The model parameers (connecion weighs and node biases) will be adjused ieraively by a process of minimizing he forecasing errors. For ime series forecasing, according o he previous compuaion process he relaionship beween he oupu (y ) and he inpus (y 1, y 2,, y p ) has he following mahemaical represenaion. where q p 0 + a j j f w j + w i ij y = 1 0 = 1 i ) y a ( + e (7) = a j (j = 0, 1, 2,, q) is a bias on he jh uni, and w ij (i = 0, 1, 2,, p; j = 0, 1, 2,, q) is he connecion weighs beween layers of he model, f( ) is he ransfer funcion of he hidden layer, p is he number of inpu nodes and q is he number of hidden nodes. Acually, he BPNN model in (11) performs a nonlinear funcional mapping from he pas observaion (y 1, y 2,, y p ). o he fuure value (y ), i.e., y ϕ + e (8) = ( y 1, y 2, L, y p, w) where w is a vecor of all parameers and φ is a funcion deermined by he nework srucure and connecion weighs. Thus, in some senses, he BPNN model is equivalen o a nonlinear auoregressive (NAR) model. A major advanage of neural neworks is heir abiliy o provide flexible nonlinear mapping beween inpus and oupus. They can capure he nonlinear characerisics of ime series well. However, using BPNN o model linear problems may produce mixed resuls [6-7]. Therefore, we can conclude ha he relaionship beween exponenial smoohing and BPNN is complemenary. To ake full advanage of he individual srenghs of wo models, i is necessary o inegrae he exponenial smoohing and BPNN models, as menioned earlier.

5 Hybridizing Exponenial Smoohing and Neural Nework The Hybrid Forecasing Mehodology In real life, financial ime series forecasing is far from simple due o high volailiy, complexiy, irregulariy and noisy marke environmen. Furhermore, real-world ime series are rarely pure linear or nonlinear. They ofen conain boh linear and nonlinear paerns. If his is he case, here is no universal model ha is suiable for all kinds of ime series daa. Boh exponenial smoohing models and BPNN models have achieved success in heir own linear or nonlinear domains, bu neiher exponenial smoohing nor BPNN can adequaely model and predic ime series since he linear models canno deal wih nonlinear relaionships while he BPNN model alone is no able o handle boh linear and nonlinear paerns equally well [6]. On he oher hand, as previously menioned, for ime series forecasing he relaionship beween exponenial smoohing and BPNN is complemenary. Exponenial smoohing is a class of linear models ha can capure ime series linear characerisics, while BPNN models are a class of general funcion approximaors capable of modeling nonlineariy and which can capure nonlinear paerns in ime series. Hybridizing he wo models may yield a robus mehod, and more saisfacory forecasing resuls may be obained by incorporaing an exponenial smoohing model and a BPNN model. Therefore, we propose a hybrid mehodology inegraing he exponenial smoohing and he BPNN for financial ime series forecasing. Differen from decomposiion-hybrid principle described in [6], we adop he parliamenary hybridizaion sraegy [14] o creae a synergeic model, as shown in Fig. 1. Fig. 1. The hybrid synergy forecasing model From Fig. 1, he inpu is fed simulaneously ino a BPNN model and an exponenial smoohing forecasing model. The BPNN model generaes a forecas resul, while he exponenial smoohing model also generaes a ime series forecas resul. The wo forecas resuls are enered ino he hybrid forecas module and generae a synergeic forecas resul as final oupu. In he hybridizaion process, he parliamenary hybridizaion sraegy [14] is used, i.e., yˆ Hybird = αyˆ + (1 α) yˆ (9) ES BPNN

6 498 K.K. Lai e al. where ES ŷ is he forecas resul obained from exponenial smoohing model, BPNN ŷ is he forecas resul of he BPNN and α is he weigh parameer. Through inegraing linear paerns and nonlinear paerns of financial ime series, a synergeic effec will believed o be creaed o improve he predicaion performance. In Equaion (9), a criical problem is how o deermine he weigh parameer α. Generally, he value of α can be esimaed by he ordinary leas square (OLS) mehod, i.e., n Hybird 2 MinQ = ( y = yˆ 1 ) (10) However, is drawback of his approach is ha he square reamen will move he fied curve o some excepional poins and hus reducing he forecasing accuracy. One modificaion in his sudy is o minimize he sum of absolue error beween esimaed and he acual value, hen MinQ = n Hybird n y = = y 1 = ˆ e (11) The Equaion (11) can be solved by linear programming, le e + e, 0, 2 e e e e 0, e 0 u = =, v = = (12) 0, e < 0 2 e, e < 0 Clearly, e = u + v, e = u v, hen he linear programming (LP) model can be formulaed below. n = ( u + v = ) 1 ES BPNN n [ y ( yˆ + βyˆ )] ( u v ) Min Q n LP ) α = 0 = 1 = (13) α + β = 1 α 0, u 0, v 0, = 1,2, L, n ( 1 Using he simplex algorihm, an opimal hybridizaion parameer can be obained from he LP problem in Equaion (13). To verify he effeciveness of he hybridizaion approach, wo experimens abou exchange rae predicaion are performed. 5 Experimen Sudy The daa se used in his paper are daily daa from 1 January 2000 ill 31 December 2002 and are obained from Pacific Exchange Rae Service (hp://fx.sauder.ubc.ca/), provided by Professor Werner Anweiler, Universiy of Briish Columbia, Vancouver, Canada. The daily daa cover hree years of observaions of wo major inernaional currency exchange raes --- euros/us dollar (EUR/USD) and Japanese yen/us dollar (JPY/USD). In our empirical experimen, he daa se is divided ino wo sample periods --- he esimaion (in-sample) and he es (ou-of-sample) periods. The esimaion period covers observaions from 1 January 2000 ill 31 Sepember 2002 and is used o esimae and refine he forecas model parameers. Meanime we ake he daa from 1 Ocober 2002 o 31 December 2002 as es ses, which are used o evaluae he good or bad performance. For space, he original daa are omied, and deailed daa can be obained from he websie. 1

7 Hybridizing Exponenial Smoohing and Neural Nework 499 In his sudy, he roo mean square error (RMSE) and direcional saisics (D sa ) [13] are used as evaluaion crieria. In addiion, he individual exponenial and BPNN models are seleced as benchmark models for comparison purposes. Finally, only onesep-ahead forecasing is considered in his sudy. Based on our analysis above, we examine he forecas performances of wo major currency exchange raes in erms of RMSE and D sa. The experimenal resuls are repored in Table 1. Table 1. The experimen resuls of EUR/USD and JPY/USD Currencies EUR/USD JPY/USD Models ES BPNN Hybrid ES BPNN Hybrid RMSE D sa (%) From he viewpoin of RMSE, he hybrid mehodology is he bes for boh EUR/USD and JPY/USD, followed by he individual exponenial smoohing and individual BPNN model. For example of EUR/USD, he RMSE of he individual BPNN model is , and he individual exponenial smoohing model is , while he RMSE of he hybrid mehodology is only However, he direcion predicion is more imporan han he accuracy predicion in he financial markes because he former can provide decision informaion for invesors direcly. Furhermore, he high RMSE can no lead o high D sa, as he exponenial smoohing and he BPNN reveals. Focusing on he D sa, he hybrid mehod performs he bes, followed by he individual BPNN model; he wors is he individual exponenial smoohing. For example of JPY/USD, he D sa of exponenial smoohing is only 51.61%, he D sa of he BPNN model is 58.06%, while ha of he hybrid mehodology arrives a 66.13%. The main reason is ha he hybrid mehodology inegraing linear paerns and nonlinear paerns creaes a synergeic effec and hus improves he predicion performance. 6 Conclusions In his paper, we propose a hybrid synergy mehodology incorporaing exponenial smoohing and neural nework for financial ime series forecasing and explore he forecasing capabiliy of he proposed hybrid synergy mehodology from he poin of level predicion and direcion predicion. Experimenal resuls obained reveal ha he hybrid mehodology performs beer han he wo benchmark models, implying ha he proposed hybrid synergy approach can be used as an alernaive soluion o he financial ime series forecasing. Acknowledgemens This work is parially suppored by Naional Naural Science Foundaion of China (NSFC No ); Chinese Academy of Sciences; Key Research Insiue of

8 500 K.K. Lai e al. Humaniies and Social Sciences in Hubei Province-Research Cener of Modern Informaion Managemen and Sraegic Research Gran of Ciy Universiy of Hong Kong (SRG No ). References 1. Brown, R.G.: Saisical Forecasing for Invenory Conrol. New York, McGraw-Hill, Gardner, E.S.: Exponenial Smoohing: The Sae of he Ar. Journal of Forecasing 4 (1985) Leung, M.T., Daouk, H., Chen, A.S.: Forecasing Sock Indices: A Comparison of Classificaion and Level Esimaion Models. Inernaional Journal of Forecasing 16 (2000) Lapedes, A., Farber, R.: Nonlinear Signal Processing Using Neural Nework Predicion and Sysem Modeling. Theoreical Division, Los Alamos Naional Laboraory, NM Repor. No. LA-UR , Refenes, A.N., Azema-Barac, M., Chen, L., Karoussos, S.A.: Currency Exchange Rae Predicion and Neural Nework Design Sraegies. Neural Compuing Applying 1 (1993) Zhang, G.P.: Time Series Forecasing Using a Hybrid ARIMA and Neural Nework Model. Neurocompuing 50 (2003) Denon J.W.: How Good are Neural Neworks for Causal Forecasing? Journal of Business Forecasing 14 (1995) Lilien, G.L., Koler, P.: Markeing Decision Making: A Model Building Approach. New York, Harper and Row Publishers (1983) 9. Mills, T.C.: Time Series Techniques for Economiss. Cambridge Universiy Press (1990) 10. Foser, B., Collopy, F., Ungar, L.: Neural Nework Forecasing of Shor, Noisy Time Series. Compuers and Chemical Engineering 16 (1992) Hornik, K., Sinchocombe, M., Whie, H.: Mulilayer Feedforward Neworks are Universal Approximaors. Neural Neworks 2 (1989) Whie, H.: Connecionis Nonparameric Regression: Mulilayer Feedforward Neworks can Learn Arbirary Mappings. Neural Neworks 3 (1990) Yu, L., Wang, S.Y., Lai, K.K.: A Novel Nonlinear Ensemble Forecasing Model Incorporaing GLAR and ANN for Foreign Exchange Raes. Compuers and Operaions Research 32 (2005) Wedding II, D.K., Cios, K.J.: Time Series Forecasing by Combining RBF Neworks, Cerainy Facors, and he Box-Jenkins Model. Neurocompuing 10 (1996)