Stock Index Modeling using EDA based Local Linear Wavelet Neural Network

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1 Stoc Idex odelg usg EDA bsed Locl Ler Wvelet Neurl Networ Yuehu Che School of Iformto Scece d Egeerg J Uversty Jwe rod 06, J , P.R.Ch E-ml: yhche@uj.edu.c Xohu Dog School of Iformto Scece d Egeerg J Uversty Jwe rod 06, J , P.R.Ch E-ml: cop2@uj.edu.c You Zho School of Iformto Scece d Egeerg J Uversty Jwe rod 06, J , P.R.Ch E-ml: you_zho@yhoo.com.c Abstrct The use of tellget systems for stoc mret predctos hs bee wdely estblshed. I ths pper, we vestgte how the seemgly chotc behvor of stoc mrets could be well represeted usg Locl Ler Wvelet Neurl Networ (LLWNN techque. To ths ed, we cosdered the Nsdq-00 dex of Nsdq Stoc ret S d the S&P CNX NIFTY stoc dex. We lyzed 7-yer Nsdq-00 m dex vlues d 4-yer NIFTY dex vlues. Ths pper vestgtes the developmet of ovel relble d effcet techques to model the seemgly chotc behvor of stoc mrets. The LLWNN re optmzed by usg Estmto of Dstrbuto Algorthm (EDA. Ths pper vestgtes whether the proposed method c provde the requred level of performce, whch s suffcetly good d robust so s to provde relble forecst model for stoc mret dces. Expermet results show tht the model cosdered could represet the stoc dces behvor very ccurtely. I. INTRODUCTION Predcto of stocs s geerlly beleved to be very dffcult ts - t behves le rdom wl process d tme vryg. The obvous complexty of the problem pves the wy for the mportce of tellget predcto prdgms. Durg the lst decde, stocs d futures trders hve come to rely upo vrous types of tellget systems to me trdg decsos [][2]. Severl tellget systems hve recet yers bee developed for modelg expertse, decso support d complcted utomto tss [3][4]. I ths pper, we lyzed the seemgly chotc behvor of two well-ow stoc dces mely the Nsdq-00 dex of Nsdq S [5] d the S&P CNX NIFTY stoc dex [6]. The Nsdq-00 dex reflects Nsdq s lrgest compes cross mjor dustry groups, cludg computer hrdwre d softwre, telecommuctos, retl/wholesle trde d botechology [5]. The Nsdq-00 dex s modfed cptlzto weghted dex, whch s desged to lmt domto of the Idex by few lrge stocs whle geerlly retg the cptlzto rg of compes. Through vestmet the Nsdq-00 dex trcg stoc, vestors c prtcpte the collectve performce of my of the Nsdq stocs tht re ofte the ews or hve become household mes. Smlrly, S&P CNX NIFTY s well-dversfed 50 stoc dex ccoutg for 25 sectors of the ecoomy [6]. It s used for vrety of purposes such s bechmrg fud portfolos, dex bsed dervtves d dex fuds. The CNX Idces re computed usg mret cptlzto weghted method, where the level of the Idex reflects the totl mret vlue of ll the stocs the dex reltve to prtculr bse perod. The method lso tes to ccout costtuet chges the dex d mporttly corporte ctos such s stoc splts, rghts, etc. wthout ffectg the dex vlue. I our prevous wor, the eurl etwor (NN d flexble eurl tree (FNT hve bee employed for stoc dex modelg [7][9]. Ths reserch s to vestgte the performce lyss of LLWNN for modelg the Nsdq-00 d the NIFTY stoc mret dces. The prmeters of the LLWNN model re optmzed by EDA. We lyzed the Nsdq-00 dex vlue from Jury 995 to Jury 2002 [5] d the NIFTY dex from 0 Jury 998 to 03 December 200 [6]. For both the dces, we dvded the etre dt to lmost two equl prts. No specl rules were used to select the trg set other th esurg resoble represetto of the prmeter spce of the problem dom [2]. II. LOCAL LINEAR WAVELET NEURAL NETWORK I terms of wvelet trsformto theory, wvelets the followg form x b { 2 ( :, b R, Z} ( x ( x,x2,...,x (, 2,..., ( b,b,...,b b re fmly of fuctos geerted from oe sgle fucto (x by the operto of dlto d trslto. (x, /05/$ IEEE 646

2 whch s loclzed both the tme spce d the frequecy spce, s clled mother wvelet d the prmeters d b re med the scle d trslto prmeters, respectvely. The x represets puts to the WNN model. I the stdrd form of wvelet eurl etwor, the output of WNN s gve by x b f(x 2 (x ( (2 where the ` s the wvelet ctvto fucto of th ut of s the weght coectg the th ut of the hdde lyer to the output lyer ut. Note tht for the -dmesol put spce, the multvrte wvelet bss fucto c be clculted by the tesor product of sgle wvelet bss fuctos s follows (x (x (3 Obvously, the loclzto of the th uts of the hdde lyer s determed by the scle prmeter d the b trslto prmeter. Accordg to the prevous reserches, the two prmeters c ether be predetermed bsed upo the wvelet trsformto theory or be determed by trg lgorthm. Note tht the bove wvelet eurl etwor s d of bss fucto eurl etwor the sese of tht the wvelets cossts of the bss fuctos. Note tht trsc feture of the bss fucto etwors s the loclzed ctvto of the hdde lyer uts, so tht the coecto weghts ssocted wth the uts c be vewed s loclly ccurte pecewse costt models whose vldty for gve put s dcted by the ctvto fuctos. Compred to the multlyer perceptro eurl etwor, ths locl cpcty provdes some dvtges such s the lerg effcecy d the structure trsprecy. However, the problem of bss fucto etwors s lso led by t. Due to the crudeess of the locl pproxmto, lrge umber of bss fucto uts hve to be employed to pproxmte gve system. A shortcomg of the wvelet eurl etwor s tht for hgher dmesol problems my hdde lyer uts re eeded. I order to te dvtge of the locl cpcty of the wvelet bss fuctos whle ot hvg too my hdde uts, here we propose ltertve type of wvelet eurl etwor. The rchtecture of the proposed LLWNN [8] s show Fg.. Its output the output lyer s gve by y ( 0 x... x (x x b ( x... x 2 0 ( (4 where x [ x, x2,..., x ]. Isted of the strghtforwrd weght (pecewse costt model, ler model v x... x (5 0 s troduced. The ctvtes of the ler models v (,2,... re determed by the ssocted loclly ctve wvelet fuctos (x(,2,..., thus v s oly loclly sgfct. The motvtos for troducg the locl ler models to WNN re s follows: ( Locl ler models hve bee studed some eurofuzzy systems d show good performces [8], [9]; d (2 Locl ler models should provde more prsmoous terpolto hgh-dmeso spces whe modelg smples re sprse. The scle d trslto prmeters d locl ler model prmeters re rdomly tlzed t the begg d re optmzed by EDA dscussed the followg secto. Fg.. A locl ler wvelet eurl etwor III. LLWNN TRAINING Estmto of dstrbuto lgorthms (EDAs [] [2] [4] [6] [7] re ew clss of evolutory lgorthms. Le other evolutory lgorthms, EDAs mt d successvely mprove populto of potetl solutos utl some stoppg codto s met. However, EDAs do ot use crossover or mutto. Isted, they select the best solutos from the curret populto d explctly extrct globl sttstcl formto from the selected solutos. A 647

3 posteror probblty dstrbuto model of promsg solutos s bult, bsed o the extrcted formto. The ew solutos re smpled from the model thus bult d fully or prt replce solutos the curret populto. ore precsely, EDAs wor s follows: S0 Rdomly pc set of solutos to form the tl populto. S Select some solutos from the curret populto ccordg to selecto method. Buld the probblty model of the selected solutos. S2 Replce some or ll of the members of the curret populto by ew solutos smpled from the probblty model. S3 If the stoppg codto re ot met, go to Step. Severl EDAs hve bee proposed for solvg globl optmzto problems. I these exstg lgorthms, the probblty dstrbuto of the promsg solutos re modeled by Guss dstrbuto, Guss mxture or hstogrm. Sce my pots re eeded to buld good probblty model, these lgorthms re ofte very tme-cosumg prctce. Oe of the mjor ssues EDAs s how to select prets. A wdely used selecto method EDA s the tructo selecto. I the tructo selecto, dvduls re sorted ccordg to ther objectve fucto vlues. Oly the best dvduls re selected s prets. Aother mjor ssue EDAs s how to buld probblty dstrbuto model p(x. I EDAs for the globl cotuous optmzto problem, the probblstc model p(x c be Guss dstrbuto [3], Guss mxture [4][], hstogrm [5], or Guss model wth dgol covrce mtrx (G/DC [4]. G/DC s used our lgorthm. I G/DC, the jot desty fucto of the -th geerto s wrtte s follows: where p (x N(x;, x 2 N(x;, exp( ( (2 2 I (2, the -dmesol jot probblty dstrbuto s fctorzed s product of uvrte d depedet orml dstrbutos. There re two prmeters for ech vrble requred to be estmted the -th geerto: the me,, d the stdrd devto,. They c be estmted s follows: ˆ x x j (8 j (6 (7 ˆ (9 2 (x j x j j Before descrbg detls of the lgorthm for trg LLWNN, the ssue of codg s preseted. Codg cocers the wy the weghts, dlto d trslto prmeters of LLWNN re represeted by dvduls or prtcles. A flot pot codg scheme s dopted here. For LLWNN codg, suppose there re odes hdde lyer d put vrbles, the the totl umber of prmeters to be coded s (2++ * = (3+. The codg of LLWNN to dvdul or prtcle s s follows: b...b0... 2b2...2b b...b0... The smple loop of the proposed trg lgorthm for locl ler wvelet eurl etwor s s follows. S Itl populto s geerted rdomly. S2 Prmeter optmzto wth EDA; S3 If the stsfctory soluto s foud or mxmum umber of geertos s reched the stop; otherwse goto step S2. TABLE I THE RSE RESULTS OF LLWNN AND WNN ODELS FOR TEST DATA SETS Idex LLWNN WNN Nsdq NIFTY TABLE II STATISTICAL ANALYSIS OF THE LEARNING ETHODS (TEST DATA LLWNN WNN Nsdq-00 CC AP APE NIFTY CC AP APE IV. EXPERIENTS We cosdered 7-yer stoc dt for the Nsdq-00 Idex d 4-yer for the NIFTY dex. Our trget s to develop effcet forecst models tht could predct the dex vlue of the followg trde dy bsed o the opeg, closg d mxmum vlues of the sme o gve dy. The ssessmet of the predcto performce of the dfferet esemble prdgms were doe by qutfyg the 648

4 predcto obted o depedet dt set. The Root e Squred Error (RSE, xmum Absolute Percetge Error (AP d e Absolute Percetge Error (APE d Correlto Coeffcet (CC were used to study the performce of the tred forecstg model for the test dt. AP s defed s follows: Pctul, Ppredcted, AP mx( 00 (0 Ppredcted, where s the ctul dex vlue o dy d s the forecst vlue of the dex o tht dy. Smlrly APE s gve s N Pctul, Ppredcted, APE ( 00 ( N Ppredcted, where N represets the totl umber of dys. We used LLWNN wth rchtecture {3-8-} for modelg the Nsdq-00 dex d LLWNN wth rchtecture {5-8-} for modelg the NIFTY dex. For comprso purpose, two WNNs tred by EDA re lso employed to predct the sme stoc dces. Tble summrzes the test results cheved for the two stoc dces usg the proposed pproch. Performce lyss of the tred forecstg models for the test dt ws show Tble 2. Fgures 2 d 3 depct the test results for the oe dy hed predcto of the Nsdq 00 dex d the NIFTY dex respectvely. V. CONCLUSIONS I ths pper, we hve demostrted how the chotc behvor of stoc dces could be well represeted by locl ler wvelet eurl etwors. Emprcl results o the two dt sets usg LLWNN models clerly revel the effcecy of the proposed techques. I terms of RSE vlues, for the Nsdq-00 dex d the NIFTY dex, LLWNN performed mrglly better th other models. For both dex (test dt, LLWNN lso hs the hghest correlto coeffcet d the lowest vlue of APE d AP vlues. A low AP vlue s crucl dctor for evlutg the stblty of mret uder uforesee fluctutos. I the preset exmple, the predctblty ssures the fct tht the decrese trde s oly temporry cyclc vrto tht s perfectly uder cotrol. Our reserch ws to predct the shre prce for the followg trde dy bsed o the opeg, closg d mxmum vlues of the sme o gve dy. Our expermet results dcte tht the most promet prmeters tht ffect shre prces re ther mmedte opeg d closg vlues. The fluctutos the shre mret re chotc the sese tht they hevly deped o the vlues of ther mmedte forerug fluctutos. Log-term treds exst, but re slow vrtos d ths formto s useful for log-term vestmet strteges. Our study focus o short term, o floor trdes, whch the rs s hgher. However, the results of our study show tht eve the seemgly rdom fluctutos, there s uderlyg determstc feture tht s drectly ecphered the opeg, closg d mxmum vlues of the dex of y dy mg predctblty possble. ACKNOWLEDGENT Ths reserch ws prtlly supported by the Ntol Hgh Techology Developmet Progrm of Ch (863 Progrm uder cotrct umber 2002AA4Z3240, d The Provcl Scece d Techology Developmet Progrm of Shdog uder cotrct umber SDSP Fg. 2. Test results showg the performce of the LLWNN for modelg the Nsdq-00 dex Fg. 3. Test results showg the performce of the LLWNN for modelg the NIFTY dex REFERENCES [] Abrhm A., Nth B. d ht P.K., Hybrd Itellget Systems for Stoc ret Alyss, Computtol Scece, Sprger-Verlg Germy, Vssl N Alexdrov et l (Edtors, USA, ( [2] Abrhm A., Phlp N.S., d Srtchdr P., odelg Chotc Behvor of Stoc Idces Usg Itellget Prdgms, Itertol Jourl of Neurl, Prllel d Scetfc Computtos, USA, Volume, Issue (,2, ( [3] Legh W., od N., Purvs R. d Roberts T., Stoc mret trdg rule dscovery usg techcl chrtg heurstcs, Expert Systems wth Applctos 23(2, ( [4] Legh W., Purvs R. d Rgus J.., Forecstg the NYSE composte dex wth techcl lyss, ptter recogzer, eurl etwor, d geetc lgorthm: cse study romtc decso support, Decso Support Systems 32(4,( [5] Nsdq Stoc rets, [6] Ntol Stoc Exchge of Id Lmted, [7] Che Y. d A. Abrhm, Hybrd Lerg ethods for Stoc Idex 649

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