Time Series Perturbation by Genetic Programming

Transcription

1 Tme Sees Petbaton by Genetc Pogammng G. Y. Lee Assstant Pofess Deatment of Comte and Infomaton Engneeng Yong-San Unvesty San 5 -Nam-R Ung-Sang-E Yang-San-Sh Kyng-Nam Soth Koea [email protected] Abstact- Ths ae esents a new algothm that combnes etbaton theoy and genetc ogammng fo modelng and foecastng eal-wold chaotc tme sees. Both etbaton theoy and tme sees modelng have to bld symbolc models fo vey comlex system dynamcs. Petbaton theoy does not wok wthot well-defned system eqaton. Dffcltes n modelng tme sees le n the fact that we can t have o assme any system eqaton. The new algothm shows how genetc ogammng can be combned wth etbaton theoy fo tme sees modelng. Detaled dscssons on sccessfl alcatons to chaotc tme sees fom actcally motant felds of scence and engneeng ae gven. Comtatonal esoces wee neglgble as comaed wth eale smla egesson stdes based on genetc ogammng. Deskto PC ovdes sffcent comtng owe to make the new algothm vey sefl fo eal-wold chaotc tme sees. Esecally t woked vey well fo detemnstc o statonay tme sees whle stochastc o nonstatonay tme sees needed extended effot as t shold be. Intodcton Tme sees s a scala seqence of nmecal data x x x.... Tme sees modelng sally stats fom geneatng vecto tme sees x x x.... An aoately aanged set of the scala tme sees data x... x x x x w constttes a vecto. The t n τ t τ t τ t sbsct t stands fo cent tme τ fo delay tme (also called lag tme o lag sacng) w = t + T fo foecast tme T fo fte tme (also called lead tme o edcton hozon) and n fo embeddng dmenson of the Ecldean state sace whee each vecto s a ont. Detemnaton of these state sace aametes fo a gven tme sees s ctcal fo good model. Howeve t s not the scoe of ths ae. See Wegend 993 fo vaos data chaactezaton technqes. Tme sees modelng s to fnd the fnctonal ~ aoxmaton f to f n Eq. () that elates x w wth emanng comonents of vecto. The aoxmaton eo e shold be mnmal fo all vectos x x x... x = f x ~ f x + e = 3... w ( t ) ( t ) () t ( ) () t n τ t τ t τ t n x x... x x x R () As ARMA (Box 994) and many othe technqes sch as GMDH (Ivakhnenko 97) do genetc ogammng aled to symbolc egesson (Koza 994) tes to fnd exlct model that s wtten n mathematcal symbols. It has acheved nteestng efomance fo statonay scentfc tme sees (Koza 994 Oakeley 994 Iba 994). Bt eal-wold tme sees s vey chaotc and sally hghly nonstatonay (Box 994). Also nose makes t dffclt to have actcally sefl model. Exensve comtatons eslted n only modeately efomng models fo them. The atho noted that tme sees modelng have some smlates wth etbaton theoy of qantm mechancs (Rae 99). They need sefl aoaches fo fomlatng nonstatonay o stochastc system dynamcs. Of cose the system behavos ae eesented dffeently. Petbaton theoy assmes well-defned system behavo e.g. the wave eqaton (Rae 99 Nayeh 993) whle tme sees modelng shold wok wthot sch knd of eqatons. Note that symbolc egesson based on genetc ogammng (Koza 994) ovdes evoltonay ways to do the essentally same woks.e. fomlaton of comlex system dynamcs as etbaton theoy does even wthot exlct system eqaton. The algothm esented hee takes seveal vewonts to model comlex system dynamcs fom etbaton theoy. And genetc ogammng s eqested to lay the ole of ll-defned system eqaton fo chaotc tme sees. Secton fomlates the new algothm followed by Secton 3 to show alcaton examles to many eal-wold chaotc tme sees. Secton 4 concldes ths ae and lsts tocs fo fthe stdy. Tme Sees Petbaton Algothm. Develoment The nheently nonstatonay dynamcs of the wave eqaton does not allow fo exact solton. Petbaton theoy (Nayeh 993) nvolves two tyes of Hamltonans fo sch eqaton. Now let Ψ ( x ) be an netbed Hamltonan and Ψ ( x ) be a etbed Hamltonan. Then Φ( x) = Ψ ( x) + Ψ ( x ) ()

2 meets the system eqaton.e. the wave eqaton. Eqaton () states that nonstatonay system behavo can be descbed by lnea combnaton of Hamltonans. The same thng holds agan fo the etbed Hamltonan. Ths means that the etbed Hamltonan Ψ ( x ) can be exanded as anothe lnea combnaton of dffeent Hamltonans and the exanson eeats fo all newly avalable seqence of etbed Hamltonans. Petbaton technqe ovde systematc ways to fnd the netbed Hamltonans that ae ntegated nto a fnal solton to the comlex system eqaton. If Eq. () meets the system eqaton any lnea combnaton of the Hamltonans also meets the system eqaton. Note that the tme sees cay system dynamcs fo soce system as the wave eqaton does fo a qantm system. Then we beleve that the same ocede fo obtanng soltons to qantm system can be aled to tme sees modelng. Gven below s the detaled analogcal develoment of the tme sees modelng algothm based on the etbaton theoy. Now the tme sees model o the fncton f n Eq. () can be exessed as a sm of the netbed and etbed Hamltonans on the analogy of Eq. (). That s f x = f x + f x (3) Eqaton (3) can be ewtten wth aoate constants a and b f( x) = af( x) + b+ f ( x) (4) snce f f ( x) and f ( x) meets the system eqaton (= system dynamcs caed by tme sees) the lnea combnaton also meets the eqaton as Hamltonans do fo the wave eqaton. The constant b s a knd of lage nmbe sch as the aveage. Seeng dffeently b s a detemnstc vale fo evey datm n the seqence of tme sees. Eqaton (4) states that tme sees consst of the netbed and the etbed tme sees. The netbed tme sees conssts of the constant and the vaable exessed by the exlct fncton f ( x). The etbed tme sees s gven by f( x) af ( x) b = f ( x) (5) Hee note that the etbed tme sees can be easly calclated f we sbstact the nmecal vales etned by f x lnea combnaton of the netbed tme sees fom the ognal tme sees f( x ). Eqaton (5) f ( x) eesents stll anothe tme sees to model. We can oceed to model t agan wth the same algothm fo the ognal tme sees. In geneal we wll have n theoy a seqence of the netbed tme sees j f x whee j stands fo j-th modelng ocede. In the long n the geneal fom of tme sees etbaton model becomes j j f( x) = a f ( x) + f ( x) whee the -th netbed tme sees f ( x) = and f x s neglgble. Now note that o oblem s edced to how we get the sees of the netbed tme sees models. As yo may aleady know t s by GP (Genetc Pogammng) see next secton.. Imlementatons Eveytme we aly the classcal GP-based symbolc egesson (Koza 994) to the seqence of the netbed tme sees we can get the fnctonal foms fo f ( x) f ( x)... f ( x) n Eq.(6). Ths ae sggests that the constants j =... ae detemned by the least sqae egesson that makes f x neglgble. a j Next seveal sbsectons descbe some geneal and secal mlementaton detals ths ae s based on. Fo moe classc technqes abot GP-based symbolc egesson see (Koza 994). Geneal Imlementaton Isses ) Constcton Selecton and Integaton of Models Fo tme sees modelng based on genetc ogammng olaton of symbolc foms s sbject to genetc evolton. Intally stctes and contents of the symbolc foms ae detemned at andom. Genetc evolton altes the stctes and contents of symbolc foms sch that they can cate the tme sees dynamcs. Usally the best symbolc fom n a olaton s taken as a model fo the tme sees. As a coollay we ae exected to have P models f we se P olatons n the evolton. Then a qeston ases. What shold we select one among the mltle models to eesent the tme sees dynamcs? A commonsense selecton may be the one that odces the mnmm aoxmaton eo fo all tme sees vectos See Eq. (). Howeve t s vey had to bld a efect model wthn actcal lmt on comtatonal esoces. The fact s that even the best one selected among P mltle models mst be only atally sccessfl n catng the tme sees dynamcs. And moeove the othe P nselected models do cate atal dynamcs of the tme sees so that we can save comtatonal esoces f we can ese them n some ways. We may easonably ntegate the atally sccessfl models. Paagahs below exlans how. g x be the model constcted n Now let olaton. Then the j-th netbed tme sees model j f x has n ths ae the fom (6)

3 P j f ( x) = α + α g ( x) = whee the nmecal coeffcents fo each g ( x) calclated by the least sqae egesson wth esect to tanng data set of the tme sees. Note that Eq. (6) can be ewtten. Inset Eq. (7) nto Eq. (6). We get P j j j f( x) = a f ( x) f ( x) a g ( x) f ( x) + = α + α + = whch becomes P j j j j j f x = a α + a α g x + f x (8) = Changng notatons fo the nmecal coeffcents Eq. (8) can be ewtten as P f( x) = β + β g ( x) + f ( x) (9) K = Fom (3) and (9) we can see that the netbed tme sees model to -th modelng ocede see Secton. takes the fom P f x = β + β g x () ) Inteetaton of the Petbaton Modelng by GP We ae now at a ont to clafy oveall ocede of the tme sees etbaton modelng ocede based on genetc ogammng. Fst the vecto tme sees Eq. () ae dvded nto thee data egons fo tanng valdaton and foecastng. Fo the fst modelng ocede genetc ogammng s n wth esect to the tanng egon and we have the netbed tme sees model as exessed by Eq. () =. Once the netbed tme sees model s detemned the j-th model bldng ocede s fomally ove. Ideally the model wold cate the tme sees dynamcs and theefoe can foecast the tme sees beyond the tanng egon. Bt o model wold fal only afte a few accetable foecasts de to seveal actcal easons. Fo examle the tanng egon may not be sffcent to ovde nfomaton needed to cate the dynamcs beyond t. Even f the tanng egon contans sffcent nfomaton comtatonal esoces may fall shot of what s necessay to make fll se of the nfomaton. Thee may be seveal ways to kee the model efomance as hgh as ossble beyond the tanng egon. The most staghtfowad one wold be to e-constct the model wth esect to newly defned tanng egon that ncldes the latest avalable te tme sees data. Bt the new model shold be constcted n tme to become sefl foecaste fo the tme sees. In ths ae the nmecal coeffcents n the lnea combnaton of the models ae smly dated wth esect to the data egon that s close to foecast ont. Fo examle assme that we have data n the tanng egon. Then the st datm s foecasted wth the nmecal coeffcents K K K K K = j (7) ae calclated wth esect to data between st and th ostons of the tme sees. The model s gven by Eq. (6) f j = and by Eq. () f j s geate than. Fo the nd datm the nmecal coeffcents ae dated wth esect to data between nd and st ostons. We can oceed fthe only f the te tme sees datm at oston v s known befoe we ty to foecast the oston f = v +. The vale S = f v s temed as the mact ste n ths ae. The data egon that comes afte the tanng egon s called the valdaton egon. The model wth changng coeffcents s sed to foecast the data n the valdaton egon. Afte foecastng the last datm n the valdaton egon the foecastng efomance e.g. Eq. () Eq. () s ecoded. The foecastng efomance n the valdaton egon s an motant cteon to detemne f we sto o oceed to anothe etbaton modelng ocede by genetc ogammng. Fo ths eason we call t the valdaton efomance. Anothe modelng ocede stats f the valdaton efomance got moved as comaed wth fome modelng ocede. By defalt the fst modelng ocede s followed by the second modelng ocede. Othewse the etbaton modelng ocede stos on the assmton that the model has stated to cate sos o excessve etbaton sch as nose o dstbance. Ths tye of cteon to temnate leanng o modelng algothm was sed n the feld of atfcal neal netwok (Geman 99) and called ealy stong olcy. Secton 4 smmazes above nteetaton of the oosed GP-based tme sees etbaton modelng ocede. Seveal Imlementaton Isses ) Pefomance of ndvdal model A model o any ndvdal n a olaton shold be gven nmecal efomance vale that meases how well the tme sees s aoxmated o smlated by the symbolc foms eesented by the ndvdal o the model. Pola efomance vale s the nomalzed mean sqaed eo (Wegend 993) and the coeffcent of vaaton (Iba 994). They ae defned by N () () CV( N) = ( x x ) x ~ N = N () () ( x ~ x ) = NMSE( N) = N () x x = = N () () ( x ~ x ) N ˆ σ 5. () MSE( N) = ˆ σ () whee ~x and x ae model evalaton and te datm at the oston. x and $σ denote the samle aveage and samle vaance of the tme sees. N s the nmbe of data ove whch NMSE( N ) o CV(N ) s calclated.

4 MSE stands fo Mean Sqaed Eo. Model ftness of an NMSE N o ndvdal s gven by the nvese of CV( N ) n ths ae. ) Se olaton and Mgaton Genetc ogammng desgned fo tme sees etbaton modelng n ths ae has a secal-ose olaton called the se olaton of whch sole sevce s to select and kee the best ndvdals g n Eq. (7) ( x) fom each of mltle olatons. The best ndvdal fom a olaton s allowed to elace a se olaton membe at the end of each geneaton f and only f ts ftness excels that of the membe beng elaced. Mgaton does occ between the mltle olatons that ndego evolton. Only those membes n the se olaton that svved the whole geneatons become the g x n Eq. (7). The se olaton s fnal model dffeent fom the mlt-agent team (Lke 996) whee agents o team membes ae somehow engaged n the evoltonay ocesses. 3) Thee Symbolc Anomaltes Symbolc stctes and contents that can not be tanslated nto mathematcally vald eqatons defne mathematcal anomalty. Dvson-by-zeo negatve agments gven to sqae oot ae tycal examles of the mathematcal abnomalty. Symbolc stctes and contents that case comte softwae eo sch as the oveflow o ndeflow defne comtatonal anomalty. The last symbolc anomalty s the semantc elcaton. When mltle symbolc foms ae mathematcally eqvalent they ae semantc elcatons of each othe. Fo examle (+ x3 (sn (/ x x))) s the semantc elcaton of (+ x3 (sn )) = x3. Esecally semantc elcaton between the best ndvdal fom a olaton and the se olaton membe shold be avoded at the end of each geneaton and each modelng ocede. If not mathematcal eo of sngla matx occs dng calclaton of the nmecal coeffcents n Eq. (6) o Eq. () sng the least sqae egesson. Any tye of symbolc anomaltes consmes comtatonal esoces becase they blemsh genetc dvesty n a olaton. Also they case nexected eo n the ogam n. Thee shold be systematc technqes to detect and ea the cases of symbolc anomaltes. If any ndvdal n a olaton has symbolc anomalty ts ftness s abtaly assgned a vey small vale n ths ae to edce the chance of beedng offsng n the next geneaton. 4) Deved Temnal Set Fncton set and temnal set fo egesson oblem based on genetc ogammng ae vey motant becase they ae basc symbols to eesent tme sees datm n exlct fom. Temnal set ovdes agment symbols fo fncton symbols. Fo examle the symbolc fom sn x 3 has the agment symbol fo the fncton symbol sn. The concet of deved temnal set (DTS) s ntodced n ths ae fo the ose of savng comtatonal esoces fo ntalzng and ocessng vaos knds of mtve fnctons sch as ( sn x 3 ) (cos x 3 ). DTS s a collecton of symbols that eesent mtve fnctons. DTS s sed along wth the temnal set and the fncton set fo ceatng the ntal olaton fo genetc ogammng. Desable chaactestcs of DTS nclde the caablty of aoxmatng a mathematcal fncton when lnealy combned. Tme sees modelng s n a sense an aoxmaton of the nknown fncton f n Eq. (). Othogonal fnctons (Sansone 99) satsfy these condtons. In addton to tgonometc fnctons we made DTS by alyng Tschebyshev fncton to elements of x t n Eq. (). The Tschebyshev temnal s gven by T ode ** [ ode accos( )] x 3 T ode cos x x ** x (3) whee ode s an ntege the doble astesks n ndcate that x shold be aoately adjsted to be n the nteval [- ]. The followng lnea mang s sed. ** MAX MIN x = sx t s= ( x x ) MAX MIN MIN t = + x x x x ** (4) MAX MIN In Eq. (4) x and x ae the global maxmm and max mn mnmm. Now let x and x be local maxmm and mnmm obsevable n the tanng egon. The global MAX MIN x and x ae estmated by ntodcng abtay exanson ato η. Eqaton (5) assme that the global nteval s η + tmes boade than the local nteval obsevable n the tanng egon. MAX max MIN mn x = x + η x = x η = x max x mn 3 Alcatons (5) 3. Hman Body Blood Flow Dynamcs Tme sees data obtaned by solvng the Mackey Glass eqaton have been sed by seveal woks (Oakeley 994 Iba 994 Casdagl 989). The eqaton smlates the nonlnea dynamcs of hman blood flow and s gven by dxt bxt = ax c t dt + x t c ( ) ( ) xt+ = a xt+ bxt + xt (6) Wth aoately assgned constant vales a b and the dffeence eqaton n Eq. (6) s sed to geneate vey chaotc tme sees fom the ntal andom seeds of

5 edetemned sze abot 4. Table below comaes eslts of ths stdy wth those of eale woks. Foecatng Eale Woks Pefomance Casdagl 993 Iba 994 Ths Stdy NMSE() NMSE(3).59.9 NMSE(4) NMSE(5) NMSE(6) Tme sees etbaton algothm otefoms the eale woks based on genetc ogammng. See secton 7.5. and secton Lee 999 fo fthe detals. 3. Santa Fe and ASHRAE Tme Sees Comettons Com- ASHRAE Santa Fe Tme Pefomances n each Regon T V F (Foecastng) Modelng Wth DTS Wth DTS No DTS Sn.5... Enegy Lase Lase Heat >> Heat Infnte C C Infnte Pat Pat Infnte Sta Sta Santa Fe Insttte and ASHRAE (Amecan Socety fo Heatng Refgeatng and A-condtonng Engnees) held woldwde comettons fo tme sees analyss and foecast. Tme sees was chosen fom vey statonay to hghly volatle system. See Wegend 993 fo moe detals on the categozaton and chaactezaton of the tme sees data fo the cometton. Ths ae has fxed comtatonal aametes fo them to the followngs: Nmbe of olatons = 5 Polaton sze = 3 Geneaton Lmt = 9 Maxmm nmbe of etbaton modelng allowed = 5 Fncton set = { + / sn cos ex log ext } Temnal set = { xt ( n )... x } τ t τ xt τ x U t { Tode ode = ~ fo each x } Deth of egesson tees = Intal 6 afte-cossove 8 Cossove facton =.8 Mtaton facton =. Reodcton facton =. Maxmm nmbe of mgatng ndvdals allowed = % of total ndvdals Lag sacng = Lead tme = Embeddng dmenson = o 4 Imact ste = Total nmbe of tme sees vecto = 4 Tanng egon T = fst data Valdaton egon V = next data afte egon T Foecast egon F = the last data afte egon V Temnatng NMSE =.. Table below smmazes the modelng and foecastng efomances of the etbaton modelng exloed n ths stdy. Symbols fo each tme sees ae sed to save sace hee and they ae : Sn = Tme sees fo the te sola beam solaton flx Enegy = Enegy consmton ate n a bldng Lase = Intensty flctaton of NH3 lase Heat = Heat ate of a hman atent C. = Cency exchange ate fo Swss fanc vs. US dollo Pat. = Qantm atcle oston n 4D otental well and Sta = Sface bghtness of a whte dwaf sta PG95. See Wegend 993. Fo Santa Fe cometton sees nmecal vales added to the symbols ae sed to eesent the embeddng dmensons. Note that Sn Lase and Sta ae all fom hyscs systems that have statonay system dynamcs. These tme sees ae categozed nto the statonay o detemnstc tme sees (Wegend 993 Box 994). Statonay o detemnstc tme sees s well modeled and foecasted by the etbaton modelng based on genetc ogammng. The temnatng NMSE fo foecastng was acheved befoe the maxmm nmbe of etbaton modelng an ot. Embeddng dmenson n lag sacng τ and lead tme T ae smlstcally assmed and ket fxed n ths stdy. Two dffeent vales of the embeddng dmenson n = and 4 ae chosen only to see f the embeddng dmenson woks dffeently fo statonay and nonstatonay tme sees. Petbaton modelng woked mch bette fo detemnstc tme sees when we se nceased embeddng dmenson. Of cose thee mst be a lmt on the embeddng dmenson ove whch model efomance deteoates. On the othe hand the moe a tme sees s nonstatonay the moe nstable and egla dynamcs t wll have. At the exteme tme sees dynamcs may be e andom. So the nceased embeddng dmenson fo nonstatonay tme sees mght ntodce nceased andomness n dynamcs that s had to cate wthn actcal small comtatonal esoce lmt. Ths ont of vew exlans the dffeent efomances n the above table between the statonay and the othe nonstatonay tme sees. The foegong table also eveals that DTS has mnmal effects on the efomances fo statonay tme sees whle t does contbte to move efomances fo nonstatonay tme sees. Intodcton of DTS saves comtatonal esoces to geneate evalate and ocess mtve fnctons. The algothm fo tme sees etbaton modelng based on genetc ogammng ses the date extenson (Smth 993) of tme sees beyond the tanng egon. In the date extenson we mst know te vales of tme sees contnaton that s S the mact ste oston behnd the foecast oston. The state sace (Kalath 996)

6 aametes wee smlstcally assmed fo all afoementoned tables. Analyss to detemne sch aametes s not the scoe of ths ae. Moeove extemely small qantty of tanng data makes vey coasely constcted state sace. The cometton contestants wee eqed to se the naway extenson (Wegend 993 Smth 993) that do not ely on the te vales to make foecastng extenson of tme sees and t s best sccessfl when mass amont of data s sed to constct vey dense state sace. Fom the actcal onts of vew each method of foecastng extenson has the long and the shot. Rnaway extenson eqes exensve modelng bdens bt the foecastng s easy wth establshed self-extendng models. It s benefcal to se the naway extenson when comtatonal tme s lmted fo tmely foecast bt thee s a lot of stoed data. On the othe hand the date extenson s good when we have lttle data bt good comtng owe. Wth moden hgh-seed comte t wll be moe actcal to se date extenson. 3.3 Majo US Economc Tme Sees Economc tme sees consttte a vey dffclt bt hghly motant categoy of tme sees. Bt most shot-tem economc tme sees ae extemely nonstatonay whch has cased eve-contnng dstes on the exstence of any ode o chaotc stcte we can smlate o model. See Dechet 996 fo elated dscsson. Tme sees etbaton modelng based on genetc ogammng was tested wth esect to 8 majo US economc tme sees data avalable fom htt:// com. Embeddng dmensons ae all fxed to based on the exeence fom Santa Fe cometton that nceased embeddng dmenson may be sky fo nonstatonay economc tme sees wth lmted comtatonal esoces. Comtatonal aametes wee same as those fo ASHRAE and Santa Fe comettons. Hee the concet of e-modelng s ntodced. Pe-modelng s the alcaton of the algothm wth vey small data sze.e. tanng data fo the ose of gasng data chaactestcs. Pemodelng s based on the exeence fom Santa Fe cometton. That s the moe statonay a tme sees s the hghe the model efomance wold be. Fedeal fnd ates FFR aanese yen to US dolla cency exchange ate YENDOL and 3-yea Teasy Constant Matty 3YTCM wee elatvely dffclt to foecast. They ae elatvely moe nonstatonay and need nceased comtatonal esoces. Table below shows how the foecastng efomances fo FFR move wth nceasng sze of tanng data. All othe aametes wee fxed. NT stands fo NMSE n the tanng (= modelng) egon and NF fo the foecastng egon. e. data afte the end of NT. Data τ = Month τ =6 Month τ = Month Sze NT NF NT NF NT NF Evolton wth nceased nmbe of tanng data. e. hgh densty state sace eslts n bette modelng and foecastng efomances. The moe dense the state sace s the moe system dynamcs can be cated (Smth 993). 4 Conclson Ths ae esented a new algothm that combnes etbaton theoy (Rae 99 Nayeh 993) wth genetc ogammng. Petbaton theoy ovdes effcent ways to get solton to comlex system eqaton that sally does not allow fo exact solton. Genetc ogammng ovdes evoltonay ocesses to get symbolc foms that model tme sees dynamcs. In the tme sees etbaton algothm the seqence of tme sees lays the ole of the system eqaton n etbaton theoy. Symbolc models obtaned by genetc ogammng lays the ole of Hamltonans n etbaton theoy. A seqence of the netbed tme sees models s obtaned fo a tme sees mch lke the netbed Hamltonans wee obtaned fo a wave eqaton by etbaton technqes. They ae lnealy combned wth nmecal coeffcents calclated wth esect to gven tme sees. Foecastng beyond the tanng egon s efomed based on the date extenson whch eqes nmecal coeffcents dated by the least sqae egesson wth esect to the latest data. The date extenson of foecast data and the ntodcton of DTS saved comtaton esoces. Statonay tme sees ae moe easly modeled and foecasted. The algothm was sccessflly aled to many ealwold chaotc tme sees coveng hyscs to economc ones. Notceable efomance was acheved even wth the smlstcally assmed vales of state sace aametes and the lmted comtatonal esoce. The algothm shold be coled wth tme sees chaactezaton technqes to get otmzed set of the state sace aametes f any. In ths ae a e-modelng o the n of the algothm wth edced sze of tanng data s sggested to classfy data chaactestcs. Tme sees wth good foecastng efomances n the e-modelng wee assmed to be statonay based on the alcaton exeence fo Santa Fe cometton. The new algothm odced consstent eslts on the tme sees data chaactestcs wth the analyses by othe technqes (Wegend 993). Effects of vaos genetc ogammng aametes on the modelng efomance shold be stded fthe.

7 Bblogahy Wegend A. S. and Geshenfeld N. A. Eds (993) Tme Sees Pedcton Foecastng the fte and Undestandng the Past SFI Stdes n the Scence of Comlexty Vol. XV Addson Wesley Pblshng Co. Box G. E. P. enkns G. M. and Rensel G. C. (994) Tme Sees Analyss 3d ed. Englewood Clffs N: Pentce Hall 994. Ivakhnenko A. G. (97) Polynomal Theoy of Comlex Systems IEEE Tans. Syst. Man Cyben. vol. (4) Koza. R. (994) Genetc Pogammng II MIT Pess. Oakeley H. (994) Two Scentfc Alcaton of Genetc Pogammng: Stack Fltes and Non-Lnea Eqaton Fttng to Chaotc Data n Advances n Genetc Pogammng K. E. Knnea. Eds. MIT Pess Iba H. de Gas H. and Sato T. (994) Genetc Pogammng sng a Mnmm Descton Length Pncle n Advances n Genetc Pogammng K. E. Knnea. Eds. MIT Pess Rae A. I. M. (99) Qantm Mechancs 3 d ed. Unvesty of Bmngham UK IOP Pblshng Ltd. Geman S. et al. (99) Neal Netwoks and the Bas / Vaance Dlemma Neal Comtaton vol Lke S. (996) and L. Secto Evolvng Teamwok and Coodnaton wth Genetc Pogammng n Genetc Pogammng 996: Poceedngs of the Fst Annal Confeence MIT Pess Sansone G. Sansome G. and Damond A. H. (99) Othogonal Fnctons Dove Pblcatons. Casdagl M. C. (989) Nonlnea edcton of chaotc tme sees Physcs D Lee G. Y. (999) Genetc Recsve Regesson fo modelng and foecastng eal-wold chaotc tme sees n Advances n Genetc Pogammng vol. 3. MIT Pess Chate 7. Kede. F. (993) eslts.asc ASHRAE Cometton ft ste ft.cs.coloado.ed/b/enegy-shootot 993. Smth L. A. (993) Does a meetng n Santa Fe mly chaos? n Tme Sees Pedcton Foecastng the Fte and Undestandng the Past A. S. Wegend and N. A. Geshenfeld Eds. SFI Stdes n the Scence of Comlexty vol. XV. Addson-Wesley Pblshng Co Kalath T. (98) Lnea Systems Englewood Clffs N: Pentce Hall. Dechet W. D. (996) Chaos Theoy n Economcs : Methods Models and Evdence Intenatonal Lbay of Ctcal Wtngs n Economcs No. 66. Edwad Elga Pb. Nayeh A. H. (993) and A. H. Nayfeh Intodcton to Petbaton Technqes ohn Wley & Sons.