Study on Financial Time Series Prediction Based on Phase Space Reconstruction and Support Vector Machine (SVM)

Transcription

1 Amerca Joural of Appled athematcs 205; 3(3): 2-7 Publshed ole Aprl 27, 205 ( do: 0.648/j.ajam ISSN: (Prt); ISSN: X (Ole) Study o Facal Tme Seres Predcto Based o Phase Space Recostructo ad Support Vector ache (SV) Hog Zhag, L Zhou, Je Zhu School of Iformato, Bejg Wuz Uversty, Bejg, Cha Emal address @qq.com (Hog Zhag) To cte ths artcle: Hog Zhag, L Zhou, Je Zhu. Study o Facal Tme Seres Predcto Based o Phase Space Recostructo ad Support Vector ache (SV). Amerca Joural of Appled athematcs. Vol. 3, No. 3, 205, pp do: 0.648/j.ajam Abstract: Aalyzg ad forecastg the facal market based o the theory of phase space recostructo of support vector regresso. The key pot of the phase space recostructo s to choose the optmal delay tme, ad to fd the optmal embeddg dmeso of space. Ths paper proposes the use of false earest eghbor method to costruct the error fucto for all the varables to determe the approprate embeddg dmeso combatos. Kerel fucto the SVR s a mportat factor for algorthm performace. Expermets show that the theory of phase space recostructo based o support vector regresso has a certa degree of predctve ablty of market value at rsk. Keywords: Phase Space Recostructo Theory, Support Vector Regresso, Facal Tme Seres Predcto. Itroducto I geeral, Tme Seres model used aalyzg ad forecastg facal markets ca be dvded to sgle varable model ad multvarate model accordg to the selecto of varables, accordg to the structure of the model ad ca be dvded to lear ad olear model. It s ow wdely used ow that lear method such as expoetal smoothg, movg average, movg average (ARIA), multple regresso ad autoregressve codtoal heteroscedastcty. Evaluato of a Tme Seres model s eeded to study the fttg valdty ad forecastg effectveess, A good model must ot oly be able to have a good uderstadg of pheomea that occur wth the Tme Seres, fdg the teral law, but also to aalyze the extet ad scope of the mpact ad the mpact of exteral factors, ad also eed to be extrapolated by model,correctly predctg the tred of Tme Seres. Nolear predcto method based o phase-space recostructo theory [-3] s a ew forecastg method developed the last 20 years. The method appled to facal Tme Seres wth olear ad o-statoary characterstcs ad ucertaty, t ca accurately predct the short-term behavor of facal markets. Based o the prcple of Takes embedded, f the selecto of embeddg dmeso ad delay tme approprate, sgle varable Tme Seres ca obta more deal predcto effect. Because practcal applcato,there always exst ose tme seres data lmted legth. It s dffcult to esure the uvarate Tme Seres cotaed suffcet formato to recostruct the power system. To resolve ths ssue, Cao ad others propose method of phase space recostructo of multvarate Tme Seres. The method of phase space recostructo of chaotc model ad support vector mache recet years the feld of artfcal tellgece s very popular, olear sequece ca be mapped to a hgh dmesoal space, the formato of olear dyamc characterstcs the sequece ca be revealed. Therefore, usg the theory of phase space recostructo of chaotc theory determe the trag samples. Fally, applyg support vector regresso to model the trag samples, realze the predcto of facal tme seres. 2. The Phase Space Recostructo of ultvarate Tme Seres Assumg that a system wth measurable varables, Tme Seres correspodg to each varable s z = ( z, z2,..., zn), =,2,..., N, Based o the phase space recostructo theory,t ca be recostructed as follows. V = ( z, z,..., z ) () r ( d ) r

2 Amerca Joural of Appled athematcs 205; 3(3): Amog them, the τ, d respectvely s tme delay of Tme Seres ad embeddg dmeso. = ( d ) τ +.If d or every d s large eough, d = d + d d > 2D, D s the attractor dmeso. Whe = for sgle varable Tme Seres, t s a specal case of multvarate Tme Seres. Auto-correlato fucto ad the mutual formato method s usually used to selected delay tme respectvely for every sgle varable Tme Seres that makes up multvarable Tme Seres. Calculated as follows: I ( τ ) P ( x x ( τ )) P ( x( ), x ( + τ )) ( ( )) P ( x ( + τ )) N = ( ), + log (2) = P x I ( τ ) 0, P ( ) s the probablty ( ) I τ represets the formato that a sequece s obtaed from aother sequece. Fraer recommeds that we could regard τ whch s reached I τ as the embeddg the frst mmum correspod to the ( ) tme delay. 2.. To Determe the Embeddg Dmeso Aother ssue of cocer s how to select approprate embeddg dmeso for multvarable Tme Seres. The false earest eghbors method, Keel proposed, s a commo method for sgle varable ad multvarable Tme-Seres. The basc dea of the method s: The Tme Seres s ormalzed, usg auto-correlato fucto ad mutual-formato to determe the delay tme of each sgle varable Tme Seres, gve a set of embeddg dmeso to obta delay vector. For every vector, fdg ts earest eghbor vector, makg that: { j τ } ( ) ( ) η = argm V V : j = max dj +,..., N, j (3) Calculatg oe step predcto error: The measuremet of error (, 2,..., ) d d d. akg ( ) choce of, 2,..., d d d depeds o the E d, d2,..., d a set of mmum embeddg dmeso as the embeddg dmeso of phase space recostructo,.e. ( ) ( ) E d, d2,..., d : d, d2,..., d Z, d, d2,..., d argm = d 0 = The reaso of the choce of embeddg dmeso that makes predcto error s mmum s that whe the embeddg dmeso s too small, wll ot be able to get a proper embeddg, the predcted results wll be very poor, so the predcto error wll be large; whe embeddg dmeso s the optmal embeddg dmeso, sutable embeddg ad may determg the mappg F reduce the predcto error; ad whe the embedded dmeso s more tha the optmal embeddg dmeso, especally whe the system exsts postve Lyapuov dex or ose, the crease of the predcto error wll be far beyod the acceptable rage, because the hstorcal data used the predcto process s assocated very small wth ow or eve completely urelated Improved ethod of Determg Embeddg Dmeso Costructo of error fucto: ( ) (5) E d, d2,..., d just for the varable z, the approprate embeddg dmeso fally got may oly apples to the varable z to the recostructo of phase space, ad t does ot ecessarly apply to other varables. Therefore, ths artcle wll buld error fucto for all varables. Frst, for each vector, fdg the adjacet vector, ad we wll defe the followg error fucto whch based o ths defto. N E d, d,..., d x x, ( ) = η ( ) N J0 + J0 J 0 ( d ) = max τ + (4) (,,,..., d ) a d d 2 = (, 2, +, ) ( ) (, 2,, η + ) (,...,,..., ) ( ) (,...,,..., ) V d d d d V d d d d (6) = V d d d V d d d η Further defto, E ( d,..., d,..., d ) = a (, d,..., d,..., d ) + (7) N J 0 = J0 akg, ( ( ) ( ) ) d,..., d,..., d = argm E d,..., d,..., d : d,..., d,..., d Z (8) The algorthm the global phase space recostructo ca determe the approprate embeddg dmeso combato, s a deal method of determg mult varable embeddg dmeso. It ca be used to calculate the phase space dmeso of sgle varable Tme Seres, s also applcable to the sgle varable case.

3 4 Hog Zhag et al.: Study o Facal Tme Seres Predcto Based o Phase Space Recostructo ad Support Vector ache (SV) 2.3. The Calculato of the axmum Lyapuov Expoet Gettg the tme delay ad embeddg dmeso ca calculate the maxmum Lyapuov dex, to test the exstece of chaotc pheomea of Tme Seres, f the Lyapuov dex s the regular dcates the exstece of chaotc pheomea. The Lyapuov dex estmato expressos are as follows: λ ( k ) where k s a costat, t j= ( + ) ( ) k d j k, = log k t (9) k d k s the sample perod, ( ) j d k s the frst j dstace of the adjacet pots o the basc track after dscrete tme step, s the umber of recostructed phase space pot. 3. Study o Nolear Predcto Algorthm of ultvarate Tme Seres Stochastc process followed a large sample theory, ad requre research sample sze teds to fty, ad obta statstcal coclusos the codtos of mass. I fact, we are uable to get a fty of sample, eve for large-scale aalyss of sample data processg also eed larger prce, usually, the codto of lmted, we wsh get accurate predctos as much as possble. The Hurst dex ad Lyapuov dex ca't be used as the ma bass of regresso predcto. I data drve, eural etwork as a oparametrc weakeg model [4], fault tolerace s excellet, for the complete data of complex system; t has the strog learg capablty, ad domate the eargs predcto of Tme Seres. Used for Tme Seres predcto of eural etwork ca be dvded to two kds: oe kd s Jorda ad Elma etwork, buldg the space-tme mechasm the addtoal layer, storage sample values the past tme pot. [5] Aother s to use tme wdow for processg the teral structure of the eural etwork does ot chage, the put sample values the past, eural etwork gves the predcted value of the ext pot tme. [6-7] The eural etwork ca effectvely predct eargs Tme Seres, but, because ths approach has heret flaws, makg ther learg geeralzato performace degradato, such as over-fttg problem caot be solved. Vapk proposed a ew eural etwork, a better soluto to ths problem. Ths eural etwork s called support vector maches (SVs), follow the structural rsk mmzato prcple, you ca get the global optmal soluto. Wth the troducto of o-sestve loss fucto, SVs exteded to the areas of retur ad t s ot prevously could oly deal wth classfcato of prmary ssues, so called support vector regresso (SVR). SVR was troduced to Tme Seres to predct stock returs. Her research proved that SVR Tme Seres forecast eargs are feasble ad effectve. Kerel fucto maly traslate olear problem to a lear problem through a certa way, the treated, the specfc j learzato method s to map the put data from low-dmesoal space to a hgh dmesoal feature space. Therefore the kerel fucto plays a decsve role to predcto performace of support vector, ts costructo or selecto mode s partcularly mportat. I the dfferet tme graularty, wavelet fucto ca descrbe arbtrary posto of the Icome Tme Seres, so t ca be used to aalyze the fluctuato characterstcs of retur Tme Seres. The SVs ad wavelet theory combed wth each other to buld wavelet kerel fucto used to predct stock returs Tme Seres s worth studyg. [8-9] 3.. Costructo of Wavelet Kerel Fucto Ths secto maly troduces some basc theory of mult-resoluto aalyss ad wavelet trasform research foudato, ad to costruct a wavelet kerel fucto. f z as Defed the cotuous wavelet trasform of ( ) follows, /2 x b WTf ( a, b) = a f ( x) ψ dx, a (0) ψ ( x) to meet the permt codtos: C ψ 2 ( ) (( )) a 0, f z L R ( ) 2 ψˆ ω = dω < () ω where ψ ( ω ) s the Fourer trasform of ψ ( x) verse trasformato s, the the ( ) = ψ ψ a b ( ) f (, ), (2) f x C t WT a b db a da, 2 where C ψ s a costat, ψ ( ω ) s a cotuous fucto of R, () shows that the value of ψ at the org s zero, that s: 3.2. Regresso Aalyss ( ) ψ ( ) ψˆ 0 = x dt = 0, (3) { Settg the trag set (, ),...,(, ),...,(, )} x y x y x y, h whch x R, y R, the regresso fucto of support vector mache ca be expressed as: ( ) ψ ( ) f x = w x + b, (4) w ad b ca be obtaed by mmzg the 2 ( ) 2 w + C ξ + ξ, (5) = I the ed ca be coverted to

4 Amerca Joural of Appled athematcs 205; 3(3): y wψ ( x ) b ε + ξ = = ξ, ξ 0 wψ ( x ) + b y ε + ξ (6) whch ξ, ξ are slack varables, ad C s push parameter. By troducg Lagrage multplers, fally get the decso fucto: a a k x x b, (7) = ( ) ( ) (, ) f x Amog them, α ad (, ) = + (, ) ψ ( ) ψ ( ) k x x = x x (8) α s the Lagrage multpler, k x x s the kerel fucto. Trag the pre-treatmet sample data by SVs regresso method to calculate the support vectors x, ad the parameters α α ad b. The obtaed results to the formula (3.3), ad the put pre-treatmet test set samples to the model. Obtag the predcto value mapped to hgh dmeso terval, ad map the resultg verse to the orgal rage to obta the fal result. I the predcto of the actual process, the ma predcto steps are as follows: ) accordg to the theory of phase space recostructo of chaotc system to obta optmal embeddg dmeso ad tme delay of trag hstorcal data to geerate trag samples. 2) ormalzed geerated trag samples to mprove the covergece speed, shorte trag tme. 3) Select the applcable SV ad kerel fucto. 4) trag geerated trag samples usg support vector mache to obta predcto model. 5) usg the traed support vector mache forecastg model to forecast The Expermetal Data ad the Preprocessg Ths expermet usg two smulated stock returs Tme Seres, each Tme Seres cotas the same umber of samples, collected 320 samples. Autoregressve tegrated movg average model (ARIA) s a mastream method of the facal Tme Seres predcto, usg the dfferece operato to complete the Tme Seres smoothg, the choce of the model to smulate the geerato frst sequece. As show Fgure 3. for the ARIA model of the Tme Seres waveform, the followg expresso descrbes the model of ARIA (,, ) ( φ B)( B) Y ( θ B) ξ,{ ξ }, N ( 0,) = + (9) ackey-glass delay dfferetal equato s a commo mathematcal model of producg smulato data. ackey-glass equato was frst used to descrbe whte blood cells produce process. [0] I the feld of facal chaos pheomeo s wdespread ad the chaotc Tme Seres o the tme axs s ether covergece or dvergece, besdes the moto trajectory s dffcult to descrbe, because t s affected by the tal codto s very serous. because ackey-glass data set has certa ablty to descrbe the chaos pheomeo, gradually at a mportat posto the feld of stadard the facal research. The defto of ackey-glass delay dfferetal equatos as follows: dx bxt d = ax + c (20) dt + x t d whch the autoregressve parameter s φ. the movg average parameters s θ ad the lag operator deoted as B. Accordg to Fgure.2 that the ARIA sequece has the characterstcs of o perodc arragemet form, also ca be observed volet decay of autocorrelato curve. Fgure 3 descrbes the waveform curve of ackey-glass data sets smulato expermet. Fgure 3 descrbes Autocorrelato characterstcs of ackey-glass data sets smulato expermet. Parameter s set to a = 0.2. Compared wth autocorrelato characterstcs of ARIA Tme Seres, ackey-glass Tme Seres has a certa perodcty, ad there s o atteuato of zero pot of autocorrelato. I summary, ackey-glass Tme Seres makes t relatvely easy to regresso predcto. The correctess of ths cojecture has bee verfed the expermet, from Table t s ot dffcult to see that the mmum stadard mea square error are appeared the sequece. There are fve compoets of the put vector, 4 of whch are yeld; aother s the closg prce after coverso. Because the data have a relatvely uform ad symmetry dstrbuto approachg ormal dstrbuto after processg, So that the predcto performace s effectvely mproved, the expermet, the closg prce of the orgal data s coverted to a correspodg retur rate. Subtractg the mea ca elmate the tred term the prce, the orgal closg prce coverso process, also ca avod the loss of useful formato. Fg.. The ARIA tme seres.

5 6 Hog Zhag et al.: Study o Facal Tme Seres Predcto Based o Phase Space Recostructo ad Support Vector ache (SV) Fg. 2. Autocorrelato fucto of ARIA tme seres. see Table. After processg the data, get the statstcal performace, accordg to these statstcal propertes ca be obtaed cocluso. The expermetal results show ths artcle: each dex all have mea close to zero, compared wth the ormal dstrbuto has some deflecto. The best predcto results are from arch 20 to arch 205 for the fourth tme sldg. As show Table 2, fve real stock returs, there all exst the mmum of three dcators (RSE, NSE, AE) the wavelet kerel. Through further comparso, the predcto performace of each kerel addto to AXJO, dcators of other real dex (SSI, CAC40, FTIB ad DAXINDX) are feror to the wavelet kerel. Therefore, wavelet kerel was compared wth the Gaussa kerel ca be as the key. For the smulated data, perhaps due to the dyamc ature of ARIA Tme Seres s weaker, ts Gauss uclear s optmal, ad so the good performace of wavelet kerel caot be fully reflected. O the whole, the optmal s wavelet kerel, followed by the Gauss kerel, thrd s polyomal kerel, ad the worst performace s the lear kerel. There s aother vew of the specal crcumstaces eed to dstgush betwee: the comparso results ca be draw based o the predcto set NSE pared t test. The t value for the oe-sded test proved that the wavelet kerel s better tha the Gauss kerel at 0. sgfcat level. Table. Sample statstcal characterstcs. Fg. 3. The ackey-glass tme seres. Stock dces ea S.D. Skewess Kurtoss ARIA GLASS FTIB SSI CAC AXJO DAXINDX Table 2. Predcto set results. Fg. 4. Autocorrelato fucto of ackey-glass tme seres Performace Idcators ad Results Aalyss Ths paper uses regular mea square error, mea absolute error, root mea square error square ad several other statstcal dexes as a measure of the predcto ad evaluato of performace stadards. Before the trag, the mea stadard devato, skewess ad kurtoss of the dexes Stock dces Lear kerel ARIA GLASS FTIB SSI CAC AXJO DAXINDX Stock dces Polyomal kerel ARIA GLASS FTIB SSI CAC AXJO DAXINDX Stock dces Gaussa kerel ARIA GLASS FTIB SSI CAC

6 Amerca Joural of Appled athematcs 205; 3(3): Stock dces Lear kerel AXJO DAXINDX Stock dces Wavelet kerel ARIA GLASS FTIB SSI CAC AXJO DAXINDX Summary Facal Tme Seres has uque features, such as dyamc characterstcs ad olear characterstcs, although the study of ts dyamc characterstcs are ot mature, coclusve, but the olear characterstcs of Facal Tme Seres has bee wdely recogzed, Therefore, based o the olear Tme Seres modelg prcple, studes the applcato of eural etwork ad support vector mache facal Tme Seres Predcto. I order to make the results fully authetc, the research object selected ths paper s maly for the foreg large stock market, selectg the followg fve stock dex data sets: FTIB, SSI, CAC40, AXJO ad DAXINDX. Two smulated stock, frst modelg usg ARIA model, through the ackey - Glass chaos dfferetal equato to get the requred data sets, ad the select the dmeso of put ad the data preprocessg, so that the data the coverso process most lkely to avod losses. Select the most approprate parameter by dvdg the data set ad some other methods, makg the predcto performace of the model s the most outstadg, Usg selected statstcal dcators to measure whether the predcto performace s the best of the model. Fally, we make a summary of the expermetal data aalyss ad draw the cocluso that wavelet kerel compared wth other gaussa kerel kerel fucto, more close to the real stuato of eargs forecast curve, so you ca thk t has better performace of eargs forecasts. Ackowledgemets Ths project (Emprcal research o Stock dex vestmet rsk model, No.68) s fuded by the " school year, Bejg Wuz Uversty, College studets' scetfc research ad etrepreeural acto pla project". Ad by Bejg Wuz Uversty, Yuhe scholars program ( /007). Ad by Bejg Wuz Uversty, aagemet scece ad egeerg Professoal group of costructo projects. (No. PX205_0424_000039) Refereces [] A Ju-ha, Wag Zh-qag, ad Che Yu-shu. Predcto Techques of Chaotc Tme Seres ad Its Applcatos at Low Nose Level. Appled athematcs ad echacs,27():8-2, 20. [2] Kugutzs D, Lgjrde 0 C, ad Chrstopherse N. Regularzed Local Lear Predcto of Chaotc Tme Seres. Physca D, 2(3-4): , 998. [3] Qgfag eg ad Yuhua Peg. A New Local Lear Predct o odel for Chaotc Tme Seres. Physcs Letters A, 370(5-6): , 20. [4] Ch-Je Lu, Tar Shyug Lee, ad Chh-Chou Chu. Faca Tme Seres Forecastg Usg Idepedet Copoet Aalyss ad Support Vector ache. Decso Support systems, 47(2009):8-24, [5] Lam Hog Lee, Rajparsac Rajkumar, ad Do lsa. Automatc Folder Allocato System Usg Bayesa-Support Vector aces Hybrd Classfcato Approach. Appled Itellgece, Vol. 36, No. 2, , 202. [6] Patrck Koch, Berd Bsch], ad Olver Flasch. Tug ad Evoluto of Support Vector Kerels. Evolulomary Itellgece, Vol. 5, No, 3, , 202 [7] asayuk Karasuyama, aoyuk Harada, asash Sugyama, ad Ichro Takouch. ult-parametrc Soluto-path Algorthm for Istace-weghted Support Vector aches. ache Learg, Vol. 88, No. 3, , 202. [8] S. Amar. Dfferetal-geometrcal ethods Statstcs. Sprger-Verlag New York, 985. [9] S. Amar. Neural Learg Structured Parameter Spaces Natural Rjcmaa Gradet. Advaces Neural Iformato Processg Systems, (9):28-32, 997. [0] C ackey ad L. Glass. Oscllato ad Chaos Physologcal Cotrol Systems. Scece, 97(4300): 287, 977.