Fast Exact Filtering in Generalized Conditionally Observed Markov Switching Models with Copulas

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1 Fast Exact Filterig i Geeralized Coditioally Observed Markov Switchig Models with Copulas Fei Zheg, Stéphae Derrode, Wojciech Pieczyski To cite this versio: Fei Zheg, Stéphae Derrode, Wojciech Pieczyski. Fast Exact Filterig i Geeralized Coditioally Observed Markov Switchig Models with Copulas. Traitemet et Aalyse de l Iformatio Méthodes et Applicatios, Apr 208, Hammamet, Tuisia. <hal > HAL Id: hal Submitted o 5 May 208 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Fast Exact Filterig i Geeralized Coditioally Observed Markov Switchig Models with Copulas Fei Zheg, Stéphae Derrode, ad Wojciech Pieczyski 2 École Cetrale de Lyo, Uiv. de Lyo, LIRIS, CNRS UMR 5205, Écully, Frace. fei.zheg@doctorat.ec-lyo.fr, stephae.derrode@ec-lyo.fr 2 Telecom Sudparis, SAMOVAR, CNRS UMR 557, Évry, Frace. wojciech.piezyski@telecom-sudparis.eu Abstract We deal with the problem of statistical filterig i the cotext of Markov switchig models. For X N hidde cotiuous process, R N hidde fiite Markov process, ad Y N observed cotiuous oe, the problem is to sequetially estimate X N ad R N from Y N. I the classical coditioal Gaussia Liear state space model CGLSSM, where R N, X N is a hidde Gaussia Markov chai, fast exact filterig is ot workable. Recetly, coditioally Gaussia observed Markov switchig model CGOMSM has bee proposed, i which R N, Y N is a hidde Gaussia Markov chai istead. This model allows fast exact filterig. I this paper, usig copula, we exted CGOMSM to a more geeral oe, i which R N, Y N is a hidde Markov chai HMC with oise of ay form ad the regimes are o eed to be all Gaussia, while the exact filterig is still workable. Experimets are coducted to show how the exact filterig results based o CGOMSM ca be improved by the use of the ew model. Key words Markov switchig model, CGLSSM, CGOMSM, GCOMSM, Copulas, Optimal filter, Triplet Markov chai. Itroductio Cosider three radom processes X N = X,..., X N, R N = R,..., R N ad Y N = Y,..., Y N. Each X, R, Y takes their values i R m, Ω = {,..., K} ad R q respectively. The problem that we deal with is to fid the uobservable or hidde processes R N, X N from the observatio Y N = y N. I the model we propose, we assume, as it is usually made, that both triplet X N, R N, Y N ad R N are Markov chais. The first markoviaity the implies that the couple X N, Y N is Markovia coditioally o R N. The distributio of X N, R N, Y N is defied by the iitial distributio p x, r, y ad the trasitios p x, r, y x, r, y which will be take of the form p r r p x, y r, x, y cosistetly for short. with the Markoviaity of R N. Here r, r is deoted by r I the coditioally Gaussia observed Markov switchig model CGOMSM proposed i [] ad applied to geeral o-liear systems i [7, 8], the trasitios p x, y r, x, y are assumed to be Gaussia with liear regimes. The

3 2 Fei Zheg, Stéphae Derrode ad Wojciech Pieczyski aim of the paper is to exted the CGOMSM, which allows exact filterig, to a more geeral oe i which p y r, y are o loger limited to be Gaussia ad the regime G x x, r are o loger ecessarily to be liear o the observatios. The ew model, called Geeralized coditioally observed Markov switchig model GCOMSM, beefits from the copulas, which has bee widely used i statistical fiace for depedece descriptio [4, ]. Copulas were firstly itroduced ito hidde Markov chai HMC with depedet oise by [3], ad importace of their role i segmetatio efficiecy is show i [5,6]. However, to our best kowledge, o work cosiders them i switchig state-space models. I the proposed GCOMSM, p y r, y is much more flexible compare to the origial CGOMSM makig use of copula. Experimets are coducted to show the iterest of the ew model with compariso to the result give by traditioal Gaussia liear assumptios. The paper is orgaized as follows. I ext Sectios, we recall CGOMSM, specify the geeral GCOMSM ad show how fast optimal filterig ad smoothig rus i this ew model. Experimets are displayed ad aalyzed i the third Sectio. Fially, the coclusio ad perspectives are give i the last Sectio four. 2 Geeralized coditioally observed Markov switchig model GCOMSM Let us cosider a CGOMSM, i which the Markov triplet X N, R N, Y N distributio is defied by p x, r, y = p r p x, y r ad trasitios of the form p x, r, y x, r, y = p r r p x, y x, y, r. Both p x, y r ad p x, y x, y, r are Gaussia. I [], the CGOMSM is described by liear regime: [ ] [ ] [ ] [ ] X F xx R F xy R X U = Y 0 F yy R +, Y V }{{} FR with FR a appropriate system trasitio matrix, ad [ U, V ] represets the idepedet Gaussia zero-mea oise which are idepedet from T = X, R, Y. We see i CGOMSM, the pair R N, Y N is a Markov chai, ad p x, y r, x, y = p y r, y p x x, r, 2 which makes p r N y N ca be computed, thus the exact filterig is feasible. Uder CGOMSM, p y r, y is Gaussia ad G x x, r is liear o x, y ad y. However, to maitai the feasibility of exact filterig, the Gaussia settig ad liear form are ot ecessary coditios.

4 Fast Exact Filterig i GCOMSM with Copulas 3 2. Defiitio of Geeralized coditioally observed Markov switchig model GCOMSM The Geeralized coditioally observed Markov switchig model GCOMSM which exteds the CGOMSM cosiders still the triplet X N, R N, Y N a Markov chai, defied by p x, r, y ad trasitio of the form p x, r, y x, r, y =p r r p 3 y r, y p x x, r, Ulike i CGOMSM, p y r p y r, y =f r y r, y is eriched by copula represeted as: c F l y r y r 4, F r y r r, to deote the probability desity where we use f l y r ad f r fuctio PDF of the left ad right margis respectively. Similarly, F l y r, F r y r are their associated cumulative distributio fuctio CDF, while c, r represets the desity of the two-dimesioal copula coditioally o switches. The copula above the completes the two margis to form a joit distributio p y which ca theoretically embrace ay distributio form. r Moreover, the simple liear regime G x x, r which correspods to p x x, r i CGOMSM is exted to x = A r x + B r + ν 5 i GCOMSM, i which A ad B ca be ay fuctio forms of r, r, y. ν N 0, V r. Itegrally, they ca be also writte as x N { } A r x + B r, V r Fast exact filterig i GCOMSM The Markov property of R N, Y N i GCOMSM leads to p x r = p x r, y. Besides, sice p x x, r is Gaussia defied as 6, we have E [ ] X x, r = A r E [X r, y ] + B r. The E [ ] X r, y is computable from E [X r, y ] with ] = E [ X r, y r p r r, y } + B r { A r E [X r, y ] 7 8

5 4 Fei Zheg, Stéphae Derrode ad Wojciech Pieczyski i which p r r is computable because of the Markoviaity of R N, Y N. More precisely, we ca write p r r, y p r = r p, r 9 ad p r ca be calculated recursively with p r = p r, r r = r p r, y p r r p y r, y, 0 Fially, the filterig is give by E [ ] X y = p [ ] r y E X r, y. r 3 Example of GCOMSM ad experimet o the matched exact filterig We preset here a example to show the flexibility of the proposed GCOMSM as well as the performace of the matched exact filterig. We focus o the timeidepedet case of the geeral GCOMSM, which meas that the parameters deped oly o the switches r. For simplificatio, we assume that R N, Y N is statioary reversible, which meas that p y r = p y r, therefore the left ad right margis i 4 are equal. Uder these assumptios the equatio 4 ad 6 ca be writte as p y r, y = fr y cr Fr y, F r y x N { A r y x + B r y 2 }, V r. 3 I place of the time depedece i origial defiitio, the depedece o switches are moved to subscript of all fuctios. For this example, we assume that R N has two compoet values Ω = {, 2}. Ad for each j, k Ω, f j y = f r=j y, c j,k Fj y, F k y = cr=j,r =k Fj y, F k y with Fj, C j,k the associated CDF i 2. I 3, the abbreviatio is take i the same way: A j,k y = A r=j,r =k, y, so as B j,k y ad V j,k. The parameters of p y r, y which are set to be o-gaussia as

6 Fast Exact Filterig i GCOMSM with Copulas 5 - Margis: f y = Beta {α = 0.9, β = 0.9, loc = 4, scale = 6}, f 2 y = Fisk 2 {β 2 = 4, loc 2 = 2.7, scale 2 = 2.4}. - Copulas: c, {, } = Arch4 3 {, α, = 3}, c 2,2 {, } = FGM {, α 2,2 = 0.5}, c,2 {, } = c 2, {, } = Clayto {, α,2 = 4.7}. The margial ad joit distributio are displayed i Figure a, b. p x x, r is set with A j,k y = a j,k x, simple o-liear fuc- = b j,k y y, ad i which the parameters are tio o y that B j,k y assiged as - a j,k : a, = 0.2, a,2 = 0.4, a 2, = 0.6, a 2,2 = 0.8, - b j,k : b, = 0.7, b,2 = 0.5, b 2, = 0.6, b 2,2 = 0.9, - V j,k : V, = V 2,2 =.0, V,2 = V 2, = 0.8. a Margis f y :Beta, f 2 y :Fisk. b joit distributio of f,2 y. c Histogram of x N. d Histogram of y N. Figure : Distributios ad histograms of simulated GCOMSM data.. α ad β are the shape parameters, loc ad scale are short for locatio ad scale. 2. β 2 represets the shape parameter, loc 2 ad scale 2 for locatio ad scale. 3. Short for Archimiedea copula, order: 4.

7 6 Fei Zheg, Stéphae Derrode ad Wojciech Pieczyski 2000 samples are simulated accordig to the above settig of GCOMSM. We see from the histograms of the simulated data illustrated i Figure c, d that they are hardly to be approximated by Gaussia mixtures with small compoet umber. Exact filterig for GCOMSM is applied o y N to restore the hidde r N decided by maximum posterior mode criterio from p r y ad x N. For compariso, we coducted also the filterig based o Gaussia assumptios both margis ad copulas are assumed to be Gaussia by usig Maximum likelihood ML ad Pseudo- Likelihood Maximizatio PLM [9] for Gaussia parameter estimatio of margis ad copulas applied o data. Restoratio results are average of 00 idepedet experimets, illustrated i Table. We ca see that the exact filterig performs well o restorig the hidde switches ad states for GCOMSM, while the Gaussia assumptio is obviously iferior comparig to the exact filter. Table : Restoratio result. Observatio Exact filterig Filterig Gaussia MSE Error Ratio MSE Error Ratio MSE % % 2.6 Their performace ca also be told from the trajectories. Figure 2 illustrates a trajectory example from oe istace amog the 00 experimet. 4 Coclusio I this work, copula is itroduced i the recet coditioally Gaussia observed Markov switchig model CGOMSM, ad fuse to a more geeral oe called geeralized coditioally observed Markov switchig model GCOMSM. Experimets verify the capability of GCOMSM to work o data uder flexible distributios. The fast exact filterig for GCOMSM ca be much less time cosumig comparig to usig other o-gaussia models which do ot allow exact filterig ad Mote- Carlo methods are eeded to be applied. The future work may cotai the model idetificatio idetifyig the margis, copulas [2, 6, 2], ad also the coditioal regime fuctios [0] of GCOMSM, ad applicatio of the model o o-gaussia o-liear data restoratio. I additio, smoothig ca also be a perspective of iterest. Refereces. Noufel Abbassi, Dalila Beboudjema, Stéphae Derrode, ad Wojciech Pieczyski. Optimal filter approximatios i coditioally Gaussia pairwise Markov switchig models. IEEE tras. o Automatic

8 Fast Exact Filterig i GCOMSM with Copulas 7 Figure 2: Trajectory example 00 samples. Cotrol, 604:04 09, Armad Kodjo Atiampo ad Georges Laussae Loum. Usupervised image segmetatio with pairwise Markov chais based o oparametric estimatio of copula usig orthogoal polyomials. Iteratioal Joural of Image ad Graphics, 64:650020, Nicolas Bruel ad Wojciech Pieczyski. Usupervised sigal restoratio usig hidde Markov chais with copulas. Sigal processig, 852: , Barbara Choroś, Rustam Ibragimov, ad Elea Permiakova. Copula estimatio. Copula theory ad its applicatios, pages 77 9, Stéphae Derrode ad Wojciech Pieczyski. Usupervised data classificatio usig pairwise Markov chais with automatic copulas selectio. Computatioal Statistics & Data Aalysis, 63:8 98, Stéphae Derrode ad Wojciech Pieczyski. Usupervised classificatio usig hidde Markov chai with ukow oise copulas ad margis. Sigal Processig, 28:8 7, Iva Goryi, Stéphae Derrode, Emmauel Mofrii, ad Wojciech Pieczyski. Exact fast smoothig i switchig models with applicatio to stochastic volatility. I Sigal Processig Coferece EUSIPCO, rd Europea, pages IEEE, Iva Goryi, Stéphae Derrode, Emmauel Mofrii, ad Wojciech Pieczyski. Fast filterig i switchig approximatios of oliear Markov systems with applicatios to stochastic volatility. IEEE tras. o Automatic Cotrol, 622: , Guky Kim, Mervy J Silvapulle, ad Paramsothy Silvapulle. Compariso of semiparametric ad parametric methods for estimatig copulas. Computatioal Statistics ad Data Aalysis, 56: , Keeth Leveberg. A method for the solutio of certai o-liear problems i least squares. Quarterly of applied mathematics, 22:64 68, Roger B Nelse. A itroductio to copulas. Spriger Sciece & Busiess Media, Hideatsu Tsukahara. Semiparametric estimatio i copula models. Caadia Joural of Statistics, 333: , 2005.