Exact Filtering and Smoothing in Markov Switching Systems Hidden with Gaussian Long Memory Noise

Transcription

1 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia 2009 xact ilteig ad Smoothig i Makov Switchig Sstems Hidde with Gaussia Log Memo oise Wojciech Pieczski oufel Abbassi ad Mohamed Be Mabouk Istitut elecom elecom SudPais Det. CII CS UM ue Chales ouie v ace -mail: Wojciech.Pieczski@it-sudais.eu Abstact. Let be a hidde eal stochastic ocess be a discete fiite Makov chai be a obseved chai. he oblem of filteig ad smoothig is the oblem of ecoveig both ad fom. I the classical models the exact comutig with liea - o eve olomial - comlexit i time idex is ot feasible ad diffeet aoximatios ae used. Diffeet alteative models i which the exact calculatios ae feasible have bee ecetl oosed he coe diffeece betwee these models ad the classical oes is that the coule is a Makov oe i the ecet models while it is ot i all the classical oes. Hee we oose a futhe extesio of these models. he coe oit of the extesio is the fact that the obseved chai is ot ecessail Makovia coditioall o ad i aticula the log-memo distibutios ca be cosideed. We show that both filteig ad smoothig ae comutable with comlexit olomial i the umbe of obsevatios i the ew model. Kewods: exact filteig exact smoothig Makov switches hidde liea sstem logmemo oise Itoductio Let... ad... be two sequeces of adom vaiables ad let... be a fiite-value Makov chai. ach q m takes its values fom while each takes its values fom. he sequeces ad ae hidde ad the sequece... is obseved. We deal with two oblems: the filteig oblem ad the smoothig oe whose fomulatio cosideed i this ae ae i calculatio of ad ; ii calculatio ad esectivel. Let us coside a simle classical Gaussia state-sace sstem which cosists of cosideig that is a Makov chai ad oughl seakig that is the classical liea sstem coditioall o. his is summaized i the followig: is a Makov chai; W ; 2 H Z

2 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia 2009 whee W Gaussia vectos W Z Z ae ideedet coditioall o ae matices of size q q deedig o switches ad H H ae matices of size q m also deedig o switches. he exact filteig ad smoothig ae ot feasible with liea - o eve olomial - comlexit i time i such models ad diffeet aoximatios must be used. Ma aes deal with this aoximatio oblem ad a ich bibliogah ca be see i ecet books Cae et al Costa et al oughl seakig thee ae two families of aoximatig methods: the stochastic oes based o the Mote Calo Makov Chais MCMC icile Adieu et al. 200 Doucet et al. 200 Cae et al Giodai et al amog othes ad detemiistic oes Costa et al Zoete et al amog othes. o emed this imossibilit of exact comutatio two diffeet models have bee ecetl oosed i Abbassi ad Pieczski 2008 Pieczski Based o the geeal tilet Makov chais Pieczski ad Desbouvies 2005 the make the exact comutatio of otimal Kalma-like filtes ossible ad the exact calculatio of smoothig is also ossible as show i Badel et al he geeal idea leadig to these models is to coside the ideedece of the ad coditioall o. he these eal models have bee exteded to moe geeal oes i which the ideedece of ad coditioall o is o loge equied Pieczski 2009 Pieczski ad Desbouvies Called Makov magial switchig hidde model MMSHM the veif: is a Makov chai; 4 W 5 q with give ad W W ideedet adom ceteed vectos i. he oieted deedece gahs of the models ad the ew oe ae give i igue x x2 x x x 2 x x2 a b c 2 x ig.. Classical model a ecet model makig exact filteig ad smoothig feasible b; oosed log coelatio model c x

3 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia Makov switchig state model with Gaussia coelated oise Let us coside the tilet as above. he coe oit of the model we oose is to coside that the distibutio of the coule is the distibutio of the atiall Makov Gaussia chai PMGC ecetl itoduced i Lachati et al A PMGC veifies 6 whee ae assumed Gaussia. Imotat i these models is that the coditioal distibutios ae comutable: see Lachati et al fo details. We see that is Makovia with esect to but is ot ecessail Makovia with esect to Makov. iall the model we oose is the followig Accodig to the esults eseted i Lachati et al we have the followig which is at the oigi of its aellatio atiall Defiitio A tilet will be said to be a hidde Makov switchig coditioall liea model HMSCLM if: is a PMGC; 7 W 8 whee each is a matix of size q q deedig o ad W q W ideedet ceteed vectos i such that W is ideedet fom fo each. he oieted deedece gah of the ew model 7-8 is give i igue c. Let us udelie the fact that thee ae o aows goig fom to which 2 meas that coditioall o the chai is ot ecessail Makovia. Let us also highlight that the mai diffeece betwee the classical models of kid a ad the models of kid b o c cosists of the fact that i a the aows go fom x x ad x to ad while i the models b 2 2 ad c the go fom ad to x x ad x. 2 2 Let us also otice that as descibed i Lachati et al icludes diffeet log memo distibutios fo.

4 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia Lemma Let be a PMGC. he the osteio magis ad tasitios ae comutable with comlexit liea i time. As a cosequece ae also comutable. xact filteig We ca state the followig esult: Poositio Let us coside a HMSCLM. he ad ae give fom ad b 9 0 Poof o show 9 we wite ad. o show 0 let us take the coditioal exectatio of 8. We get

5 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia the last equalit beig due to the ideedece of ad coditioall o see the deedece gah c igue. Othewise. eotig this quatit to the exessio of above gives 0. 4 xact smoothig We ca state the followig esult: Poositio 2 Let be a HMSCLM with give tasitios. he ca be comuted fom b:. If i additio the covaiace matices Σ Σ of W W exist the satisfies Σ 2 ad thus Cov ca also be comuted with comlexit liea i time. Poof. B assumtio W. Sice W ad ae ideedet ad W is zeo-mea we have b takig the exectatio of the both sides coditioal o. O the othe had fom model which gives. quatio 2.7 is show similal: the ideedece of W W imlies that ad W ae ideedet coditioall o so.5 gives W W Σ.

6 III Iteatioal Cofeece Alied Stochastic Models ad Data Aalsis ASMDA 2009 Jue 0- Jul Vilius Lithuaia 2009 O the othe had. Combiig it with gives 2 ad eds the oof. efeeces Abbassi. ad Pieczski W xact ilteig I Semi-Makov Jumig Sstem Sixth Iteatioal Cofeece of Comutatioal Methods i Scieces ad gieeig Setembe 25-0 Hesoissos Cete Geece. Adieu C. Dav C. M. ad Doucet A fficiet aticle filteig fo jum Makov sstems. Alicatio to time-vaig autoegessios I as. o Sigal Pocessig 57: Badel. Desbouvies. Pieczski W. ad Babaesco A exact o- Gaussia Makov switchig Baesia smoothig algoithm with alicatio to ack-befoe-detect ADA 2009 submitted. Caé O. Moulies. ad de Ifeece i hidde Makov models Sige. Costa O. L. V. agoso M. D. ad Maques. P Discete time Makov jum liea sstems ew ok Sige-Velag. Doucet A. Godo. J. ad Kishamuth V Paticle filtes fo state estimatio of Jum Makov Liea Sstems I as. o Sigal Pocessig 49: Giodai P. Koh. ad va Dijk D A uified aoach to olieait stuctual chage ad outlies Joual of coometics 7: 2-. Lachati P. Lauade-Lahogue J. ad Pieczski W Usuevised segmetatio of tilet Makov chais hidde with log-memo oise Sigal Pocessig 885: 4-5. Pieczski W. ad Desbouvies O tilet Makov chais Iteatioal Smosium o Alied Stochastic Models ad Data Aalsis ASMDA 2005 Best ace. Pieczski W xact calculatio of otimal filte i semi-makov switchig model outh Wold Cofeece of the Iteatioal Associatio fo Statistical Comutig IASC 2008 Decembe 5-8 okohama Jaa. Pieczski W xact filteig i Makov magial switchig hidde models. Submitted to Comtes edus Mathématique. Pieczski W. ad Desbouvies xact Baesia smoothig i tilet switchig Makov chais Comlex data modelig ad comutatioall itesive statistical methods fo estimatio ad edictio S. Co 2009 Setembe 4-6 Mila Ital Zoete O. ad Heskes Detemiistic aoximate ifeece techiques fo coditioall Gaussia state sace models Statistical Comutatio 6: