Georey E. Hinton. University oftoronto. Technical Report CRG-TR February 22, Abstract
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1 Parameer Esimaion for Linear Dynamical Sysems Zoubin Ghahramani Georey E. Hinon Deparmen of Compuer Science Universiy oftorono 6 King's College Road Torono, Canada M5S A4 zoubin@cs.orono.edu Technical Repor CRG-TR-96- February, 996 Absrac Linear sysems have been used exensively in engineering o model and conrol he behavior of dynamical sysems. In his noe, we presen he Expecaion Maximizaion (EM) algorihm for esimaing he parameers of linear sysems (Shumway and Soer, 98). We also poin ou he relaionship beween linear dynamical sysems, facor analysis, and hidden Markov models. Inroducion The goal of his noe is o inroduce he EM algorihm for esimaing he parameers of linear dynamical sysems (LDS). Such linear sysems can be used boh for supervised and unsupervised modeling of ime series. We rs describe he model and hen briey poin ou is relaion o facor analysis and oher daa modeling echniques. The Model Linear ime-invarian dynamical sysems, also known as linear Gaussian sae-space models, can be described by he following wo equaions: x + = Ax + w () y = Cx + v : () Time is indexed by he discree index. The oupu y is a linear funcion of he sae, x,and he sae a one ime sep depends linearly on he previous sae. Boh sae and oupu noise, w and v, are zero-mean normally disribued random variables wih covariance marices Q and R, respecively. Only he oupu of he sysem is observed, he sae and all he noise variables are hidden. Raher han regarding he sae as a deerminisic value corruped by random noise, we combine he sae variable and he sae noise variable ino a single Gaussian random
2 variable we form a similar combinaion for he oupu. Based on () and () we can wrie he condiional densiies for he sae and oupu, P (y jx ) = exp ; [y ; Cx ] 0 R ; [y ; Cx ] () ;p= jrj ;= (3) P (x jx ; ) = exp ; [x ; Ax ; ] 0 Q ; [x ; Ax ; ] () ;k= jqj ;= (4) A sequence of T oupu vecors (y y ::: y T ) is denoed by fyg a subsequence (y 0 y 0 + ::: y ) by fyg 0 similarly for he saes. By he Markov propery implici in his model, P (fxg fyg) =P (x ) TY = P (x jx ; ) Assuming a Gaussian iniial sae densiy P (x )=exp ; [x ; ] 0 V ; [x ; ] Therefore, he join log probabiliy is a sum of quadraic erms, log P (fxg fyg) = ; ; = = TY = [y ; Cx ] 0 R ; [y ; Cx ] ; T [x ; Ax ;] 0 Q ; [x ; Ax ;] P (y jx ): (5) () ;k= jv j ;= : (6) log jrj ; T ; log jqj ; [x ; ] 0 V ; [x ; ] ; log jv T (p + k) j; log : (7) Ofen he inpus o he sysem can also be observed. In his case, he goal is o model he inpu{oupu response of a sysem. Denoing he inpus by u, he sae equaion is x + = Ax + Bu + w : (8) where B is he inpu marix relaing inpus linearly o saes. We will presen he learning algorihm for he oupu-only case, alhough he exensions o he inpu{oupu case are sraighforward. If only he oupus of he sysem can be observed he problem can be seen as an unsupervised problem. Tha is, he goal is o model he uncondiional densiy of he observaions. If boh inpus and oupus are observed, he problem becomes supervised, modeling he condiional densiy of he oupu given he inpu. Relaed Mehods In is unsupervised incarnaion, his model is an exension of maximum likelihood facor analysis (Everi, 984). The facor, x,evolves over ime according o linear dynamics. In facor analysis, a furher assumpion is made ha he oupu noise along each dimension
3 is uncorrelaed, i.e. ha R is diagonal. The goal of facor analysis is herefore o compress he correlaional srucure of he daa ino he values of he lower dimensional facors, while allowing independen noise erms o model he uncorrelaed noise. The assumpion of a diagonal R marix can also be easily incorporaed ino he esimaion procedure for he parameers of a linear dynamical sysem. The linear dynamical sysem can also be seen as a coninuous-sae analogue of he hidden Markov model (HMM see Rabiner and Juang, 986, for a review). The forward par of he forward-backward algorihm from HMMs is compued by he well-known Kalman ler in LDSs similarly, he backward par is compued by using Rauch's recursion (Rauch, 963). Togeher, hese wo recursions can be used o solve he problem of inferring he probabiliies probabiliies of he saes given he observaion sequence (known in engineering as he smoohing problem). These poserior probabiliies form he basis of he E sep of he EM algorihm. Finally, linear dynamical sysems can also be represened as graphical probabilisic models (someimes referred o as belief neworks). The Kalman-Rauch recursions are special cases of he probabiliy propagaion algorihms ha have been developed for graphical models (Laurizen and Spiegelhaler, 988 Pearl, 988). The EM Algorihm Shumway and Soer (98) presened an EM algorihm for linear dynamical sysems where he observaion marix, C, is known. Since hen, many auhors have presened closely relaed models and exensions, also wih he EM algorihm (Shumway and Soer, 99 Kim, 994 Ahaide, 995). Here we presen a basic form of he EM algorihm wih C unknown, an obvious modicaion of Shumway and Soer's original work. This noe is mean as a succinc review of his lieraure for hose wishing o implemen learning in linear dynamical sysems. The E sep of EM requires compuing he expeced log likelihood, Q = E[log P (fxg fyg)jfyg]: (9) This quaniy depends on hree expecaions E[x jfyg], E[x x 0 jfyg], E[x x 0 ; jfyg] which we will denoe by he symbols: ^x E[x jfyg] (0) P E[x x 0 jfyg] () P ; E[x x 0 ;jfyg]: () Noe ha he sae esimae, ^x, diers from he one compued in a Kalman ler in ha i depends on pas and fuure observaions he Kalman ler esimaes E[x jfyg ] (Anderson and Moore, 979). We rs describe he M sep of he parameer esimaion algorihm before showing how he above expecaions are compued in he E sep. 3
4 The M sep The parameers of his sysem are A, C, R, Q,, V. Each of hese is re-esimaed by aking he corresponding parial derivaive of he expeced log likelihood, seing o zero, and solving. This resuls in he following: Oupu = ; = C new = R ; y ^x 0 + = y ^x 0 =! = R ; CP =0 (3) P! ; (4) Oupu noise covariance: Sae dynamics ; = T R = ; = R new = T = y y 0 ; C^x y 0 + CP C 0 =0 (5) = Q ; P ; + (y y 0 ; C new^x y 0 ) (6) = Q ; AP ; =0 (7) Sae noise covariance: = T ; ; A new = Q ; = T ; Q ; = = P ;! = P ;! ; (P ; AP ; ; P ; A 0 + AP ; A 0 )=0 Q new = T ; = P ; A new T X = P ;! = P ; A new T X = P ;! (8) (9) (0) Iniial sae mean: ; )V ; =0 () new = ^x () 4
5 Iniial sae ; = V ; (P ; ^x 0 ; ^x ) (3) V new = P ; ^x ^x 0 (4) The above equaions can be readily generalized o muliple observaion sequences, wih one subley regarding he esimae of he iniial sae covariance. Assume N observaion sequences of lengh T, le ^x (i) be he esimae of sae a ime given he i h sequence, and Then he iniial sae covariance is The E sep ^x = N V new = P ; ^x ^x 0 + N NX i= NX i= ^x (i) : [^x (i) ; ^x ][^x (i) ; ^x ] 0 : (5) Using x o denoe E(x jfyg ), and V Kalman ler forward recursions: o denoe Var(x jfyg ), we obain he following x ; = Ax ; ; (6) V ; = AV ; ; A0 + Q (7) K = V ; C 0 (CV ; C 0 + R) ; (8) x = x ; + K (y ; Cx ; ) (9) V = V ; ; K CV ; (30) where x 0 = and V 0 = V. Following Shumway and Soer (98), o compue ^x x T and P V T + x T x T 0 one performs a se of backward recursions using J ; = V ; ; A0 (V ; ) ; (3) x T ; = x ; ; + J ; (x T ; Ax; ;) (3) V T ; = V ; + J ; ;(V T ; V ; )J 0 : (33) ; We also require P ; V T + 0 ; xt xt ;,which can be obained hrough he backward recursions V T ; ; = V ; ; J 0 ; + J ; (V T ; ; ; AV; )J 0 ; (34) which is iniialized V T T T; =(I ; K T C)A V T ; T ; : 5
6 References Anderson, B. D. O. and Moore, J. B. (979). Opimal Filering. Prenice-Hall, Englewood Clis, NJ. Ahaide, C. R. (995). Likelihood Evaluaion and Sae Esimaion for Nonlinear Sae Space Models. Ph.D. Thesis, Graduae Group in Managerial Science and Applied Economics, Universiy ofpennsylvania, Philadelphia, PA. Everi, B. S. (984). An Inroducion o Laen Variable Models. Chapman and Hall, London. Kim, C.-J. (994). Dynamic linear models wih Markov-swiching. J. Economerics, 60:{. Laurizen, S. L. and Spiegelhaler, D. J. (988). Local compuaions wih probabiliies on graphical srucures and heir applicaion o exper sysems. J. Royal Saisical Sociey B, pages 57{4. Pearl, J. (988). Probabilisic Reasoning in Inelligen Sysems: Neworks of Plausible Inference. Morgan Kaufmann, San Maeo, CA. Rabiner, L. R. and Juang, B. H. (986). An Inroducion o hidden Markov models. IEEE Acousics, Speech & Signal Processing Magazine, 3:4{6. Rauch, H. E. (963). Soluions o he linear smoohing problem. IEEE Transacions on Auomaic Conrol, 8:37{37. Shumway, R. H. and Soer, D. S. (98). An approach o ime series smoohing and forecasing using he EM algorihm. J. Time Series Analysis, 3(4):53{64. Shumway, R. H. and Soer, D. S. (99). Dynamic linear models wih swiching. J. Amer. Sa. Assoc., 86:763{769. 6
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