Maximum likelihood esimaion of sae-space models Florian Chevassu 1 and Juan-Pablo Orega Absrac The use of he Kalman filer for esimaion purposes is no always an easy ask despie he obvious advanages in many siuaions of he sae-space represenaion This is in par due o he fac ha he compuaion of he corresponding score gradien of he log-likelihood is someimes complicaed and numerically challenging We presen an algorihm for he exac consrucion of he score ha is wrien in he form of explici and ready o be used Markov ype marix recursions ha are very efficien from he memory managemen poin of view As an example, his general resul is applied o he esimaion of ARMA models in sae-space represenaion Keywords: Kalman filer, score, model esimaion, maximum likelihood esimaion, ARMA 1 Inroducion Linear sae-space models and he associaed Kalman recursions are a very imporan ool of much use in, for example, engineering, conrol heory, and ime series analysis This scheme provides a compac and numerically efficien way o encode, filer, and forecas a large array of dynamic models A sandard procedure o esimae he parameers of a sae-space model is maximizing he associaed quasi-loglikelihood The main ingredien when carrying ou maximum likelihood esimaion is he gradien of he quasi-loglikelihood funcion, also called score; his objec is used a he ime of seing up many cusomarily used opimizaion procedures and, addiionally, is he main building block of he Fisher informaion marix ha describes he asympoic normaliy properies of his esimaor The main goal of his paper is he consrucion of a se of explici and ready o be used marix recursions ha deermine uniquely he gradien of he quasi-loglikelihood associaed o any linear dynamic sae-space model These recursions, conained in Theorem 1, are very efficien from he memory managemen poin of view for hey are buil in such way ha he updae of he gradien of he log-likelihood a ime + 1 needs only he values of hese marices a ime Inermediae resuls leading o he resuls ha we presen in his paper appeared already in several places in he lieraure; for example [1] presened a se of recursive relaions called filer and Riccaiype sensiiviy equaions ha can be used o compue he differenial of he log-likelihood The use of he adjoin o make explici he gradien appeared in [1] Alernaive ways o compue he score based, for example, on expecaion maximizaion, can be found in [11] As an example of applicaion of he resuls in he paper we have explicily sudied he ARMA parameric family in sae-space represenaion in Secion 3 The proofs of he resuls in he paper are all conained in appendices in Secion 5 1 Tea-Cegos eploymen 11 rue enis Papin F-5 Besançon fchevassu@deploymenorg 3 Corresponding auhor Cenre Naional de la Recherche Scienifique, éparemen de Mahémaiques de Besançon, Universié de Franche-Comé, UFR des Sciences e Techniques 16, roue de Gray F-53 Besançon cedex France Juan-PabloOrega@univ-fcomefr 1
Maximum likelihood esimaion of sae-space models simaion of sae-space models A linear dynamic sae-space model consiss of wo relaions, namely, he observaion and he sae equaions: { Y = G X + W, W WN, R, 1 X +1 = F X + V, V WN, Q, where he symbols WN, R and WN, Q denoe vecor whie noises of zero mean and covariance marices R and Q, respecively The firs equaion represens he observaion Y R w as a linear funcion of he sysem sae X R r plus a noise erm W The second one describes he sae dynamics as a one-lagged vecor auoregression perurbed by a noise erm V More specifically, if we denoe by M n,m he se of real n m marices, in 1 we have ha Y M w,1 = R w, X M r,1 = R r, G M w,r, and F M r,r Linear sochasic sae-space models, wih heir associaed Kalman filer [8], have been inensively used for applicaions in fields such as conrol, communicaions, signal processing, and ime series forecasing The main building block of he Kalman filer is he one-sep predicor X of X in erms of he informaion se F 1 generaed by he observaions {Y 1,, Y 1 } When he driving noises in 1 are Gaussian and he processes {Y } and {X } are hence joinly Gaussian, we will be using he condiional expecaion X := [X F 1 ], while in more general siuaions we will conen wih he bes[ linear predicor X := P X Y 1,, Y 1 of X in erms of {Y 1,, Y 1 } If we denoe by Ω = X X X X ] he corresponding error covariance marices, he Kalman predicion recursions [8] provide an ieraive consrucion of he one sep predicors X and he corresponding error covariance marices Ω, namely: { ˆX+1 = F ˆX + Θ 1 Y G ˆX, Ω +1 = F Ω F + Q Θ 1 Θ, where { = G Ω G + R, Θ = F Ω G, and he ieraions are sared using he iniial condiions [ ˆX 1 = P X 1 Y, and Ω 1 = X 1 ˆX 1 X 1 ˆX ] 1 Suppose now ha he srucural consiuens of he sae-space model 1, ha is, he marices G and F, and he covariances R and Q, are deermined by a se of parameers θ R k The esimaion problem consiss of deermining θ ou of a sample {Y 1,, Y n } of observaions A sandard approach o he soluion of his quesion consiss of maximizing he Gaussian log-likelihood l θ; Y 1,, Y n associaed o 1 wih respec o he parameer θ, ha is, l θ; Y 1,, Y n := nw log π 1 n log de θ 1 n I θ 1 θ I θ, =1 =1 where { I θ = Y P 1Y = Y G θ ˆX θ, θ = G θ Ω θ G θ + R θ 3
Maximum likelihood esimaion of sae-space models 3 In his expressions we obviously have ha I M w,1 and M w,w The maximum likelihood esimaor θ of θ is given by soluion of he opimizaion problem θ := arg min l θ; Y 1,, Y n 4 θ R k In he presence of independen observaions i has been shown [1] ha under very general condiions he maximum likelihood esimaor is consisen and asympoically normal wih covariance marix given by he inverse of he Fisher informaion marix [4] J := [ θ l θ l ] The use of any seepes descen mehod in he soluion of 4 and he Fisher informaion marix require he compuaion of he gradien θ l of he log-likelihood ; his is he main goal of his paper Before saing a resul ha provides an ieraive consrucive soluion o his problem and whose proof can be found in he appendix, we inroduce some noaion ha will be used all along he paper Firs of all, insead of working wih he log-likelihood, we will deal mosly wih he condiional loglikelihoods l θ; Y 1,, Y n a a fixed ime insan {1,, n}, ha is, l θ; Y 1,, Y n := w log π 1 log de θ 1 I θ 1 θ I θ 5 Given ha he sample {Y 1,, Y n } remains consan in our compuaions we will omi his dependence in mos expressions The differenial l : R k R k of he condiional log-likelihood wih respec o he variable θ is he map ha assigns o any θ R k he linear form θ l : R k R defined by θ l ψ = d d l θ + ψ; Y 1,, Y n = If we denoe by, he uclidean inner produc in R k, we define he gradien l θ R k of l as he vecor uniquely deermined by he relaion l θ, ψ = θ l ψ, for any ψ R k Given a linear map A : V,, V W,, W beween wo vecor spaces V,, V and W,, W he dual or adjoin mapping A : W V is defined by he relaion w, Av W = A w, v V for any v V, w W When dealing wih he vecor space of marices of a given size M n,m, we will be using he inner produc given by he Frobenius conracion, ha is, A, B = r A B, for any A, B M n,m Finally, for any inner produc space V,, V we will idenify he dual vecor space V wih V by using he linear isomorphism v, v, for any v V Theorem 1 The score l θ of he condiional log-likelihood l is given by l θ = θg 1 θi 1 θ G θ Ω θ 1 θω G θ 1 θ G θ 1 θr 1 θ θ I θ 1 θ 1 I θ I θ 6 where I θ and θ are given by 3 and he dual linear maps θ Ω : M r,r M k,1, θ Θ : M r,w M k,1, θ 1 : M w,w M k,1, ˆX θ : M r,1 M k,1, and θ I : M w,1 M k,1, can be
Maximum likelihood esimaion of sae-space models 4 recursively consruced by using he relaions θω ρ = θf 1 ρf 1 θ Ω 1 θ + ρ F 1 θ Ω 1 θ + θω 1 F1 θ ρf 1 θ + θq ρ θθ 1 ρθ1 θ 1 1 θ + ρ Θ 1 θ 1 1 θ θ 1 1 Θ1 θ ρθ 1 θ, 7 θθ ρ = θf ρg θ Ω θ + θω F θ ρg θ + θg ρ F θ Ω θ, 8 θ 1 ρ = θg 1 ρ 1 G θ Ω θ θω G θ 1 ρ 1 G θ θg 1 ρ 1 G θ Ω θ θr 1 ρ 1, 9 + ˆX θ 1 F 1 θ ρ + θθ 1 θ ˆX ρ = θf 1 ρ ˆX 1 θ θ 1 1 Θ 1 θ ρ Y 1 ˆX 1 θ G 1 θ ρ Y 1 ˆX 1 θ G 1 θ 1 1 θ θg 1 1 1 θ Θ 1 θ ρ ˆX 1 θ θ ˆX 1 G 1 θ 1 1 θ Θ 1 θ ρ, 1 θ I ρ = θg ρ ˆX θ θ ˆX G θ ρ 11 The use of he relaions 7 11 in Theorem 1 in order o deermine he score 6 righ as hey are saed, may prove o be numerically inefficien because a each ime, he evaluaion of he dual linear maps θ Ω, θ Θ, θ 1, ˆX θ, and θ I a he differen poins where hey are needed in 6 requires ieraing 7 11 all he way o he iniial condiions a ime = In order o circumven his problem, we will rewrie 7 11 as pure marix recursions and no as ideniies ha ieraively define he evaluaion of a collecion of linear maps on prescribed values of heir domains The main ool ha we will be using in he following resul are he algebraic properies of he vec and he ma operaors see for example [9, Secion A11] We recall ha vec : M m,n R mn is he linear operaor ha assigns a vecor o a marix by sacking is columns; ma m,n : R mn M m,n is is inverse Addiionally, we use he commuaion marices K m,n as defined, for insance in Appendix A of [9]; we recall ha given any A M m,n he commuaion K m,n M mn,mn is such ha veca = K m,n veca The symbol I r M r,r denoes he ideniy marix Finally, in order o spell ou he marix recursions ha we announced above, we will use he following definiions: f M k,r is he marix corresponding o he linear map θ F ma r,r : R r R k g M k,rw is he marix corresponding o he linear map θ G ma w,r : R rw R k q M k,r is he marix corresponding o he linear map θ Q ma r,r : R r R k r M k,w he marix corresponding o he linear map θ R ma w,w : R w R k ω M k,r he marix corresponding o he linear map θ Ω ma r,r : R r R k θ M k,rw he marix corresponding o he linear map θ Θ 1 ma r,w : R rw R k δ 1 M k,w he marix corresponding o he linear map θ 1 ma w,w : R w R k ˆx M k,r he marix corresponding o he linear map θ ˆX ma r,1 : R r R k i M k,w he marix corresponding o he linear map θ I ma w,1 : R w R k The following resul, whose proof can be found in he appendix, reformulaes he ieraive definiions 7 11 in Theorem 1 as recursions on he associaed marix represenaions This feaure improves dramaically memory managemen issues a he ime of he programming implemenaion of he associaed algorihm for i allows us o updae he log-likelihood gradien a ime + 1 jus by knowing he values of hese marices a ime, wihou going back all he way o ime =
Maximum likelihood esimaion of sae-space models 5 Proposiion The ieraive definiions 7 11 in Theorem 1 are uniquely deermined by he marix recursions: ω = f 1 Ω1 F 1 I r Ir + K r,r + ω 1 F1 F 1 θ 1 1 1 Θ 1 I r Ir + K r,r δ1 1 Θ1 Θ 1 + q, 1 θ = f Ω G I r + ω G F + g Ω F I w Kr,w, 13 δ 1 = g Ω G 1 1 ω G 1 G 1 g 1 Ω G 1 r 1 1, 14 Y 1 G 1 ˆX1 Θ 1 ˆx = f 1 ˆX1 I r + ˆx 1 I1 F 1 + δ 1 1 g 1 ˆX1 1 1 Θ 1 ˆx 1 I1 G 1 1 1 Θ 1, 15 i = g ˆX I w ˆx I1 G 16 3 Applicaion: esimaion of univariae ARMA ime series models The convenience of he Kalman filer in he reamen of ime series relaed quesions has been profusely exploied in he lieraure from he very beginning of his field see for insance [5, 7] and references herein someimes wih much success in conexs ha go beyond he linear caegory [6] In his secion we will use he univariae ARMA parameric family as an applicaion example of he score compuaion echnique inroduced in Theorem 1 and Proposiion The exension of he expressions ha follow o he mulidimensional VARMA seup can easily be carried ou Le φ = φ 1,, φ p and θ = θ 1,, θ q be wo parameer ses such ha he associaed polynomials φz = 1 φ 1 z φ p z p and θz = 1 + θ 1 z + + θ q z q do no have any common roos and hey are all ouside he uni circle A discree ime process {U } is called ARMAp,q whenever i saisfies he recursive relaion: U φ 1 U 1 φ p X p = ɛ + θ 1 ɛ 1 + + θ q ɛ q, {ɛ } WN, σ, where he symbol WN, σ denoes a zero-mean whie noise wih variance σ ARMA processes admi a sae-space represenaion see for example [3, ] for a self-conained presenaion ha can be achieved by choosing he auonomous ha is, do no depend on ime srucural marices G θ M 1,r and F θ M r,r, r = maxp, q + 1, θ = [ φ 1 φ p θ 1 θ q σ ], given by I r1 F θ =, G θ = [ ] θ q θ, 31 φ p φ 1 wih θ = 1 In his case, he noise covariances are given by R = and Q = 3 σ
Maximum likelihood esimaion of sae-space models 6 In order o obain he ARMA log-likelihood and he associaed gradien ou of he expressions in Theorem 1 and Proposiion, i suffices o compue he marices f, g, q, and r Firs, as R =, we obviously have he r = Regarding he oher marices, we find explici expressions for θ F ma r,r, θ G ma w,r, and θ Q ma r,r The linear map θ F ma r,r : R r R k and is marix expression f M k,r I is clear ha in his siuaion k = p + q + 1 Moreover, using 31, i is easy o see ha θ F : R p+q+1 M r,r ψ 1 ψ p+q+1 r1,r, an hence, as θ F ρ, ψ = ρ, θ F ψ, we have ha θ F : M r,r R p+q+1 ρ ψ p ψ 1 ρ r,r ρ r,rp+1, and hence Consequenly, θ F ma r,r : R r R p+q+1 ρ r ρ ρ rr1 ρ rrp+1 e r f = p+q+1,rrp e r e r q+1,rp e r wih e r = [ 1 ] M 1,r The linear map θ G ma 1,r : R r R k and is marix expression g M k,r Using 31 i is
Maximum likelihood esimaion of sae-space models 7 easy o see ha θ G : R p+q+1 M 1,r ψ 1 ψ p+q+1 As θ G ρ, ψ = ρ, θ G ψ, we have ha [ ψ p+q ψ p+1 ] θ G : M 1,r R p+q+1 ρ ρ 1,r1, ρ 1,rq and hence Consequenly, θ G ma w,r : R r R p+q+1 ρ ρ r1,1 ρ rq,1 p,rq1 p,q p,1 1 g = q,rq1 O q,1 1 1,rq1 1,q The linear map θ Q ma r,r : R r R p+q+1 and is marix expression q M p+q+1,r In view of 3 i is easy o see ha θ Q : M p+q+1,1 M r,r ψ 1 ψ p+q+1 ψ p+q+1
Maximum likelihood esimaion of sae-space models 8 As θ Q ρ, ψ = ρ, θ Q ψ, we conclude ha θ Q : M r,r M p+q+1,1 ρ, ρ r,r and hence Consequenly, θ Q ma r,r : R r R p+q+1 ρ ρ r q = 1 4 Conclusions In his paper we have consruced a se of explici and ready o be used marix recursions ha deermine uniquely he gradien of he quasi-loglikelihood associaed o any linear dynamic sae-space model This gradien is much needed a he ime of model esimaion and when compuing he corresponding Fisher informaion marix These recursions are very efficien from he memory managemen poin of view for hey are buil in such way ha he updae of he gradien of he log-likelihood a ime + 1 needs only he values of hese marices a ime As an example of applicaion of he resuls in he paper we have explicily sudied he ARMA parameric family in sae-space represenaion 5 Appendix 51 Proof of Theorem 1 In order o compue he gradien of he log-likelihood, we need o know he differenials of he elemens i is made of We will do so in such an order ha when we differeniae a variable a ime, all he elemens i depends on have already been compued a ime 1; his requiremen imposes compuing firs he differenial of Ω, and hen hose of Θ,, 1, ˆX and I In he following paragraphs we will use he fac ha Ω and are symmeric, wihou necessarily recalling i ifferenial of Ω : θ Ω ψ = θ F 1 ψ Ω 1 θ F 1 θ + F 1 θ θ Ω 1 ψ F 1 θ + F 1 θ Ω 1 θ θ F 1 ψ ifferenial of Θ : + θ Q ψ θ Θ 1 ψ 1 1 θ Θ 1 θ Θ 1 θ θ 1 1 ψ Θ 1 θ Θ 1 θ 1 1 θ θθ 1 ψ
Maximum likelihood esimaion of sae-space models 9 ifferenial of : θ Θ ψ = θ F ψ Ω θ G θ + F θ θ Ω ψ G θ + F θ Ω θ θ G ψ θ ψ = θ G ψ Ω θ G θ + G θ θ Ω ψ G θ + G θ Ω θ θ G ψ + θ R ψ ifferenial of 1 : θ 1 ψ = 1 θ ψ 1 ifferenial of ˆX : θ ˆX ψ = θ F 1 ψ ˆX 1 θ + F 1 θ θ ˆX1 ψ + θ Θ 1 ψ 1 1 Y θ 1 G 1 θ ˆX 1 θ + Θ 1 θ θ 1 1 Y ψ 1 G 1 θ ˆX 1 θ + Θ 1 θ 1 1 h θ θ G 1 ψ ˆX i 1 θ G 1 θ θ ˆX1 ψ ifferenial of I : θ I ψ = θ G ψ ˆX θ G θ θ ˆX ψ ifferenial of he log-likelihood: We now use he previous expressions and he chain rule and conclude ha: θ l ψ = 1 1 1 I θ θ 1 = 1 1 1 θ, θ ψ 1 θi ψ 1 θ I θ ψ I θ 1 I θ 1 θ θ I ψ θ, θ G ψ Ω θ G θ + G θ θ Ω ψ G θ + G θ Ω θ θ G ψ + θ R ψ θ I ψ, 1 θ I θ 1 I θ, θ 1 ψ I θ 1 1 θ I θ, θ I ψ = 1 r 1 θ ˆ θ G ψ Ω θ G θ + G θ θ Ω ψ G θ + G θ Ω θ θ G ψ + θ R ψ θ I ψ, 1 = θ G θ I 1 1 θ I θ 1 I θ, θ 1 θ G θ Ω θ, ψ 1 θ Ω θ I θ, ψ 1 θ 1 ψ I θ G θ 1 `I θ I θ, ψ Since he gradien l θ R k of l is he vecor uniquely deermined by he relaion for any ψ R k, we have ha l θ = θ G 1 θ I 1 l θ, ψ = θ l ψ, θ G θ Ω θ 1 θ Ω θ I θ G θ 1 1 θ 1 `I θ I θ, θ G θ, ψ 1 θ R θ G θ 1 θ R 1 1 θ θ, ψ which he equaliy 6 We now provide recursive relaions ha deermine he various dual linear maps ha appear in his expression by compuing he adjoins of he differenials in he previous paragraphs We recall ha when dealing wih he vecor space of marices of a given size M n,m, we will be using he inner produc given by he Frobenius conracion, ha is, A, B = r `A B, for any A, B M n,m
Maximum likelihood esimaion of sae-space models 1 Compuaion of θ Ω: Hence, θ Ω ρ, ψ = ρ, θω ψ = ρ, θ F 1 ψ Ω 1 θ F 1 θ + ρ, F 1 θ θ Ω 1 ψ F 1 θ + ρ, F 1 θ Ω 1 θ θ F 1 ψ + ρ, θ Q ψ ρ, θ Θ 1 ψ 1 1 θ Θ 1 θ ρ, Θ 1 θ θ 1 1 ψ Θ 1 θ ρ, Θ 1 θ 1 1 θ θθ 1 ψ = r `ρ θ F 1 ψ Ω 1 θ F 1 θ + r `ρ F 1 θ θ Ω 1 ψ F 1 θ + r `ρ F 1 θ Ω 1 θ θ F 1 ψ + θ Q ρ, ψ r ρ θ Θ 1 ψ 1 1 θ Θ 1 θ r ρ Θ 1 θ θ 1 1 ψ Θ 1 θ r ρ Θ 1 θ 1 1 θ θθ 1 ψ = ρf 1 θ Ω 1 θ, θ F 1 ψ + F 1 θ ρf 1 θ, θ Ω 1 ψ + ρ F 1 θ Ω 1 θ, θ F 1 ψ + θ Q ρ, ψ ρθ 1 θ 1 1 θ, θθ 1 ψ Θ 1 θ ρθ 1 θ, θ 1 1 ψ ρ Θ 1 θ 1 1 θ, θθ 1 ψ = θ F 1 `ρf1 θ Ω 1 θ + ρ F 1 θ Ω 1 θ, ψ + θ Ω 1 `F1 θ ρf 1 θ, ψ + θ Q ρ, ψ θ Θ 1 ρθ 1 θ 1 1 θ + ρ Θ 1 θ 1 1 θ, ψ θ 1 1 `Θ1 θ ρθ 1 θ, ψ θ Ω : Mr,r M k,1 ρ θ F 1 ρf 1 θ Ω 1 θ + ρ F 1 θ Ω 1 θ + θ Ω 1 F 1 θ ρf 1 θ + θ Q ρ θ Θ 1 ρθ 1 θ 1 1 θ + ρ Θ 1 θ 1 1 θ θ 1 1 Θ 1 θ ρθ 1 θ Compuaion of θ Θ: Hence, θ Θ ρ, ψ = ρ, θθ ψ = ρ, θ F ψ Ω θ G θ + F θ θω ψ G θ + F θ Ω θ θg ψ = r `ρ θ F ψ Ω θ G θ + r `ρ F θ θ Ω ψ G θ + r `ρ F θ Ω θ θ G ψ = r `Ω θ G θ ρ θ F ψ + r `G θ ρ F θ θ Ω ψ + r ` θ G ψ ρ F θ Ω θ = θ F ρg θ Ω θ + θ Ω `F θ ρg θ + θ G `ρ F θ Ω θ, ψ θ Θ : Mr,w M k,1 ρ θ F ρg θ Ω θ + θ Ω F θ ρg θ + θ G ρ F θ Ω θ Compuaion of θ 1 : θ 1 ρ, ψ = ρ, θ 1 ψ = ρ, 1 θ ψ 1 = ρ, 1 θ G ψ Ω θ G θ 1 + ρ, 1 G θ θ Ω ψ G θ 1 + ρ, 1 G θ Ω θ θ G ψ 1 + ρ, 1 θ R ψ 1 = r ρ 1 θ G ψ Ω θ G θ 1 + r ρ 1 G θ θ Ω ψ G θ 1 + r ρ 1 G θ Ω θ θ G ψ 1 + r ρ 1 θ R ψ 1 = θ G 1 ρ 1 G θ Ω θ, ψ + θ Ω G θ 1 ρ 1 G θ, ψ + θ G 1 ρ 1 G θ Ω θ, ψ + θ R 1 ρ 1, ψ
Maximum likelihood esimaion of sae-space models 11 Hence, θ 1 : M w,w M k,1 ρ θ G 1 ρ 1 G θ Ω θ θ Ω G θ 1 ρ 1 G θ θ G 1 ρ 1 G θ Ω θ θ R 1 ρ 1 Compuaion of ˆX θ : ˆX θ ρ, ψ = ρ, θ ˆX ψ = ρ, θ F 1 ψ ˆX 1 θ + ρ, F 1 θ θ ˆX1 ψ + ρ, θ Θ 1 ψ 1 1 Y θ 1 G 1 θ ˆX 1 θ + ρ, Θ 1 θ θ 1 1 Y ψ 1 G 1 θ ˆX 1 θ + ρ, Θ 1 θ 1 1 h θ θ G 1 ψ ˆX i 1 θ G 1 θ θ ˆX1 ψ = θ F 1 ρ ˆX 1 θ, ψ + ˆX θ 1 `F 1 θ ρ, ψ + θ Θ 1 ρ Y 1 ˆX 1 θ G 1 θ 1 1 θ, ψ + θ 1 1 Θ 1 θ ρ Y 1 ˆX 1 θ G 1 θ, ψ θ G 1 1 1 θ Θ 1 θ ρ ˆX 1 θ, ψ ˆX θ 1 G 1 θ 1 1 θ Θ 1 θ ρ, ψ Hence, ˆX θ : M r,1 M k,1 ρ θ F 1 ρ ˆX 1 θ + ˆX θ 1 `F 1 θ ρ + θ Θ 1 ρ Y 1 ˆX 1 θ G 1 θ 1 1 θ θ 1 1 Θ 1 θ ρ Y 1 ˆX 1 θ G 1 θ θ G 1 1 1 θ Θ 1 θ ρ ˆX 1 θ ˆX θ 1 G 1 θ 1 1 θ Θ 1 θ ρ Compuaion of θ I : θ I ρ, ψ = ρ, θi ψ = ρ, θ G ψ ˆX θ G θ θ ˆX ψ = r ρ θ G ψ ˆX θ r ρ G θ θ ˆX ψ = θ ρ G ˆX θ, ψ ˆX θ `G θ ρ, ψ Hence, θ I : M w,1 M k,1 ρ θ G ρ ˆX θ θ ˆX G θ ρ 5 Proof of Proposiion The goal of his proposiion is obaining recursive consrucive formulas for he marices associaed o he dual linear maps θ Ω, θ Θ, θ 1, ˆX θ, and θi ou of he recursions provided in Theorem 1 ha define he evaluaions of hose linear maps
Maximum likelihood esimaion of sae-space models 1 Compuaion of θ Ω mar,r : Rr R k : θ Ω mar,r v, ψ = θ F 1 `mar,r v F 1 θ Ω 1 θ + ma r,r v F 1 θ Ω 1 θ, ψ Hence, expression 1 follows from: + θ Ω 1 `F 1 θ ma r,r v F 1 θ, ψ + θq mar,r v, ψ θ Θ 1 ma r,r v Θ 1 θ 1 1 θ + mar,r v Θ 1 θ 1 1 θ, ψ θ 1 1 `Θ1 θ ma r,r v Θ 1 θ, ψ = θ F 1 ma r,r ``Ω1 θ F 1 θ I r `v + vec `mar,r v, ψ + θ Ω 1 ma r,r ``F 1 θ F 1 θ v, ψ + θq mar,r v, ψ θ Θ 1 ma r,w 1 1 θ Θ 1 θ I r `v + vec `mar,r v, ψ θ 1 1 ``Θ1 maw,w θ Θ 1 θ v, ψ = θ F 1 ma r,r ``Ω1 θ F 1 θ I r Ir + K r,r v, ψ + θ Ω 1 ma r,r ``F 1 θ F 1 θ v, ψ + θq mar,r v, ψ θ Θ 1 ma r,w 1 1 θ Θ 1 θ I r I r + K r,r v, ψ θ 1 1 ``Θ1 maw,w θ Θ 1 θ v, ψ θ Ω mar,r : Rr R k v θ F 1 ma r,r ``Ω1 θ F 1 θ Ir Ir + K r,r v + θ Ω 1 ma r,r ``F 1 θ F 1 θ v + θq mar,r v θ Θ 1 ma r,w 1 1 θ Θ 1 θ I r I r + K r,r v θ 1 1 maw,w Θ 1 θ Θ 1 θ v Compuaion of θ Θ mar,w : Rrw R k : θ Θ mar,w v, ψ = θ F mar,w v G θ Ω θ + θ Ω `F θ ma r,w v G θ Hence, expression 13 follows from: = = + θ `mar,w G v F θ Ω θ, ψ θ F ``Ω mar,r θ G θ I r v + θ Ω ma r,r ``G θ F θ v + θ G ``Ω maw,r θ F θ Iw vec `mar,w v, ψ θ F ``Ω mar,r θ G θ I r v + θ Ω ma r,r ``G θ F θ v + θ G ``Ω maw,r θ F θ Iw Kr,wv, ψ θ Θ mar,w : Rrw R k v θ F mar,r Ω θ G θ Ir v + θ Ω mar,r G θ F θ v + θ G maw,r Ω θ F θ Iw Kr,wv
Maximum likelihood esimaion of sae-space models 13 Compuaion of θ 1 ma w,w : R w R k : θ 1 ma w,w v, ψ = θ G 1 ma w,w v 1 G θ Ω θ, ψ θ Ω G θ 1 ma w,w v 1 G θ, ψ θ G Ω θ G θ 1 ma w,w v 1, ψ θ R 1 ma w,w v 1, ψ = θ G maw,r Ω θ G θ 1 1 v, ψ θ Ω mar,r G θ 1 G θ 1 v, ψ θ G mar,w 1 Ω θ G θ 1 v, ψ θ R maw,w 1 1 v, ψ Hence, expression 14 follows from: θ 1 ma w,w : R w R k ρ θ G maw,r Ω θ G θ 1 1 v θ Ω mar,r G θ 1 G θ 1 v θ G mar,w 1 Ω θ G θ 1 v θ R maw,w 1 1 v Compuaion of ˆX θ ma r,1 : R r R k : ˆX θ ma r,1 v, ψ = θ F 1 ma r,1 v ˆX 1 θ, ψ + ˆX θ 1 `F 1 θ ma r,1 v, ψ + θ Θ 1 ma r,1 v Y 1 ˆX 1 θ G 1 θ 1 1 θ, ψ + θ 1 1 Θ 1 θ ma r,1 v Y 1 ˆX 1 θ G 1 θ, ψ θ G 1 1 1 θ Θ 1 θ ma r,1 v ˆX 1 θ, ψ ˆX θ 1 G 1 θ 1 1 θ Θ 1 θ ma r,1 v, ψ = θ F 1 ma r,r ˆX1 θ I r v, ψ + ˆX θ 1 ma r,1 ``I1 F 1 θ v, ψ + θ Θ 1 ma r,w 1 1 Y θ 1 G 1 θ ˆX 1 θ I r v, ψ + θ 1 1 Y maw,w 1 G 1 θ ˆX 1 θ Θ 1 θ v, ψ θ G 1 ma w,r ˆX1 θ 1 1 θ Θ 1 θ v, ψ ˆX θ 1 ma r,1 I 1 G 1 θ 1 1 θ Θ 1 θ v, ψ Hence, expression 15 follows from: ˆX θ ma r,1 : R r R k ρ θ F 1 ma r,r ˆX1 θ I r v + ˆX θ 1 ma r,1 ``I1 F 1 θ v + θ 1 1 Y mar,w 1 G 1 θ ˆX 1 θ Θ 1 θ v θ G 1 ma w,r ˆX1 θ 1 1 θ Θ 1 θ v ˆX θ 1 ma r,1 I 1 G 1 θ 1 1 θ Θ 1 θ v Compuaion of θ I ma w,1 : R w R k : θ I ma w,1 v, ψ = θ ma G w,1 v ˆX θ ˆX θ `G θ ma w,1 v, ψ = θ G maw,r ˆX θ I w v ˆX θ ma r,1 ``I1 G θ v, ψ
Maximum likelihood esimaion of sae-space models 14 Hence, expression 16 follows from: θ I ma w,1 : R w R k v θ G maw,r ˆX θ I w v ˆX θ ma r,1 I 1 G θ v References [1] Åsröm, K J Maximum likelihood and predicion error mehods Auomaica 16, 5 Sep 198, 551 574 [] Brockwell, P J, and avis, R A Inroducion o ime series and forecasing Springer, [3] Brockwell, P J, and avis, R A Time Series: Theory and Mehods Springer-Verlag, 6 [4] Cramér, H Mahemaical Mehods of Saisics Princeon Universiy Press, 1946 [5] Gardner, G, Harvey, A C, and Phillips, G A An Algorihm for xac Maximum Likelihood simaion of Auoregressive-Moving Average Models by Means of Kalman Filering Journal of he Royal Saisical Sociey Series C Applied Saisics 9, 3 198, 311 3 [6] Harvey, A C, Ruiz,, and Shephard, N Mulivariae sochasic variance models Review of conomic Sudies 61 1994, 47 64 [7] Jones, R H Maximum Likelihood Fiing of ARMA Models o Time Series wih Missing Observaions Technomerics, 3 198, 389 395 [8] Kalman, R A new approach o linear filering and predicion problems Trans ASM, J Basic ngineering 8 196, 35 45 [9] Lükepohl, H New inroducion o muliple ime series analysis Springer-Verlag, Berlin, 5 [1] Sandell, N R, and Yared, K I Maximum likelihood idenificaion of sae space models for linear dynamic sysems lecronic Sysems Laboraory, ep of lecrical ngineering and Compuer Science, Massachuses Insiue of Technology R-814 1978 [11] Segal, M; Weinsein, A new mehod for evaluaing he log-likelihood gradien, he Hessian, and he Fisher informaion marix for linear dynamic sysems I Trans Inform Theory 35, 3 1989, 68 687 [1] Wald, A Noe on he consisency of he maximum likelihood esimae The Annals of Mahemaical Saisics 1949, 595 61