Time series model fitting via Kalman smoothing and EM estimation in TimeModels.jl

Transcription

1 Time series model fiing via Kalman smoohing and EM esimaion in TimeModels.jl Gord Sephen Las updaed: January 206 Conens Inroducion 2. Moivaion and Acknowledgemens Noaion Kalman Smoohing 3 2. Predicion and Filering Smoohing Lag- Covariance Smooher Linear Consrain Paramerizaion 4 3. Sysem Marix Decomposiion Linear Consrain Formulaion Sysem Equaion Represenaion Expecaion-Maximizaion Parameer Esimaion 7 4. Basic Log-likelihood Derivaion Log-likelihood Derivaion wih Zero-variance Saes or Observaions Solving for p A Solving for p B Solving for p Q Solving for p C Solving for p D Solving for p R Solving for Solving for p S

2 Inroducion. Moivaion and Acknowledgemens While he expecaion-maximizaion (EM) opimizaion approach is well-recognized in ime-series analysis exs as a useful mehod for obaining local maximum-likelihood (ML) parameer esimaes of linear sae-space model sysem marices, avoiding he need o resor o general gradien-free opimizaion echniques, he procedure is generally only described as a means o generae maximum likelihood esimaes of full sysem marices. In many pracical cases, however, mos elemens of a sae-space model s sysem marices are se deerminisically, wih only a small subse requiring esimaion (one noable example of such a case is he fiing of ARIMA models). In such siuaions, allowing all elemens of he sysem marix o vary during he fiing process is clearly undesirable. A common soluion o his challenge is o simply disregard analyical knowledge of he sysem and fall back on gradien-free nonlinear opimizaion echniques o compue maximumlikelihood esimaes for he unknown parameers. This can be avoided, however, by paramerizing sysem marices as reshaped linear ransformaions of smaller parameer vecors, hen applying he EM esimaion algorihm o he resuling sysem equaions o generae ML esimaes for he parameer vecors alone. This model fiing approach was firs oulined by Holmes and implemened as he MARSS R package (hps://cran. r-projec.org/web/packages/marss/index.hml). This documen re-derives hese echniques and describes heir implemenaion in he TimeModels.jl Julia package (hps://gihub.com/juliasas/timemodels.jl). The resuls presened here are based very heavily on Holmes original derivaions (available a hps:// cran.r-projec.org/web/packages/marss/vignees/emderivaion.pdf), wih some gaps in derivaion seps filled, and changes o noaion and he reamen of exernal sysem inpus ha will be more familiar o readers approaching sae-space modelling from a conrol-heory background. Like TimeModels.jl, his documen is very much a work in progress. In paricular, general conrains for solvable sysems (while usually alluded o in he various derivaions) are no ye deailed explicily. The reader is insead referred o secion 0 of Holmes derivaions, which provides a relevan summary of such consrains, albei wih he differen noaion and exernal inpu expressions menioned above. In some cases where derivaions are idenical o hose given by Holmes (noaional differences aside), only he final resuls are given..2 Noaion We define a general mulivariae sae-space process wih Gaussian noise according o he following noaion: x = A x + B u + v, v N (0, V ), = 2... n 2

3 y = C x + D u + w, w N (0, W ), =... n x N (x 0, P 0 ) Here, x R nx is an unobserved sae vecor, y R ny is an observaion vecor wih possible missing values, and u R nu is a deerminisic inpu vecor. A is he sae ransiion marix, B is he conrol inpu marix, and V is he sae ransiion noise covariance marix. C is he observaion marix, D he feed-forward marix, and W he observaion noise covariance marix. Each sysem marix can be ime-varying. x 0 and P 0 are iniial sae value and covariance condiions needed o iniiae he recursive process evoluion. 2 Kalman Smoohing Given he observaion ime series y (possibly conaining missing values), he inpu ime series u, and values for he sysem marices, we can applying Kalman smoohing o compue he probabilisic disribuion of he laen sae ime series x. Afer Kalman smoohing, he esimaed laen sae ime series values will be disribued as x N (ˆx, P ). This disribuion is obained by aking ino accoun all observaions in y. Obaining hese esimaes also requires compuing and soring a prediced sae disribuion x + N (x +, P + ), based only on observaions y... y. 2. Predicion and Filering x +, P +, he Kalman gain marix K, and he marginal log-likelihood ll of he observed series given he sysem marices can be compued via forward recursion saring from iniial condiions x 0, P 0 : ɛ = y C x D u Σ = C P C T + W K = A P C T Σ ll = ɛ T Σ ɛ + ln Σ x + = A x + B u + K ɛ P+ = A P (A K C ) T + V 3

4 2.2 Smoohing Now, given x, P, K, and Σ, smoohed values ˆx and P can be compued via backwards recursion, wih iniial condiions r n = 0 nx and N n = 0 nx,nx : L = A K C r = C T Σ ɛ + L T r N = C T Σ C + L T N L ˆx = x P = P + P r P N P 2.3 Lag- Covariance Smooher The EM parameer esimaion procedure described below also requires P,, he lag- sae covariance beween x and x. This can be compued by smoohing a corresponding sacked sae space model given as follows: I nx 0 nx,n x x = [ ] A 0 Ã = nx,nx, B = [ x x [ B 0 nx,n u ] ], Ṽ = C = [ C 0 ny,n x ] [ V 0 nx,n x 0 nx,n x 0 nx,n x ] x = Ã x + B u + ṽ, ṽ N (0, Ṽ ) y = C x + D u + w, w N (0, W ) The resuling x sae covariance P will be a block marix of he form: from which P, can be easily exraced. [ ] P P P =, P, P 4

5 3 Linear Consrain Paramerizaion 3. Sysem Marix Decomposiion To paramerize and esimae he ime-dependen sysem marices, we firs facor hem ino componens ha are eiher ime-independen and paramerized, or ime-dependen and explicily defined: A = A A 2 pa A 3, B = B B 2 pb, V = G Q pq G T C = C C 2 pc C 3, D = D D 2 pd, W = H R pr H T where p A, p B, p Q, p C, p D, and p R are parameer vecors o be esimaed. We require ha Q pq and R pr be inverible in addiion o being valid covariance marices (symmeric and posiive-semidefinie). 3.2 Linear Consrain Formulaion We formalize he linear consrains on he paramerized marices as: vec(z pz ) = f Z + D Z p Z where f Z and D Z represen he consan and parameer coefficien erms making up Z. 3.3 Sysem Equaion Represenaion We define Φ = (G T G ) G T such ha: Φ x = Φ A x + Φ B u + Φ G q Φ x = Φ A x + Φ B u + q Using Kronecker produc ideniies, we can now represen he general sysem equaions in erms of he parameers o be esimaed: vec(φ x ) = vec(φ A x ) + vec(φ B u ) + vec(q ) Using Ab = vec(ab) = (b T I na )vec(a), where A is an n a n b marix and b is a lengh-n b vecor: Φ x = Φ vec(a x ) + Φ vec(b u ) + q 5

6 Φ x = Φ (x T I nx )vec(a ) + Φ (u T I nx )vec(b ) + q Φ x = Φ (x T I nx )vec(a A 2 pa A 3 ) + Φ (u T I nx )vec(b B 2 pb ) + q Using vec(abc) = (C T A)vec(B): Φ x = Φ (x T I nx )(A T 3 A )vec(a 2 pa )+Φ (u T I nx )vec(b B 2 pb )+q Using vec(ab) = (I nc A)vec(B) where B is an n b n c marix:: Φ x = Φ (x T I nx )(A T 3 A )vec(a 2 pa )+Φ (u T I nx )(I nu B )vec(b 2 pb )+q Using he linear consrain definiion vec(z pz ) = f Z + D Z p Z : Φ x = Φ (x T I nx )(A T 3 A )(f A +D A p A )+Φ (u T I nx )(I nu B )(f B +D B p B )+q Finally, we can consolidae he deerminisic elemens by defining f A, = (A T 3 A )f A D A, = (A T 3 A )D A f B, = (I nu B )f B D B, = (I nu B )D B which resuls in Φ x = Φ (x T I nx )(f A, +D A, p A )+Φ (u T I nx )(f B, +D B, p B )+q The second sysem equaion follows from he same process and gives: Ξ = (H T H ) H T f C, = (C3 T C )f C D C, = (C3 T C )D C f D, = (I nu D )f B D D, = (I nu D )D B Ξ y = Ξ (x T I ny )(f C, + D C, p C ) + Ξ (u T I ny )(f D, + D D, p D ) + r 6

7 The paramerized iniial sae and error covariances are given similarly, as follows: x = x 0 + ζ, ζ N (0, P 0 ), P 0 = JSJ x = x 0 + Js, s N (0, S) Inroducing he usual linear parameer consrains we ge x 0 = f x 0 + D x 0, S = f S + D S p S Like Q and R, S mus be boh inverible and a valid covariance marix. Finally, defining Π = (J J) J gives Πx = Πx 0 + s = Π(f x 0 + D x 0 ) + s 4 Expecaion-Maximizaion Parameer Esimaion 4. Basic Log-likelihood Derivaion Given he paramerized sysem equaions and aking errors o be normally-disribued, he log-likelihood of he saes and observaions given he marix parameers can be represened: ɛ = Φ x Φ (x T I nx )(f A, + D A, p A ) Φ (u T I nx )(f B, + D B, p B ) ɛ = Φ (x (x T I nx )f A, (x T I nx )D A, p A (u T I nx )f B, (u T I nx )D B, p B ) f = x (x T I nx )f A, (u T I nx )f B, a = (x T I nx )D A, b = (u T I nx )D B, ɛ = Φ (f a p A b p B ) η = Ξ y Ξ (x T I ny )(f C, + D C, p C ) Ξ (u T I ny )(f D, + D D, p D ) η = Ξ (y (x T I ny )f C, (x T I ny )D C, p C (u T I ny )f D, (u T I ny )D D, p D ) g = y (x T I ny )f C, (u T I ny )f D, c = (x T I ny )D C, d = (u T I ny )D D, 7

8 η = Ξ (g c p C d p D ) ξ = Πx Π(f x 0 + D x 0 ) LL = 2 =2 ɛ Q ɛ 2 ln Q 2 =2 = η R η 2 ln R 2 ξ S ξ 2 ln S n ln 2π 2 For his and he following secions, i will be useful o define Ṽ = Φ Q Φ, W = Ξ R Ξ, and P 0 = Π S Π. Expanding he above gives: LL = = =2 = (f + a p A + b p B ) T Ṽ (f + a p A + b p B ) + 2 (g + c p C + d p D ) T W (g + c p C + d p D ) + 2 ln Q =2 ln R + 2 (f x 0 + D x 0p x 0) P 0 (fx 0 + D x 0p x 0) + 2 ln S + n ln 2π 2 = 4.2 Log-likelihood Derivaion wih Zero-variance Saes or Observaions We also need o accoun for he possibiliy ha some sae or observaion rows are deerminisicallydefined (wih G or H having a one or more all-zero-valued rows), and hus do no make any probabilisic conribuion o he log-likelihood calculaion and canno be esimaed via maximum-likelihood mehods. While he values of elemens of G and H can vary in ime, i is assumed heir overall rows will remain consisenly eiher all-zero or no-all-zero. There is addiional nuance needed in considering ha errors inroduced in non-deerminisic sae or observaion rows have he poenial o propagae o seemingly-deerminisic rows via he A and C marices. Holmes provides he full deails of his derivaion (along wih explanaions for associaed esimaion limiaions) and so for breviy only he broad srokes are presened here. Given an n x n x selecion marix I d ha zeros ou any rows ha are all-zero in G, he fully-deerminisic saes x d 2 can be expressed as: x d 2 = I d 2x 2 = I d 2(A x + B u ) x d 2 = I d 2(A x + (u T I nx )f B, + (u T I nx )D B, p B ) Similarly: x d 3 = I d 3(A 2 x 2 + (u T 2 I nx )f B,2 + (u T 2 I nx )D B,2 p B ) 8

9 Now, recursing on x 2 : x d 3 = I d 3(A 2 (A x +(u T I nx )f B, +(u T I nx )D B, p B )+(u T 2 I nx )f B,2 +(u T 2 I nx )D B,2 p B ) x d 3 = I d 3(A 2 A x +(u T 2 I nx )f B,2 +A 2 (u T I nx )f B, +((u T 2 I nx )D B,2 +A 2 (u T I nx )D B, )p B ) More generally, x d can be expressed via he recursion relaions x d = I d (A x 0 + f Bu, + D Bu, p B ) A = A A, A 0 = I nx f Bu, = (u I nx )f B, + A f Bu,, f Bu,0 = 0 D Bu, = (u I nx )D B, + A D Bu,, D Bu,0 = 0 Wih his recursion we will also need o accoun for he probabilisic and non-deerminisic componens of he expecaion of x. Specifically, defining I d as a selecion marix zeroingou non-deerminisic rows in x 0 (corresponding o non-zero rows in x 0 J): ˆx 0 = (I nx I d x )ˆx 0 + I d x x 0 0 = (I nx I d x )ˆx 0 + I d x(f 0 x 0 + D x 0 ) We can now consider he general log-likelihood formula wih deerminisic rows se o zero. Φ will auomaically zero ou likelihood conribuions from deerminisic x rows in he process evoluion erms, while Ξ will do he same for y in he observaion erms and Π for he iniial sae x 0. We can apply he xd relaion derived above o replace x and x in he process and observaion erms respecively. ɛ = Φ (x A x B u ) ɛ = Φ (x A (I nx I d )x A I d x B u ) ɛ = Φ (x A (I nx I d )x A x d B u ) ɛ = Φ (x A (I nx I d )x A I d (A 2x 0 + fbu, 2 + DBu, 2p B ) B u ) η = Ξ (y C x D u ) η = Ξ (y C (I nx I d )x C I d x D u ) η = Ξ (y C (I nx I d )x C x d D u ) η = Ξ (y C (I nx I d )x C I d (A x 0 + fbu, + DBu, p B ) D u ) ξ = Π(x f x 0 D x 0 ) 9

10 LL = 2 =2 ɛ T Q ɛ + 2 ln Q + 2 =2 = + 2 ξ S ξ + 2 ln S + n ln 2π 2 η T R η + 2 ln R From his poin we can compue derivaives wih respec o o-be-esimaed parameer vecors, and se he expeced values (expecaion) o zero (maximizaion). Solving for all parameers and using hem o recompue expeced values gives a new model wih improved fi log-likelihood. This process can be repeaed ieraively unil he log-likelihood value converges sufficienly. = 4.3 Solving for p A Same for finie-variance rows, doesn work for fixed rows. Final resul is: ( p A = DA,( x x Q ) )D A, = = D A,( vec( Q x x ) ( x x Q )f A, vec( Q B u ˆx ) ) 4.4 Solving for p B We sar by aking he derivaive of negaive log-likelihood wih respec o p B. Using he marix derivaive rule a a Ca = 2a C: ( LL) = 2 ( LL) = =2 ɛ T Q ɛ + 2 ɛ T Q ɛ + =2 = η T R η η T R η = ɛ = Φ (A I d D Bu, 2 + (u T I nx )D B, ) η = Ξ C I d D Bu, ( LL) = ɛ T Q Φ (A I d DBu, 2+(u T I nx )D B, ) η T R Ξ C I d DBu, =2 = ( LL) ( = x A (I nx I d )x A I d (A 2x + fbu, 2 + DBu, 2p B ) =2 0

11 (u T I nx )f B, (u T )T I nx )D B, p B Ṽ (A I d DBu, 2 + (u T I nx )D B, ) ( y C (I nx I d )x C I d (A x + fbu, + DBu, p )T B ) D u W C I d DBu, = [ ] Taking expecaions and seing E ( LL) = 0: 0 = (ˆx A (I nx I d )x A I d (A 2ˆx 0 + fbu, 2 + DBu, 2p B ) =2 (u T I nx )f B, (u T )T I nx )D B, p B Ṽ (A I d DBu, 2 + (u T I nx )D B, ) (ŷ C (I nx I d )x C I d (A ˆx 0 + fbu, + DBu, p )T B ) D u W C I d DBu, + = =2 0 = (ˆx A (I nx I d )x A I d (A 2ˆx 0 + fbu, 2) =2 (u T I nx )f B, )T Ṽ (A I d D Bu, 2 + (u T I nx )D B, ) ( A I d D Bu, 2p B +(u T I nx )D B, p B )T Ṽ (A I d D Bu, 2+(u T I nx )D B, ) +p B =2 = (ŷ C (I nx I d )x C I d (A ˆx 0 + fbu, ) )T D u W C I d DBu, 0 = + = ( C I d DBu, p )T B W C I d DBu, (ˆx A (I nx I d )x A I d (A 2ˆx 0 + fbu, 2) =2 (u T I nx )f B, )T Ṽ (A I d D Bu, 2 + (u T I nx )D B, ) ( A I d D Bu, 2+(u T I nx )D B, ) Ṽ (A I d D Bu, 2+(u T I nx )D B, ) = (ŷ C (I nx I d )x C I d (A ˆx 0 + fbu, ) )T D u W C I d DBu, For clariy we can define: +p B = ( C I d DBu, )T W C I d DBu,, = A I d D Bu, 2 + (u T I nx )D B,

12 2, = ˆx A (I nx I d )x A I d (A 2ˆx 0 + f Bu, 2) (u T I nx )f B, This gives: 0 = =2 3, = C I d D Bu, 4, = ŷ C (I nx I d )x C I d (A ˆx 0 + f Bu, ) D u 2,Ṽ, + p B =2,Ṽ, = 4, W 3, + p B = 3, W 3, ( p B =,Ṽ, + =2 = 3, W 3, ) ( =2,Ṽ 2, + = 3, W 4, ) 4.5 Solving for p Q The soluion for p Q is no general, bu works in a wide range of special cases, mos noably when Q is a simple diagonal marix of parameers, which when muliplied wih any arbirary G can serve mos purposes. Aside from noaional differences, he derivaion is idenical o he one presened in Holmes, so for breviy only he final resul is presened here: ( ) p Q = DQ,D Q, = DQ,vec(S ) = S = Φ ( x x + x x A + A x x ˆx u B B u ˆx + A x x A +A ˆx u B + B u ˆx A + B u u B )Φ 4.6 Solving for p C The soluion for p C is analogous o p A : ( p C = DC,( x x R ) )D C, = = DC, ( vec( R ŷ x ) ( x x R )f C, vec( R D u ˆx ) ) 2

13 4.7 Solving for p D Finie-variance rows works as normal, wih consrains This derivaion isn in Holmes: ( ) p D = DD,(u I ny ) R (u I ny )D D, DD,(u I ny ) R (ŷ C ˆx (u ) I ny )f A, = = 4.8 Solving for p R p R is fully analogous o p Q, and he same marix propery consrains apply. ( ) p R = D R, DR, DR,vec(T ) = T = Ξ ( y y + ŷx C + C x y ŷ u D D u ŷ + C x x C = +C ˆx u D + D u ˆx C + D u u D )Ξ 4.9 Solving for Solving for is paricularly messy given how ofen i appears in he log-likelihood equaion! The derivaion for he final resul is very similar o he one used o solve for p B : ( LL) = 2 ( LL) = =2 ɛ T Q ɛ + 2 ɛ T Q =2 ɛ + = = η T R η + 2 η T R ξ S ξ η + ξ S ξ ɛ = Φ A I d p A 2I d x D 0 x 0 x 0 Φ 5, η = Ξ C I d A p I d x D 0 x 0 x 0 Ξ 7, ξ = ΠD x 0 ( LL) = =2 ɛ Q Φ 5, = η R Ξ 7, ξ S ΠD x 0 3

14 [ ( LL) ] E = 0 = =2 E[ɛ ] Q Φ 5, = E[η ] R Ξ 7, E[ξ] S ΠD x 0 E[ɛ ] = Φ (ˆx A (I nx I d )ˆx A I d (A 2( (Inx I d x)ˆx 0 +I d x (f 0 x 0 +D x 0 ) ) +fbu, 2+ D Bu, 2p B ) B u ) E[η ] = Ξ (ŷ C (I nx I d )ˆx C I d (A ( (Inx I d x)ˆx 0 +I d x (f 0 x 0 +D x 0 ) ) +fbu, +D Bu, p B ) D u ) E[ξ] = Π(ˆx f x 0 D x 0 ) 0 = = =2 (ˆx A (I nx I d )ˆx A I d (A 2( (Inx I d x)ˆx 0 +I d x (f 0 x 0 +D x 0 ) ) +fbu, 2+ D Bu, 2p B ) B u ) Ṽ 5, (ŷ C (I nx I d )ˆx C I d (A ( (Inx I d x)ˆx 0 +I d x (f 0 x 0 +D x 0 ) ) +fbu, +D Bu, p B ) D u ) W 7, (ˆx f x 0 D x 0 ) P 0 Dx 0 0 = = =2 (ˆx A (I nx I d )ˆx A I d (A 2( (Inx I d x)ˆx 0 + I d ) x f 0 x 0 + f Bu, 2 + D Bu, 2p B ) B u ) Ṽ 5, + (A I d A 2I d x D 0 x 0 ) Ṽ 5, =2 (ŷ C (I nx I d )ˆx C I d (A ( (Inx I d x)ˆx 0 +I d ) x f 0 x 0 +f Bu, +DBu, p B ) D u ) W 7, + (C I d A I d x D 0 x 0 ) W 7, = (ˆx f x 0 ) P 0 Dx 0 + (D x 0p x 0) P 0 Dx 0 6, ˆx A (I nx I d )ˆx A I d (A 2( (Inx I d x)ˆx 0 +I d ) x f 0 x 0 +f Bu, 2 +DBu, 2p B ) B u 4

15 8, ŷ C (I nx I d )ˆx C I d (A ( (Inx I d x)ˆx 0 +I d ) x f 0 x 0 +f Bu, +DBu, p B ) D u =2 + 0 = = =2 6,Ṽ 5, + ( 7, ) W ( 5, ) Ṽ 5, + =2 =2 ( 5, ) Ṽ 5, 7, (ˆx f x 0 ) = 6,Ṽ 5, + ( 7, ) W = 8, Transposing he equaion and isolaing : ( =2 =2 5,Ṽ 5, + 5,Ṽ 6, + = = ( = 5,Ṽ 5, + ( =2 =2 5,Ṽ 6, + = P 0 Dx 0 = 8, + (D x 0 ) W 7, 7, + D x 0 + (D x 0 ) W 7, + (ˆx f x 0 ) P 0 Dx 0 7, W 7, W = 7, + D x 0 P 0 Dx 0 P 0 Dx 0 ) = 8, + D x P 0 0 (ˆx f x 0 7, W 7, W 7, + D x 0 P 0 Dx 0 P 0 Dx 0 = ) ) 8, + D x P 0 ) 0 (ˆx f x 0 ) 4.0 Solving for p S p S can be solved for in a very similar fashion o p Q and p R, alhough his is no done explicily in Holmes: p S = (D S D S ) D S vec ( Π( x x ˆx x 0 x 0 ˆx + x 0 x 0 )Π ) 5