Approximate Conditional-mean Type Filtering for State-space Models - PowerPoint PPT Presentation

Approximate Conditional-mean Type Filtering for State-space Models Bernhard Spangl, Universit¨ at f¨ ur Bodenkultur, Wien joint work with Peter Ruckdeschel, Fraunhofer Institut, Kaiserslautern , and Rudi Dutter, Technische Univerist¨ at, Wien UseR! 2008, Dortmund, Germany B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 1/20

Contents State-space models & Kalman filter Multivariate ACM-type filter rLS filter Simulation study Results R package robKalman Remarks & Outlook B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 2/20

Linear State Space Models State equation: = Φ x t − 1 + ε t x t Observation equation: = Hx t + v t y t Ideal model assumptions: x 0 ∼ N p ( µ 0 , Σ 0 ) , ε t ∼ N p ( 0 , Q ) , v t ∼ N q ( 0 , R ) , all independent B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 3/20

Classical Kalman Filter Initialization ( t = 0) : x 0 | 0 = µ 0 , P 0 = Σ 0 Prediction ( t ≥ 1) : = x t | t − 1 Φ x t − 1 | t − 1 Φ P t − 1 Φ ⊤ + Q = Cov ( x t | t − 1 ) = M t Correction ( t ≥ 1) : x t | t − 1 + K t ( y t − Hx t | t − 1 ) = x t | t M t − K t HM t = Cov ( x t | t ) = P t with K t = M t H ⊤ ( HMH ⊤ + R ) − 1 (Kalman gain) B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 4/20

Types of Outliers Innovational Outliers (IO’s): state equation is contaminated not considered here Additive Outliers (AO’s): observations are contaminated error process v t is affected possible model: CN q ( γ, R , R c ) = (1 − γ ) N q ( 0 , R ) + γ N q ( µ c , R c ) Other Types of Outliers: substitutive outliers (SO’s) patchy outliers B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 5/20

Masreliez’s Theorem (1975) If x t | Y t − 1 ∼ N p ( x t | t − 1 , M t ) , t ≥ 1 , then x t | t = E ( x t | Y t ) , t ≥ 1 , is generated by the recursions x t | t − 1 + M t H ⊤ Ψ t ( y t ) = x t | t M t − M t H ⊤ Ψ ′ = t ( y t ) HM t P t Φ P t Φ ⊤ + Q , = M t +1 with (Ψ t ( y )) i = − ( ∂/∂y i ) log f y t ( y | Y t − 1 ) and (Ψ ′ t ( y )) ij = ( ∂/∂y j )(Ψ t ( y )) i . Ψ t ( y ) is called the score function. Note: If f y t ( . | Y t − 1 ) is Gaussian, Masreliez’s filter reduces to the Kalman filter. B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 6/20

The Score Function Ψ t (a) (b) 4 2 0.10 y 0 0.05 −2 4 2 0 −4 y −4 −2 −2 0 2 −4 x 4 −4 −2 0 2 4 x (c) (d) 2 4 1 2 d/dx d/dy 0 0 −1 −2 4 4 2 2 −2 −4 0 0 y y −4 −4 −2 −2 −2 −2 0 0 2 −4 2 −4 x x 4 4 B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 7/20

Multivariate ACM-type Filter approximate conditional-mean (ACM) type filter proposed by B. Spangl and R. Dutter (2008) modified correction step: x t | t − 1 + M t H ⊤ S t ψ ( S t ( y t − Hx t | t − 1 )) = x t | t M t − M t H ⊤ S t ψ ′ ( S t ( y t − Hx t | t − 1 )) S t HM t = P t for an S t and a ψ -function appropriately chosen in the case univariate observations equivalent to Martin’s ACM type filter (Martin, 1979) B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 8/20

Huber’s Multivariate Psi-function (a) (b) 2 2 1 1 x−coord. y−coord. 0 0 −1 −1 4 4 −2 −2 2 2 0 0 y y −4 −4 −2 −2 −2 −2 0 0 2 2 −4 −4 x x 4 4 B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 9/20

Hampel’s Multivariate Psi-function (a) (b) 2 2 1 1 x−coord. y−coord. 0 0 −1 −1 4 4 −2 −2 2 2 0 0 y y −4 −4 −2 −2 −2 −2 0 0 2 2 −4 −4 x x 4 4 B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 10/20

Approximating the Score Function (a) (b) 1 2 d/dx d/dy 0 0 −1 −2 4 4 2 2 0 0 y y −4 −4 −2 −2 −2 −2 0 0 2 −4 2 −4 x x 4 4 (c) (d) 2 1 x−coord. y−coord. 0 0 −1 −2 4 4 2 2 0 0 y y −4 −4 −2 −2 −2 −2 0 0 2 −4 2 −4 x x 4 4 B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 11/20

rLS Filter proposed by P . Ruckdeschel (2001) modified correction step: = x t | t − 1 + H b ( K t ( y t − Hx t | t − 1 )) x t | t with H b ( z ) = z min { 1 , b/ � z � 2 } and � . � 2 the Euclidean norm optimal for SO’s in some sense B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 12/20

Simulation State Space Process: simulate state space process using two different sets of hyper parameters and AO’s from two different contamination setups: � ✂ � ✂ � ✂ � ✂ 0 100 0 25 0 . 9 0 N 2 ( ) or N 2 ( ) . ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ , , 0 0 100 30 0 0 . 9 vary contamination γ from 0% to 20% by 5% each 400 times Filtering: robust filtering (ACM, rLS) Evaluation: compare with true state process via MSE B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 13/20

Simulation (cont.) Example I: � ✂ � ✂ � ✂ 0 0 . 5 0 . 3 3 2 µ 0 = Φ = Q = ✁ ✄ ✁ ✄ ✁ ✄ , , , 0 0 . 6 0 . 5 2 3 � ✂ � ✂ � ✂ 0 0 1 − 1 2 − 0 . 2 Σ 0 = H = R = ✁ ✄ ✁ ✄ ✁ ✄ , , . 0 0 0 1 − 0 . 2 0 . 5 Example II: � ✂ � ✂ � ✂ 20 1 1 0 0 µ 0 = Φ = Q = ✁ ✄ ✁ ✄ ✁ ✄ , , , 0 0 0 0 9 � ✂ � ✂ � ✂ 0 0 0 . 3 1 9 0 Σ 0 = H = R = ✁ ✄ ✁ ✄ ✁ ✄ , , . 0 0 − 0 . 3 1 0 9 B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 14/20

Results contamination level 0% contamination level 10% contamination level 20% 20 20 20 1. coordinate of state process 1. coordinate of state process 1. coordinate of state process 10 10 10 0 0 0 −10 −10 −10 true true true class. class. class. rLS rLS rLS −20 −20 −20 ACM ACM ACM 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 time time time contamination level 0% contamination level 10% contamination level 20% 2. coordinate of state process 2. coordinate of state process 2. coordinate of state process 10 10 10 0 0 0 true true true −10 −10 −10 class. class. class. rLS rLS rLS ACM ACM ACM 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 time time time B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 15/20

Results (cont.) contamination level 0% contamination level 10% contamination level 20% 60 60 60 true true true class. class. class. 1. coordinate of state process 1. coordinate of state process 1. coordinate of state process 50 50 50 rLS rLS rLS ACM ACM ACM 40 40 40 30 30 30 20 20 20 10 10 10 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 time time time contamination level 0% contamination level 10% contamination level 20% true true true 20 20 20 class. class. class. 2. coordinate of state process 2. coordinate of state process 2. coordinate of state process rLS rLS rLS 15 15 15 ACM ACM ACM 10 10 10 5 5 5 0 0 0 −5 −5 −5 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 time time time B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 16/20

Results (cont.) (a) (b) 70 class. class. 60 300 rLS rLS ACM ACM 50 200 40 MSE MSE 30 100 20 10 0 0 0 5 10 15 20 0 5 10 15 20 contamination in % contamination in % B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 17/20

The R package robKalman general function recursiveFilter with parameters: observations state-space model (hyper parameters) functions for the init./pred./corr. step available filters: KalmanFilter, rLSFilter, ACMfilter, mACMfilter all: wrappers to recursiveFilter B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 18/20

Remarks & Outlook ACM performs better than rLS for both contamination situations rLS yields larger errors in the case of 0% contamination because it was calibrated to a loss of efficiency δ = 10% all simulations were made with R R -package robKalman for filtering already exists ( but is still under construction! ) http://r-forge.r-project.org/projects/ robkalman/ S4 classes for state-space models and filtering results B. Spangl et al., Approximate Conditional-mean Type Filtering for State-space Models – p. 19/20