Approximate Conditional-mean Type Filtering for State-space Models

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1 Approimate Conditional-mean Tpe Filtering for State-space Models Bernhard Spangl, Universität für Bodenkultur, Wien joint work with Peter Ruckdeschel, Fraunhofer Institut, Kaiserslautern, and Rudi Dutter, Technische Univeristät, Wien UseR! 8, Dortmund, German B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

2 Contents State-space models & Kalman filter Multivariate -tpe filter filter Simulation stud Results R package robkalman Remarks & Outlook B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

3 Linear State Space Models State equation: t = Φ t 1 + ε t Observation equation: t = H t + v t Ideal model assumptions: N p (µ,σ ), ε t N p (,Q), v t N q (,R), all independent B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 3/

4 Classical Kalman Filter Initialization (t = ): = µ, P = Σ Prediction (t 1): t t 1 = Φ t 1 t 1 M t = ΦP t 1 Φ + Q = Cov( t t 1 ) Correction (t 1): t t = t t 1 + K t ( t H t t 1 ) P t = M t K t HM t = Cov( t t ) with K t = M t H (HMH + R) 1 (Kalman gain) B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

5 Tpes 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 (,R) + γn q (µ c,r c ) Other Tpes of Outliers: substitutive outliers (SO s) patch outliers B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 5/

6 Masreliez s Theorem (1975) If t Y t 1 N p ( t t 1,M t ), t 1, then t t = E( t Y t ), t 1, is generated b the recursions t t = t t 1 + M t H Ψ t ( t ) P t = M t M t H Ψ t( t )HM t M t+1 = ΦP t Φ + Q, with (Ψ t ()) i = ( / i ) log f t ( Y t 1 ) and (Ψ t ()) ij = ( / j )(Ψ t ()) i. Ψ t () is called the score function. Note: If f t (. Y t 1 ) is Gaussian, Masreliez s filter reduces to the Kalman filter. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 6/

7 The Score Function Ψ t (a) (b).1.5 (c) (d) 1 d/d d/d 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 7/

8 Multivariate -tpe Filter approimate conditional-mean () tpe filter proposed b B. Spangl and R. Dutter (8) modified correction step: t t = t t 1 + M t H S t ψ(s t ( t H t t 1 )) P t = M t M t H S t ψ (S t ( t H t t 1 ))S t HM t for an S t and a ψ-function appropriatel chosen in the case univariate observations equivalent to Martin s tpe filter (Martin, 1979) B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 8/

9 Huber s Multivariate Psi-function (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 9/

10 Hampel s Multivariate Psi-function (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

11 Approimating the Score Function (a) (b) 1 d/d d/d 1 (c) (d) 1 coord. coord. 1 B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 11/

12 Filter proposed b P. Ruckdeschel (1) modified correction step: t t = t t 1 + H b (K t ( t H t t 1 )) with H b (z) = z min{1,b/ z } and. the Euclidean norm optimal for SO s in some sense B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

13 Simulation State Space Process: simulate state space process using two different sets of hper parameters and AO s from two different contamination setups: N (, 1 1 ) or N ( 5 3,.9.9 ). var contamination γ from % to % b 5% each s Filtering: robust filtering (, ) Evaluation: compare with state process via MSE B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 13/

14 Simulation (cont.) Eample I: µ =, Φ = , Q = 3 3, Σ =, H = 1 1 1, R =...5. Eample II: µ =, Φ = 1 1, Q = 9, Σ =, H = , R = 9 9. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 1/

15 Results contamination level % contamination level 1% 1. coordinate of state process contamination level %. coordinate of state process coordinate of state process 1 1 contamination level 1%. coordinate of state process contamination level % contamination level % coordinate of state process 1. coordinate of state process B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 15/

16 Results (cont.) contamination level % contamination level 1% 1. coordinate of state process contamination level %. coordinate of state process coordinate of state process contamination level 1%. coordinate of state process contamination level % contamination level % coordinate of state process 1. coordinate of state process B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 16/

17 Results (cont.) MSE (a) MSE 1 3 (b) contamination in % contamination in % B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 17/

18 The R package robkalman general function recursivefilter with parameters: observations state-space model (hper parameters) functions for the init./pred./corr. step available filters: KalmanFilter, Filter, filter, mfilter all: wrappers to recursivefilter B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 18/

19 Remarks & Outlook performs better than for both contamination situations ields larger errors in the case of % contamination because it was calibrated to a loss of efficienc δ = 1% all simulations were made with R R-package robkalman for filtering alread eists (but is still under construction!) robkalman/ S classes for state-space models and filtering results B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. 19/

20 References R.D. Martin (1979). Approimate Conditional-mean Tpe Smoothers and Interpolators. In Gasser and Rossenblatt (eds.), Smoothing Techniques for Curve Estimation, , Springer, Berlin. C.J. Masreliez (1975). Approimate non-gaussian filtering with linear state and observation relations. IEEE Transactions on Automatic Control,, R Development Core Team (5). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. P. Ruckdeschel (1). Ansätze zur Robustifizierung des Kalman-Filters. PhD thesis, Universität Bareuth, Bareuth. P. Ruckdeschel and B. Spangl (7). robkalman: An R package for robust Kalman filtering. Web: B. Spangl and R. Dutter (8). Approimate Conditional-mean Tpe Filtering for Vector-valued Observations. Technical Report TR-AS-8-1, Universität für Bodenkultur, Vienna. B. Spangl et al., Approimate Conditional-mean Tpe Filtering for State-space Models p. /

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