Approximate Conditionalmean Type Filtering for Statespace Models


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1 Approimate Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace Models p. 1/
2 Contents Statespace models & Kalman filter Multivariate tpe filter filter Simulation stud Results R package robkalman Remarks & Outlook B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace Models p. 7/
8 Multivariate tpe Filter approimate conditionalmean () 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 ψfunction appropriatel chosen in the case univariate observations equivalent to Martin s tpe filter (Martin, 1979) B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace Models p. 8/
9 Huber s Multivariate Psifunction (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace Models p. 9/
10 Hampel s Multivariate Psifunction (a) (b) 1 1 coord. coord. 1 1 B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Conditionalmean Tpe Filtering for Statespace Models p. 16/
17 Results (cont.) MSE (a) MSE 1 3 (b) contamination in % contamination in % B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace Models p. 17/
18 The R package robkalman general function recursivefilter with parameters: observations statespace 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 Conditionalmean Tpe Filtering for Statespace 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 Rpackage robkalman for filtering alread eists (but is still under construction!) robkalman/ S classes for statespace models and filtering results B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace Models p. 19/
20 References R.D. Martin (1979). Approimate Conditionalmean Tpe Smoothers and Interpolators. In Gasser and Rossenblatt (eds.), Smoothing Techniques for Curve Estimation, , Springer, Berlin. C.J. Masreliez (1975). Approimate nongaussian 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 KalmanFilters. 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 Conditionalmean Tpe Filtering for Vectorvalued Observations. Technical Report TRAS81, Universität für Bodenkultur, Vienna. B. Spangl et al., Approimate Conditionalmean Tpe Filtering for Statespace Models p. /
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