Classical setup: Linear state space models (SSMs) robKalman a - PowerPoint PPT Presentation

Robust Kalman Filtering Robust Kalman Filtering Classical setup Implementation proposal Implementation proposal Robustification Demonstration Demonstration Approaches Classical setup: Linear state space models (SSMs) robKalman — a package on Robust Kalman Filtering State equation: X t = F t X t − 1 + v t Peter Ruckdeschel 1 Bernhard Spangl 2 Observation equation: Y t = Z t X t + ε t 1 Fakult¨ at f¨ ur Mathematik und Physik Peter.Ruckdeschel@uni-bayreuth.de Ideal model assumption: www.uni-bayreuth.de/departments/math/org/mathe7/RUCKDESCHEL X 0 ∼ N p ( a 0 , Σ 0 ) , v t ∼ N p (0 , Q t ) , ε t ∼ N q (0 , V t ) , 2 Universit¨ at f¨ ur Bodenkultur, Wien Bernhard.Spangl@boku.ac.at all independent www.rali.boku.ac.at/statedv.html (preliminary ?) simplification: Hyper parameters F t , Z t , V t , Q t constant in t 16.06.2006 Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Classical setup: Linear state space models (SSMs) Classical setup: Linear state space models (SSMs) State equation: State equation: X t = F t X t − 1 + v t X t = F t X t − 1 + v t Observation equation: Observation equation: Y t = Z t X t + ε t Y t = Z t X t + ε t Ideal model assumption: Ideal model assumption: X 0 ∼ N p ( a 0 , Σ 0 ) , v t ∼ N p (0 , Q t ) , ε t ∼ N q (0 , V t ) , X 0 ∼ N p ( a 0 , Σ 0 ) , v t ∼ N p (0 , Q t ) , ε t ∼ N q (0 , V t ) , all independent all independent (preliminary ?) simplification: Hyper parameters F t , Z t , V t , Q t (preliminary ?) simplification: Hyper parameters F t , Z t , V t , Q t constant in t constant in t Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering

Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Classical setup: Linear state space models (SSMs) Problem and classical solution State equation: X t = F t X t − 1 + v t Problem: Reconststruction of X t by means of Y s , s ≤ t Observation equation: Criterium: MSE Y t = Z t X t + ε t general solution: E X t | ( Y s ) s ≤ t � Computational difficulties: Ideal model assumption: ⇒ restriction to linear procedures = / or: Gaussian assumptions X 0 ∼ N p ( a 0 , Σ 0 ) , v t ∼ N p (0 , Q t ) , ε t ∼ N q (0 , V t ) , classical Kalman Filter � all independent (preliminary ?) simplification: Hyper parameters F t , Z t , V t , Q t constant in t Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Problem and classical solution Problem and classical solution Problem: Reconststruction of X t by means of Y s , s ≤ t Problem: Reconststruction of X t by means of Y s , s ≤ t Criterium: MSE Criterium: MSE general solution: E X t | ( Y s ) s ≤ t general solution: E X t | ( Y s ) s ≤ t � � Computational difficulties: Computational difficulties: ⇒ restriction to linear procedures ⇒ restriction to linear procedures = = / or: Gaussian assumptions / or: Gaussian assumptions classical Kalman Filter classical Kalman Filter � � Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering

Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Kalman filter Kalman filter Initialization ( t = 0): Initialization ( t = 0): 0 0 X 0 | 0 = a 0 , Σ 0 | 0 = Σ 0 X 0 | 0 = a 0 , Σ 0 | 0 = Σ 0 Prediction ( t ≥ 1): Prediction ( t ≥ 1): 1 1 Cov ( X t | t − 1 ) = Σ t | t − 1 = F Σ t − 1 | t − 1 F ′ + Q Cov ( X t | t − 1 ) = Σ t | t − 1 = F Σ t − 1 | t − 1 F ′ + Q X t | t − 1 = FX t − 1 | t − 1 , X t | t − 1 = FX t − 1 | t − 1 , Correction ( t ≥ 1): Correction ( t ≥ 1): 2 2 X t | t = X t | t − 1 + K t ( Y t − ZX t | t − 1 ) X t | t = X t | t − 1 + K t ( Y t − ZX t | t − 1 ) Σ t | t − 1 Z ′ ( Z Σ t | t − 1 Z ′ + V ) − , Σ t | t − 1 Z ′ ( Z Σ t | t − 1 Z ′ + V ) − , = (Kalman gain) = (Kalman gain) K t K t Σ t | t = Σ t | t − 1 − K t Z Σ t | t − 1 Σ t | t = Σ t | t − 1 − K t Z Σ t | t − 1 Cov ( X t | t ) = Cov ( X t | t ) = Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Kalman filter Types of outliers and robustification Initialization ( t = 0): 0 X 0 | 0 = a 0 , Σ 0 | 0 = Σ 0 IOs (system intrinsic): state equation is distorted — not considered here Prediction ( t ≥ 1): 1 AO/SOs (exogeneous): observations are distorted: Cov ( X t | t − 1 ) = Σ t | t − 1 = F Σ t − 1 | t − 1 F ′ + Q either error ε t is affected (AO) X t | t − 1 = FX t − 1 | t − 1 , or observations Y t are modified (SO) Correction ( t ≥ 1): a robustifications as to AO/SOs is to 2 retain recursivity (three-step approach) X t | t = X t | t − 1 + K t ( Y t − ZX t | t − 1 ) modify correction step ✥ bound influence of Y t Σ t | t − 1 Z ′ ( Z Σ t | t − 1 Z ′ + V ) − , retain init./pred.step but with modified filter past X t − 1 | t − 1 = (Kalman gain) K t Σ t | t = Σ t | t − 1 − K t Z Σ t | t − 1 Cov ( X t | t ) = Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering

Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Types of outliers and robustification Types of outliers and robustification IOs (system intrinsic): state equation is distorted IOs (system intrinsic): state equation is distorted — not considered here — not considered here AO/SOs (exogeneous): observations are distorted: AO/SOs (exogeneous): observations are distorted: either error ε t is affected (AO) either error ε t is affected (AO) or observations Y t are modified (SO) or observations Y t are modified (SO) a robustifications as to AO/SOs is to a robustifications as to AO/SOs is to retain recursivity (three-step approach) retain recursivity (three-step approach) modify correction step ✥ bound influence of Y t modify correction step ✥ bound influence of Y t retain init./pred.step but with modified filter past X t − 1 | t − 1 retain init./pred.step but with modified filter past X t − 1 | t − 1 Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Robust Kalman Filtering Classical setup Robust Kalman Filtering Classical setup Implementation proposal Robustification Implementation proposal Robustification Demonstration Approaches Demonstration Approaches Considered approaches Considered approaches Approximate conditional mean (ACM): [Martin(79)] Approximate conditional mean (ACM): [Martin(79)] dim Y t = 1 dim Y t = 1 particular model: Y t ∼ AR ( p ) particular model: Y t ∼ AR ( p ) ✥ X t = ( Y t , . . . , Y t − p +1 ), ✥ X t = ( Y t , . . . , Y t − p +1 ), hyper parameters Z = (1 , 0 , . . . , 0), V id = 0, F , Q unknown hyper parameters Z = (1 , 0 , . . . , 0), V id = 0, F , Q unknown estimation of F , Q by means of GM -Estimators estimation of F , Q by means of GM -Estimators modified Corr.step: for suitable location influence curve ψ modified Corr.step: for suitable location influence curve ψ X t | t − 1 + Σ t | t − 1 Z ′ ψ ( Y t − ZX t | t − 1 ) X t | t − 1 + Σ t | t − 1 Z ′ ψ ( Y t − ZX t | t − 1 ) X t | t = X t | t = Σ t | t − 1 − Σ t | t − 1 Z ′ ψ ′ ( Y t − ZX t | t − 1 ) Z Σ t | t − 1 Σ t | t − 1 − Σ t | t − 1 Z ′ ψ ′ ( Y t − ZX t | t − 1 ) Z Σ t | t − 1 Σ t | t = Σ t | t = Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering Peter Ruckdeschel, Bernhard Spangl robKalman — a package on Robust Kalman Filtering