A Kalman Filter for Robust Outlier Detection

Transcription

1 A Kalman Filter for Robust Outlier Detection Jo-Anne Ting, Evangelos Theodorou, Stefan Schaal Computational Learning & Motor Control Lab University of Southern California IROS 2007 October 31, 2007

2 Outline Motivation Quic review of the Kalman filter Robust Kalman filtering with Bayesian weights Experimental results Conclusions IROS

3 Motivation Consider real-time applications where storing data samples may not be a viable option due to high frequency of sensory data In systems where high quality sensory data is needed, reliable detection of outliers is essential for optimal performance (e.g. legged locomotion): The Kalman filter (Kalman, 60) is commonly used for real-time tracing, but it is not robust to outliers! IROS

4 Previous Methods Robust Kalman filter approach 1) Use non-gaussian distributions for random variables (Sorenson & Alspach 71, West 82) Drawbac Complicated resulting parameter estimation for systems with transient disturbances 2) Model observation & state noise as non-gaussian, heavy-tailed distributions (Masreliez 75) Difficult & involved filter implementation 3) Use resampling or numerical integration (Kitagawa 87) 4) Use a robust least squares approach & model weights with heuristic functions (e.g., Durovic & Kovacevic, 99) Heavy computation not suitable for real-time applications Need to determine the optimal values of open parameters IROS

5 Outline Motivation Quic review of the Kalman filter Robust Kalman filtering with Bayesian weights Experimental results Conclusions IROS

6 A Quic Review of the Kalman Filter The system equations for the Kalman filter are as follows: Observation matrix z = C! + v Observation noise: ( ) v ~ Normal 0,R! = A! "1 + s State transition matrix State noise: s ~ Normal( 0,Q) IROS

7 Standard Kalman Filter Equations Propagation:! # Update: S K = A! "1 = A# "1 A T + Q ( ) "1 = C# C T + R = # C T S! =! + K # = I " K ( C)# ( z " C! ) Can use ML framewor to estimate system dynamics (Myers & Tapley, 1976) IROS

8 Outline Motivation Quic review of the Kalman filter Robust Kalman filtering with Bayesian weights Experimental results Conclusions IROS

9 Robust Kalman Filtering with Bayesian Weights Use a weighted least squares approach & learn the optimal weights: ( ) ( ) z!,w ~ Normal C!,R / w!! "1 ~ Normal A! "1,Q ( ) w ~ Gamma a w,b w Kalman filter Robust Kalman filter with Bayesian weights! "1!! "1! A!1 Q!1 A Q z!1 z z!1 w!1 z w C!1 R!1 C R IROS

10 Inference Procedure We can treat this as an EM learning problem (Dempster & Laird, 77): Maximize log p! 1:,z i,w 1: N " i=1 ( ) We use a variational factorial approximation of the true posterior distribution: ( ) = Q w i Q w,! N i=1 ( ) ( ) " " Q! i! i#1 Q! 0 to get analytically tractable inference (e.g., Ghahramani & Beal, 00). N i=1 ( ) IROS

11 Robust Kalman Filter Equations Propagation:! # Update: S K = A! "1 = Q $ = C # C T + R & % w ) ( = # C T S! =! + K ( z " C! ) # = I " K ( C )# "1 Compare to standard Kalman filter Propagation:! # Update: S K = A! "1 = A# "1 A T + Q ( ) "1 = C# C T + R = # C T S! =! + K # = I " K ( C)# ( z " C! ) w = a w b w 0 + ( z! C " ) T R!1 ( z! C " ) IROS

12 Important Things to Note Our robust Kalman filter: 1) Has the same computational complexity as the standard Kalman filter 2) Is principled & easy to implement (no heuristics) 3) Offers a natural framewor to incorporate prior nowledge of the presence of outliers IROS

13 Outline Motivation Quic review of the Kalman filter Robust Kalman filtering with Bayesian weights Experimental results Conclusions IROS

14 Real-time Outlier Detection on LittleDog Outliers Our robust KF performs as well as a handtuned KF (that required prior nowledge and, hence, is near-optimal) IROS

15 Outline Motivation Quic review of the Kalman filter Robust Kalman filtering with Bayesian weights Experimental results Conclusions IROS

16 Conclusions We have introduced an outlier-robust Kalman filter that: 1) Is principled & easy to implement 2) Has the same computational complexity as the Kalman filter 3) Provides a natural framewor to incorporate prior nowledge of noise This framewor can be extended to other more complex, nonlinear filters & methods in order to incorporate automatic outlier detection abilities. IROS

17 Final Posterior EM Update Equations E-step: ( ) "1 ( )! = w C T R "1 C + Q "1 # =! Q "1 A # "1 + w C T R "1 z w = a w b w 0 + ( z " C # ) T R "1 ( z " C # ) Need to be written in incremental form M-step: # T & # T C = " w i z i! i $ % ( " w i! i! i $ % i=1 i=1 # T & # T A = "! i! i)1 $ % ( "! i)1! i)1 $ % r m = 1 q n = 1 i=1 " i=1 " i=1 w i i=1 ( z i ) C (m,:)! i ) 2 w i (! i ) A (n,:)! i)1 ) 2 )1 & ( )1 & ( These are computed once for each time step (e.g., Ghahramani & Hinton, 1996) IROS

18 Incremental Version of M-step Equations Gather sufficient statistics to re-write M-step equations in incremental form (i.e., only using values observed or calculated in the current time step, ): M-step: C = A = r m = 1 q n = 1 (" wz! T w!! " T ( ) #1!! T!! " " $ %& )#1 { } wzz wz! w!! " # 2C m (m,:)" + diag C m (m,:)" T C (m,:) T $ %&! " 2 # 2A (n,:) n!! " + diag{!! A n (n,:)" A (n,:) T } () () IROS