The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions

The state-space model with random noises is widely used to model dynamic systems. In many practical problems, the noise terms will have unknown probability distributions, which is contrary to the usual assumption of Gaussian noises made in state-vector estimation; this assumption is usually made for reasons of mathematical tractability. A problem arises, however, in that inference based on the incorrect Gaussian assumption can lead to misleading or erroneous conclusions. This note shows how the Kantorovich inequality from probability theory has potential for characterizing the estimation error of a Kalman filter in such a non-Gaussian (distribution-free) setting.

[1]  S. Gupta,et al.  ON ANDERSON'S PROBABILITY INEQUALITY , 1983 .

[2]  Morris L. Eaton,et al.  Concentration inequalities for Gauss-Markov estimators , 1988 .

[3]  R. J. Kelly.  Reducing geometric dilution of precision using ridge regression , 1990 .

[4]  Achim Clausing.  Kantorovich-Type Inequalities , 1982 .

[5]  Han-Fu Chen,et al.  Lp‐stability of estimation errors of kalman filter for tracking time‐varying parameters , 1991 .

[6]  James C. Spall.  [That BLUP is a Good Thing: The Estimation of Random Effects]: Comment: The Kalman Filter and BLUP , 1991 .

[7]  Y. Tong.  Probability Inequalities in Multivariate Distributions , 1980 .

[8]  J. Spall.  THE DISTRIBUTION OF NONSTATIONARY AUTOREGRESSIVE PROCESSES UNDER GENERAL NOISE CONDITIONS , 1993 .

[9]  Y. Tong.  Some majorization inequalities in multivariate statistical analysis , 1988 .

[10]  H. Sorenson,et al.  Recursive bayesian estimation using gaussian sums , 1971 .

[11]  Samuel Karlin,et al.  Studies in econometrics, time series, and multivariate statistics , 1983 .

[12]  C. Masreliez.  Approximate non-Gaussian filtering with linear state and observation relations , 1975 .

[13]  Ludwik Kurz,et al.  An adaptive robustizing approach to kalman filtering , 1983, Autom..

[14]  G. Kitagawa.  Non-Gaussian State—Space Modeling of Nonstationary Time Series , 1987 .

[15]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  D. Varberg.  Convex Functions , 1973 .

[17]  Robert Kohn,et al.  Prediction mean squared error for state space models with estimated parameters , 1986 .

[18]  G. S. Watson,et al.  A method for discovering Kantorovich-type inequalities and a probabilistic interpretation , 1987 .

[19]  James Ting-Ho Lo,et al.  Finite-dimensional sensor orbits and optimal nonlinear filtering , 1972, IEEE Trans. Inf. Theory.

[20]  Guanrong Chen.  Approximate Kalman filtering , 1993 .

[21]  J. Spall,et al.  Asymptotic distribution theory for the kalman filter state estimator , 1984 .

[22]  Sylvia Schnatter.  Integration-based Kalman-filtering for a dynamic generalized linear trend model , 1992 .

[23]  S. Gupta,et al.  A generalization of Anderson’s theorem on unimodal functions , 1976 .

[24]  M. E. Welch,et al.  Bayesian analysis of time series and dynamic models , 1990 .

[25]  Joel M. Morris,et al.  The Kalman filter: A robust estimator for some classes of linear quadratic problems , 1976, IEEE Trans. Inf. Theory.

[26]  W. Chan,et al.  Unimodality, convexity, and applications , 1989 .