Confidence Bounds for the Estimation Error in Adaptive Kalman-Type Filters

摘要

Consider the setting where one wishes to produce a state estimate in a linearstate-space model, but where the parameters of the model are not fully known. Often, one estimates the state and parameters simultaneously or estimates the state using a Kalman filter (or similar procedure) with estimated parameters used in place of the true (unknown) parameters. One of the issues in such adaptive filtering is that the inherent nonlinearity of the recursion leads to difficulties in determining the confidence bounds for the state estimation errors. While some authors have constructed approximate covariance matrices for the estimation errors, no one seems to have determined the distributions for the errors. Thus, constructing confidence bounds has been an unsolved problem. This paper shows how the theory on autoregressive processes in Spall (1993) can be used to construct confidence bounds for the state estimation error in a broad class of adaptive filters. This class includes, e.g., the extended Kalman filter and other joint state/parameter estimation procedures that have appeared in the statistics or econometrics literature (such as due to Ansley and Kohn 1986, Hamilton 1986, or Watanabe 1985). State-space model, Adaptive filter, Autoregressive process, Confidence intervals.