Adaptive Kalman Filter for Systems with Unknown Initial Values

TL;DR

This paper develops adaptive Kalman filters for partially observed linear stochastic systems with unknown initial states, addressing both deterministic and Gaussian random initial conditions, and discusses their asymptotic optimality.

Contribution

It introduces a method combining preliminary estimation with recursive MLE to adapt Kalman filters for systems with unknown initial values.

Findings

The proposed filters are asymptotically optimal.

The method effectively estimates unknown initial states.

The approach applies to both deterministic and Gaussian initial conditions.

Abstract

The models of partially observed linear stochastic differential equations with unknown initial values of the non-observed component are considered in two situations. In the first problem, the initial value is deterministic, and in the second problem, it is assumed to be a Gaussian random variable. The main problem is the computation of adaptive Kalman filters and the discussion of their asymptotic optimality. The realization of this program for both models is done in several steps. First, a preliminary estimator of the unknown parameter is constructed by observations on some learning interval. Then, this estimator is used for the calculation of recurrent one-step MLE estimators, which are subsequently substituted in the equations of Kalman filtration.