Smoothing Dynamic Systems with State-Dependent Covariance Matrices

TL;DR

This paper extends Kalman smoothing to handle state-dependent covariance matrices, proposing a generalized Gauss-Newton algorithm that maintains computational efficiency for dynamic systems inference.

Contribution

It introduces a novel formulation and algorithm for inference in systems with covariance matrices depending on unknown parameters, enhancing Kalman smoothing capabilities.

Findings

Algorithm preserves computational efficiency of classic Kalman smoother

Effective in modeling covariance matrices dependent on the state sequence

Demonstrated with a synthetic numerical example

Abstract

Kalman filtering and smoothing algorithms are used in many areas, including tracking and navigation, medical applications, and financial trend filtering. One of the basic assumptions required to apply the Kalman smoothing framework is that error covariance matrices are known and given. In this paper, we study a general class of inference problems where covariance matrices can depend functionally on unknown parameters. In the Kalman framework, this allows modeling situations where covariance matrices may depend functionally on the state sequence being estimated. We present an extended formulation and generalized Gauss-Newton (GGN) algorithm for inference in this context. When applied to dynamic systems inference, we show the algorithm can be implemented to preserve the computational efficiency of the classic Kalman smoother. The new approach is illustrated with a synthetic numerical example.