On-line Bayesian parameter estimation in general non-linear state-space models: A tutorial and new results

TL;DR

This paper introduces a Bayesian on-line estimation method for non-linear state-space models that effectively handles missing data, improves computational efficiency, and adapts dynamically using KL divergence-based optimization.

Contribution

It proposes a novel adaptive SIR filter with kernel density estimation for real-time joint state-parameter estimation, including missing data handling, with an optimal tuning strategy.

Findings

Effective handling of missing data in real-time estimation

Improved speed and non-degeneracy of on-line algorithms

Validated through numerical examples demonstrating accuracy

Abstract

On-line estimation plays an important role in process control and monitoring. Obtaining a theoretical solution to the simultaneous state-parameter estimation problem for non-linear stochastic systems involves solving complex multi-dimensional integrals that are not amenable to analytical solution. While basic sequential Monte-Carlo (SMC) or particle filtering (PF) algorithms for simultaneous estimation exist, it is well recognized that there is a need for making these on-line algorithms non-degenerate, fast and applicable to processes with missing measurements. To overcome the deficiencies in traditional algorithms, this work proposes a Bayesian approach to on-line state and parameter estimation. Its extension to handle missing data in real-time is also provided. The simultaneous estimation is performed by filtering an extended vector of states and parameters using an adaptive sequential-importance-resampling (SIR) filter with a kernel density estimation method. The approach uses an on-line optimization algorithm based on Kullback-Leibler (KL) divergence to allow adaptation of the SIR filter for combined state-parameter estimation. An optimal tuning rule to control the width of the kernel and the variance of the artificial noise added to the parameters is also proposed. The approach is illustrated through numerical examples.