A particle filter approach to approximate posterior Cram\'er-Rao lower bound

TL;DR

This paper introduces a particle filter-based recursive method to approximate the posterior Cramér-Rao lower bound in non-linear, non-Gaussian systems where true states are unmeasured, demonstrated through simulations including ballistic target tracking.

Contribution

A novel particle filter approach for recursive approximation of the PCRLB in non-linear, non-Gaussian systems with hidden states, overcoming previous computational challenges.

Findings

Effective in non-linear, non-Gaussian scenarios

Applicable to practical problems like ballistic tracking

Demonstrated through simulation examples

Abstract

The posterior Cram\'er-Rao lower bound (PCRLB) derived in Tichavsk\'y et al., 1998, provides a bound on the mean square error (MSE) obtained with any non-linear state filter. Computing the PCRLB involves solving complex, multi-dimensional expectations, which do not lend themselves to an easy analytical solution. Furthermore, any attempt to approximate it using numerical or simulation based approaches require a priori access to the true states, which may not be available, except in simulations or in carefully designed experiments. To allow recursive approximation of the PCRLB when the states are hidden or unmeasured, a new approach based on sequential Monte-Carlo (SMC) or particle filters (PF) is proposed. The approach uses SMC methods to estimate the hidden states using a sequence of the available sensor measurements. The developed method is general and can be used to approximate the PCRLB in non-linear systems with non-Gaussian state and sensor noise. The efficacy of the developed method is illustrated on two simulation examples, including a practical problem of ballistic target tracking at re-entry phase.