A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

Abstract

The posterior Cramér-Rao lower bound (PCRLB) derived in [1] 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.

Tracking With Sparse and Correlated Measurements via a Shrinkage-Based Particle Filter

“…The PCRLB provides a lower bound on the mean square error obtained with any non-linear filter and is equivalent to the inverse of the posterior Fisher information matrix (PFIM) [27]. The implementation of the PCRLB requires knowledge of the true state.…”

“…After comparing (48) with the EKF covariance matrix computed in (27), by replacing J k by P −1 k and by applying the matrix inversion lemma, the following expression is obtained:…”

Approximating Posterior Cramér–Rao Bounds for Nonlinear Filtering Problems Using Gaussian Mixture Models

“…However, such an assumption may be inappropriate when the system is highly nonlinear or the state PDF becomes multimodal. More recently, Tulsyan et al proposed a sequential Monte Carlo (SMC)-based method to approximate the PCRB [21]. The authors first computed the expectations with respect to the measurement-conditioned state PDF by using the SMC approach and then with respect to the measurement PDF by the standard Monte Carlo method.…”

Event-triggered particle filtering and Cramér–Rao lower bound computation

“…Although the posterior CRLB does not fully demonstrate the accuracy of nonlinear filters, it is an important benchmark for assessing and comparing the quality of different nonlinear filters. 28 A conditional posterior CRLB is then introduced in Zuo et al 29 to improve the posterior CRLB such that all the observation data up to the current time are considered and therefore the corresponding prior knowledge is more recent and relevant one. The resulting posterior CRLB is conditioned on the measurements up to the reset initial time.…”

“…To overcome this problem, it is proposed to compute the conditional posterior CRLB using particle filter. 28,30 In this context, in order to evaluate the performance of the proposed PF-based event-triggered state estimator, its conditional posterior CRLB using Monte Carlo simulations is computed in this article. We have proposed the preliminary version of this work without any performance evaluation in our previous work in Sadeghzadeh-Nokhodberiz et al 31…”