In practical nonlinear filtering, the assessment of achievable filtering performance is important. In this paper, we focus on the problem of efficiently approximate the posterior Cramer-Rao lower bound (CRLB) in a recursive manner. By using Gaussian assumptions, two types of approximations for calculating the CRLB are proposed: An exact model using the state estimate as well as a Taylor-series-expanded model using both of the state estimate and its error covariance, are derived. Moreover, the difference between the two approximated CRLBs is also formulated analytically. By employing the particle filter (PF) and the unscented Kalman filter (UKF) to compute, simulation results reveal that the approximated CRLB using mean-covariance-based model outperforms that using the mean-based exact model. It is also shown that the theoretical difference between the estimated CRLBs can be improved through an improved filtering method.
Deep Dive into Error Analysis of Approximated PCRLBs for Nonlinear Dynamics.
In practical nonlinear filtering, the assessment of achievable filtering performance is important. In this paper, we focus on the problem of efficiently approximate the posterior Cramer-Rao lower bound (CRLB) in a recursive manner. By using Gaussian assumptions, two types of approximations for calculating the CRLB are proposed: An exact model using the state estimate as well as a Taylor-series-expanded model using both of the state estimate and its error covariance, are derived. Moreover, the difference between the two approximated CRLBs is also formulated analytically. By employing the particle filter (PF) and the unscented Kalman filter (UKF) to compute, simulation results reveal that the approximated CRLB using mean-covariance-based model outperforms that using the mean-based exact model. It is also shown that the theoretical difference between the estimated CRLBs can be improved through an improved filtering method.
It is well known that optimal estimators for the nonlinear filtering of the discrete-time dynamic systems is an active area of research and that a large number of suboptimal approximated approaches were developed [1]. It is important to quantify the accuracy of estimates obtained for the design of algorithms such as the interacting multiple models (IMM) where weighted estimates from multiple estimators are simultaneously employed.
During the past thirty years many attempts have been made to theoretically derive the achievable performance of nonlinear filters. Deriving performance bounds are important since such bound serve as indicators to measure system performance, and can be used to determine whether imposed performance requirements are realistic or not.
For dynamical statistical models, a commonly used bound is the CRLB that has been investigated by various researchers: Van Trees [2] presented the batch form of a posterior CRLB for random parameter vectors and a pre-1989 review [3] summarized several lower bounds for nonlinear filtering, which heavily emphasized the continuous time case. Bobrovsky [4] applied CRLB to discrete time problems and Galdos [6] generalized it to the multi-dimensional case. The main shortcoming of these formulations is the batch form of implementation resulting high computational loads. Tichavsky [7] was the first to derive a recursive CRLB for updating the posterior Fisher information matrix (FIM) from one time instance to the next while keeping the FIM constant in size.
Subsequently, CRLB theory was extended to many applications, e.g., introducing the CRLB to multiple target tracking [9], incorporating data association for tracking with the CRLB [10], target detection for the case having a detection probability less than unit [8], etc.
It is well known that the matrices in recursive form of FIM, can only be theoretically determined by the true value of state. Unfortunately, we cannot obtain the true state online in practice, except in some well-designed experiments where true value of the state is given as a prior knowledge. Therefore we naturally focus on how to determine an approximate CRLB by using online state estimates (as opposed to the true state values).
We have mainly two ways to approximate the CRLB [5]: 1) Make full use of the first-two order moments of the state estimate, i.e., expectation and covariance, by incorporating them with the Taylor series expansion of the dynamics. 2) Combine the expectation of the state with the exact dynamic model directly. The first method use both estimates and is rather complex while the second method is considerably simple, but depends heavily on an exact model. The second method is mostly preferred in practice for its simpleness and is sufficient to obtain an usable approximated CRLB.
The following question therefore needs to be addressed: By how much the CRLB employed the two kinds of approximations differ from, and which one is a better approximation to the true CRLB. This is the main motivation of this investigation. In addition, determining the accuracy of the estimated CRLB by using a state estimate, rather than the true state under a recursive framework for a general nonlinear dynamics, has not been addressed previously.
In this paper, we show how the state estimates can be applied to determine the difference between the two estimated CRLBs. By using Monte Carlo simulations, we show that the proposed method achieve a satisfactory approximation, and the accuracy of estimated CRLB can be explicitly improved by increasing the accuracy of filtering.
Consider the following discrete-time nonlinear dynamics with additive Gaussian noise:
where the nonlinear vector-valued functions f k ∈ R n×1 and h k ∈ R m×1 be used to model the state kinematics and measurement respectively, and generally n > m. x k ∈ R n×1 is the state vector, z k ∈ R m×1 is the measurement vector, w k ∈ R n×1 is a zero-mean white Gaussian process noise with known covariance Q k , and v k ∈ R m×1 a zero-mean Gaussian white measurement noise with variance R k . The initial state x 0 is assumed as a Gaussian distribution with mean x0 and variance P 0 . Moreover, a general accepted assumption like cov(x 0 , v k ) = 0, cov(x 0 , w k ) = 0.
Let xk and C k denote the unbiased state estimate and its error covariance at time instant k. We therefore have
where xk = x k -xk is the prediction error of state. J -1 k is the posterior CRLB (PCRLB), defined to be the inverse of FIM, J k . The superscript (•) ′ in (3) denotes the transpose of a vector or a matrix, and the inequality in (3) means that the difference C k -J -1 k is a positive semidefinite matrix. From [7], [11] we know that the sequential FIM J k can be recursively calculated by
here let ∇ and ∆ be operators of the first and second-order partial derivatives, i.e.,
Note that all the above expectations are taken with respect to the joint probability density function (PDF) p(x 0:k+1 |z 1:k+1 ), where x 0
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