Error Analysis of the Stochastic Linear Feedback Particle Filter

Abstract

This paper is concerned with the convergence and long-term stability analysis of the feedback particle filter (FPF) algorithm. The FPF is an interacting system of N particles where the interaction is designed such that the empirical distribution of the particles approximates the posterior distribution. It is known that in the mean-field limit (N = ∞), the distribution of the particles is equal to the posterior distribution. However little is known about the convergence to the mean-field limit. In this paper, we consider the FPF algorithm for the linear Gaussian setting. In this setting, the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although these algorithms have been numerically evaluated and widely used in applications, their convergence and long-term stability analysis remains an active area of research. In this paper, we show that, (i) the mean-field limit is well-defined with a unique strong solution; (ii) the mean-field process is stable with respect to the initial condition; (iii) we provide conditions such that the finite-N system is long term stable and we obtain some mean-squared error estimates that are uniform in time.

Error Bounds for Locally Optimal Distributed Filters with Random Communication Graphs

“…Remark 2. In Theorem 3, we basically study convergence of the differential equations ( 12) which contain quadratic 1 When taking the norm of a matrix, we refer to Frobenius norm.…”

“…In the literature on stochastic systems, the filter design, or analysis, has received continued attention. An important question addressed in these works is to quantify the performance of the filters by analyzing the bounds on the evolution of error covariance matrix [1]. Several design techniques have now emerged in the literature with varying degree of analysis.…”

Optimality vs Stability Trade-off in Ensemble Kalman Filters

“…For detailed analysis of P-EnKF algorithm, see Tugaut 2018, Bishop and, and extension to nonlinear setting [Del Moral et al 2017, de Wiljes et al 2018]. Analysis of S-EnKF and D-EnKF appears in [Taghvaei and Mehta 2018b] and [Taghvaei and Mehta 2018a] respectively.…”

Error estimate for the approximate solution to multivariate feedback particle filter

“…Theoretically, the general error analysis of the FPF for general NLF systems has much less research literature as far as we know. In [TM1,TM2], they discussed the convergence analysis of FPF in the setting of the linear Gaussian systems and heavily relied on the assumption that the posterior density is Gaussian. Chen, Luo, Shi, and Yau in [CLSY] for the first time studied the error bound between the empirical distribution and real posterior distribution in the continuous-discrete FPF.…”