feedback pf

Feedback Particle Filter with Data-Driven Gain-Function Approximation https://www.merl.com/publications/docs/TR2018-034.pdf

Abstract

This paper addresses the continuous-disctrete time nonlinear filtering problem for stochastic dynamical systems using feedback particle filter. The main difficulty in the FPF is to approximate the gain function that controls the particles.

This paper proposed a novel Galerkin-based method inspired by high-dimensional data-analysis techniques.

Introduction

One problem with traditional PFs is the inevitable particle degeneracy. The resampling step makes PFs practically usefull, but introduces other negative effects, such as sample improverishment and increased variance.

Background: Feedback particle filter

In conventional PFs, the measurement update is implemented as a point-wise multiplication between likelihood and prior, where the prior is represented by a set of (N) weighted particles, where the weights are computed using the likelihood conditioned on the respective particle. The FPF approximates the posterior with (N) unweighted samples, or particles.

At time (t_k), a new observation (y_k) arrives. To incorporate (y_k), a particle flow ({Sk^i(lambda) }{i=1}^N) defined by differential equations is introduced

[
frac{dS_k^i(lambda)}{dlambda} = K(S_k^i(lambda), lambda) I_k^i + frac{1}{2} Omega(S_k^i(lambda), lambda)
]

In particular, for linear and Gaussian systems, the gian function becomes the Kalman gain and the Wong-Zakai term vanished.

The main difficulty in the FPF is to find (K), and only in limited cases can an exact solution be computed.

Constant-Gain Approximation

[
K approx [c_1 dots c_m] R^{-1}
]