Advances in Particle Flow Filters with Taylor Expansion Series

TL;DR

This paper introduces high-order polynomial expansion techniques for particle flow filters, improving their accuracy by deriving drift and diffusion terms directly on polynomial representations, leading to two novel filter variants.

Contribution

It presents a new derivation of particle flow filters using differential algebra and high-order Taylor expansions, enhancing performance over existing methods.

Findings

High-order terms improve filter accuracy

New filters outperform Gromov and exact flows

Polynomial-based derivation simplifies implementation

Abstract

Particle Flow Filters perform the measurement update by moving particles to a different location rather than modifying the particles' weight based on the likelihood. Their movement (flow) is dictated by a drift term, which continuously pushes the particle toward the posterior distribution, and a diffusion term, which guarantees the spread of particles. This work presents a novel derivation of these terms based on high-order polynomial expansions, where the common techniques based on linearization reduce to a simpler version of the new methodology. Thanks to differential algebra, the high-order particle flow is derived directly onto the polynomials representation of the distribution, embedded with differentiation and evaluation. The resulting technique proposes two new particle flow filters, whose difference relies on the selection of the expansion center for the Taylor polynomial evaluation. Numerical applications show the improvement gained by the inclusion of high-order terms, especially when comparing performance with the Gromov flow and the "exact" flow.