An adaptive split-combine Gaussian mixture filter for nonlinear and multimodal state estimation

2026-08-05

AI Summary

This work addresses the challenge of accurately approximating non-Gaussian, multimodal, or highly skewed posterior probability density functions commonly arising in nonlinear systems, which are poorly captured by conventional single-Gaussian filtering approaches. To this end, the paper proposes an Adaptive Merging-and-Splitting Gaussian Mixture Filter (AMF) that dynamically adjusts the number of mixture components through a novel splitting mechanism—reducing variance along target isocontour directions—and an adaptive merging strategy. Built upon a Gaussian mixture model, AMF efficiently approximates complex posteriors without requiring online optimization and supports parallel computation, significantly enhancing computational efficiency. Experimental results on benchmark nonlinear systems, including the Van der Pol oscillator and Lorenz attractor, demonstrate that AMF consistently outperforms existing methods in both estimation accuracy and practical applicability.

Abstract

Filtering combines model predictions with measurements to estimate the probability density function (PDF) of a system state over time. The PDF often becomes highly asymmetric and even multimodal in nonlinear systems with oscillatory or chaotic dynamics. Such non-Gaussian features violate the single-Gaussian assumption underlying Kalman-type filters. To address this problem, Gaussian mixture filtering has been proposed. However, accurately propagating mixture components and adaptively adjusting their number and weights over time remain open challenges. Here, we develop an adaptive split-combine Gaussian mixture filter (AMF) that estimates the time evolution of asymmetric and multimodal PDFs by adaptively splitting and combining Gaussian particles without auxiliary online numerical optimization. Notably, the proposed splitting method guarantees a reduction in variance along a target level-set-point direction of a Gaussian particle. This enables accurate and efficient propagation of particles. We show that AMF consistently outperforms various baseline filters across diverse benchmarks, including single and coupled slow-fast Van der Pol oscillators and the Lorenz attractor. We also propose a parallel implementation of AMF, allowing high-fidelity PDF estimation with practical computational cost.

Problem

• nonlinear filtering
• multimodal state estimation
• Gaussian mixture filter
• non-Gaussian PDF
• adaptive filtering

Innovation

• adaptive Gaussian mixture filter
• split-combine strategy
• nonlinear state estimation
• multimodal PDF
• parallel implementation