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Chicken Swarm Kernel Particle Filter: A Structured Rejuvenation Approach with KLD-Efficient Sampling

Hangshuo Tian

TL;DR

The paper addresses the efficiency of particle filters by studying the interaction between CSO-based rejuvenation and KLD-based adaptive sampling. It models CSO as inducing a mean-square contraction toward high-posterior regions, which, via majorization and Karamata's inequality, suggests a CPF can achieve the same statistical error with fewer particles than a baseline PF. Theoretical arguments in a simplified 1D setting are supported by empirical observations showing substantial particle-count reductions with CPF while preserving estimation accuracy and consistency. The work provides a conceptual framework for designing more efficient adaptive PFs and highlights avenues for extending the analysis to higher-dimensional tracking problems.

Abstract

Particle filters (PFs) are often combined with swarm intelligence (SI) algorithms, such as Chicken Swarm Optimization (CSO), for particle rejuvenation. Separately, Kullback--Leibler divergence (KLD) sampling is a common strategy for adaptively sizing the particle set. However, the theoretical interaction between SI-based rejuvenation kernels and KLD-based adaptive sampling is not yet fully understood. This paper investigates this specific interaction. We analyze, under a simplified modeling framework, the effect of the CSO rejuvenation step on the particle set distribution. We propose that the fitness-driven updates inherent in CSO can be approximated as a form of mean-square contraction. This contraction tends to produce a particle distribution that is more concentrated than that of a baseline PF, or in mathematical terms, a distribution that is plausibly more ``peaked'' in a majorization sense. By applying Karamata's inequality to the concave function that governs the expected bin occupancy in KLD-sampling, our analysis suggests a connection: under the stated assumptions, the CSO-enhanced PF (CPF) is expected to require a lower \emph{expected} particle count than the standard PF to satisfy the same statistical error bound. The goal of this study is not to provide a fully general proof, but rather to offer a tractable theoretical framework that helps to interpret the computational efficiency empirically observed when combining these techniques, and to provide a starting point for designing more efficient adaptive filters.

Chicken Swarm Kernel Particle Filter: A Structured Rejuvenation Approach with KLD-Efficient Sampling

TL;DR

The paper addresses the efficiency of particle filters by studying the interaction between CSO-based rejuvenation and KLD-based adaptive sampling. It models CSO as inducing a mean-square contraction toward high-posterior regions, which, via majorization and Karamata's inequality, suggests a CPF can achieve the same statistical error with fewer particles than a baseline PF. Theoretical arguments in a simplified 1D setting are supported by empirical observations showing substantial particle-count reductions with CPF while preserving estimation accuracy and consistency. The work provides a conceptual framework for designing more efficient adaptive PFs and highlights avenues for extending the analysis to higher-dimensional tracking problems.

Abstract

Particle filters (PFs) are often combined with swarm intelligence (SI) algorithms, such as Chicken Swarm Optimization (CSO), for particle rejuvenation. Separately, Kullback--Leibler divergence (KLD) sampling is a common strategy for adaptively sizing the particle set. However, the theoretical interaction between SI-based rejuvenation kernels and KLD-based adaptive sampling is not yet fully understood. This paper investigates this specific interaction. We analyze, under a simplified modeling framework, the effect of the CSO rejuvenation step on the particle set distribution. We propose that the fitness-driven updates inherent in CSO can be approximated as a form of mean-square contraction. This contraction tends to produce a particle distribution that is more concentrated than that of a baseline PF, or in mathematical terms, a distribution that is plausibly more ``peaked'' in a majorization sense. By applying Karamata's inequality to the concave function that governs the expected bin occupancy in KLD-sampling, our analysis suggests a connection: under the stated assumptions, the CSO-enhanced PF (CPF) is expected to require a lower \emph{expected} particle count than the standard PF to satisfy the same statistical error bound. The goal of this study is not to provide a fully general proof, but rather to offer a tractable theoretical framework that helps to interpret the computational efficiency empirically observed when combining these techniques, and to provide a starting point for designing more efficient adaptive filters.

Paper Structure

This paper contains 32 sections, 2 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bayesian Filtering and Bootstrap Particle Filter
  4. KLD-Based Adaptive Particle Sizing
  5. Chicken Swarm Optimization as a Rejuvenation Kernel
  6. Empirical Observation of KLD Behavior under CSO-Based Rejuvenation
  7. Experimental setup.
  8. Theoretical Analysis of CPF and KLD-Sampling
  9. Setup and Notation
  10. Displacement.
  11. Histogram and occupied bins.
  12. Mean-Square Contraction of CSO-Based Rejuvenation
  13. Roosters
  14. Hens
  15. Chicks
  16. ...and 17 more sections

Key Result

Lemma 1

Suppose Theorems th:rooster, th:hen, and th:chick hold, and the rooster variance is uniformly bounded. Then there exist constants $\rho \in (0, 1)$ and $B \ge 0$ such that:

Figures (4)

  • Figure 1: Illustration of the CSO-based rejuvenation step in CPF.
  • Figure 2: Effect of noise levels on the KLD-selected particle numbers for PF and CPF. Subfigure (a) shows the dependence on the process noise, whereas subfigure (b) illustrates the dependence on the measurement noise.
  • Figure 3: Role-based particle contraction in CPF.
  • Figure 4: Results of the noise robustness analysis (Experiment 2) on the stable CV model. (A) Position RMSE vs. noise. (B) Average particle count vs. noise. (C) Efficiency improvement (particle reduction %) vs. noise. (D) Filter consistency (average NEES) vs. noise.

Theorems & Definitions (7)

  • Lemma 1: Global mean-square bound
  • Lemma 2: Majorization and occupied bins
  • proof : Proof of Theorem \ref{['th:rooster']}
  • proof : Proof of Theorem \ref{['th:hen']}
  • proof : Proof of Theorem \ref{['th:chick']}
  • proof : Proof of Lemma \ref{['lem:global-ms']}
  • proof : Proof of Lemma \ref{['lem:majorization']}