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Nudged Particle Filter with Optimal Resampling Applied to the Duffing Oscillator

Ryne Beeson, Uwe Hanebeck

TL;DR

This work tackles the challenge of filtering chaotic dynamical systems with separatrix structures, where standard particle filters struggle due to degeneracy and multi-modal posteriors. It introduces the intermediate resampling nudged particle filter (IRnPF), which couples a control-based nudging toward the future observation with a deterministic resampling step that minimizes the modified Cramér-von Mises distance $D(\mu,\nu)$ to control weight variance. The method is demonstrated on the 2D stochastic Duffing oscillator, showing that IRnPF consistently outperforms both the standard PF and the original nudged PF at the same particle count, particularly across separatrix boundaries. The results suggest IRnPF offers a more robust and efficient filtering strategy for chaotic, sparsely observed systems and may extend to higher-dimensional problems.

Abstract

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Nudged Particle Filter with Optimal Resampling Applied to the Duffing Oscillator

TL;DR

This work tackles the challenge of filtering chaotic dynamical systems with separatrix structures, where standard particle filters struggle due to degeneracy and multi-modal posteriors. It introduces the intermediate resampling nudged particle filter (IRnPF), which couples a control-based nudging toward the future observation with a deterministic resampling step that minimizes the modified Cramér-von Mises distance to control weight variance. The method is demonstrated on the 2D stochastic Duffing oscillator, showing that IRnPF consistently outperforms both the standard PF and the original nudged PF at the same particle count, particularly across separatrix boundaries. The results suggest IRnPF offers a more robust and efficient filtering strategy for chaotic, sparsely observed systems and may extend to higher-dimensional problems.

Abstract

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Paper Structure

This paper contains 10 sections, 1 theorem, 21 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Duffing Oscillator
  3. Standard and Nudged Particle Filter
  4. Standard Particle Filter with Resampling (PF)
  5. Nudged Particle Filter (nPF)
  6. Intermediate Resampling Nudged Particle Filter (IRnPF)
  7. The IRnPF Method
  8. Numerical Experiments
  9. Monte Carlo Results
  10. Conclusions

Key Result

Theorem 4.1

With $\mkern 1.5mu\overline{\mkern-1.5mu b \mkern-1.5mu}\mkern 1.5mu \gg 1$, $w(b) = 1 / b^{n - 1}$, $\mu, \nu$ Dirac mixtures with support $(x_i), (y_i)$ and weights $(w^i_x), (w^i_y)$ respectively, and $\mathcal{K}$ given according to equation: separable Gaussian kernel, $D(\mu, \nu)$ is approxima where where $\Gamma \approx 0.5772$ is the Euler gamma constant and $\operatorname{xlog}(s) \equiv

Figures (11)

  • Figure 1: Phase portrait for the deterministic unforced nonlinear Duffing oscillator, highlighting realms of dynamical significance.
  • Figure 2: Global view for a realization of the advected initial PF distribution, started with a deterministic initial condition and separated from the true hidden signal state, to the first observation time.
  • Figure 3: Local view for a realization of the advected initial PF distribution, started with a deterministic initial condition and separated from the true hidden signal state, to the first observation time.
  • Figure 4: Global view of the PF behavior from the second to last observation to the last observation of the simulation.
  • Figure 5: Same as Fig. \ref{['figure: global view, separated by stable manifold, prediction flow 8, PF']}, but for the nPF.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Definition 4.1: Localized Cumulative Distribution Hanebeck:2008.ieee.cmfiis
  • Definition 4.2: Modified Cramér-von Mises Distance Hanebeck:2008.ieee.cmfiis
  • Theorem 4.1