Particle Filter

Problem Statement

When posterior distributions are multi-modal or strongly non-Gaussian, Kalman-family filters become brittle. Particle filters approximate the full posterior with weighted samples and remain effective in nonlinear, non-Gaussian settings.

Model and Formulation

The posterior is approximated by particles:

$$ p(x_k|z_{1:k}) \approx \sum_{i=1}^{N} w_k^{(i)}\delta\left(x_k - x_k^{(i)}\right) $$

Weights are updated from measurement likelihood:

$$ w_k^{(i)} \propto w_{k-1}^{(i)}p(z_k|x_k^{(i)}) $$

Resampling is triggered by effective sample size:

$$ N_{eff} = \frac{1}{\sum_i (w_k^{(i)})^2} $$

Algorithm Procedure

1. Sample particles from transition model.
2. Evaluate measurement likelihood for each particle.
3. Normalize weights and compute N_eff.
4. Resample when degeneracy threshold is crossed.

Tuning Guidance

• Increase particle count for higher-dimensional states.
• Match proposal noise to platform maneuver envelope.
• Use stratified/systematic resampling to reduce variance.

Failure Modes and Diagnostics

• Particle impoverishment occurs with frequent resampling and narrow proposals.
• Sparse particle sets miss low-probability but valid hypotheses.
• Computational load scales with particle count and measurement complexity.

Implementation and Execution

python -m uav_sim.simulations.estimation.particle_filter

Evidence

References

• Arulampalam et al., A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking (2002)
• Doucet and Johansen, A Tutorial on Particle Filtering and Smoothing (2009)

Related Algorithms

• Unscented Kalman Filter
• EKF-SLAM