Understanding Classic Particle Filter Algorithms

Particle Filter is a state estimation technique based on Monte Carlo methods for nonlinear non-Gaussian systems. It approximates probability distributions using a set of random samples (particles) and is suitable for tracking and prediction problems in complex systems. The classic particle filter algorithm primarily consists of the following key steps:

Initialization phase: Randomly generate a set of particles according to the prior distribution, where each particle represents a possible system state hypothesis and is assigned equal weight. In code implementation, this typically involves using random number generators to sample from the initial state distribution.

Importance sampling: Propagate particles according to the system model, advancing each particle to the new time step using the state transition equation. This step requires designing an appropriate importance density function. Algorithmically, this involves implementing the state transition model, often using system dynamics equations with process noise.

Weight update: When new observation data is obtained, recalculate each particle's weight based on the observation likelihood function. Particles that better match the observations receive higher weights. This is typically implemented by computing the probability of observations given each particle's state using measurement models and noise characteristics.

Resampling: To address particle degeneracy (where few particles dominate most weights), redistribute particle resources by replicating high-weight particles and eliminating low-weight ones. Common resampling methods include multinomial resampling, systematic resampling, and stratified resampling. Code implementation often involves cumulative sum calculations and random sampling techniques.

State estimation: Finally, compute the system state estimate based on the weighted particle set, typically using weighted average or maximum a posteriori probability estimation. This can be implemented through statistical calculations on the particle population.

Understanding particle filters also requires mastering several key concepts: The choice of importance density function directly affects filtering performance, where optimal importance density minimizes weight variance; Degeneracy phenomenon is inevitable and needs monitoring through metrics like effective sample size; While resampling solves degeneracy, it introduces sample impoverishment issues.

Variant algorithms of particle filters such as SIR filter (Sampling Importance Resampling), auxiliary particle filter, and others have been developed to better handle these challenges. These variants often involve modifications in proposal distribution design and resampling strategies to improve computational efficiency and estimation accuracy.