Particle Filter with Resampling Method Comparison

In this document, we discuss particle filters and several different resampling methods, including SIR resampling, auxiliary resampling, and regular resampling. We will subsequently examine the working principles, advantages, and disadvantages of each method in detail.

SIR (Sequential Importance Resampling) resampling is a commonly used method based on importance sampling and random sampling principles. It effectively addresses particle degeneracy issues in particle filters through systematic resampling of particles according to their weights. In code implementation, SIR typically involves calculating cumulative weights and using random numbers to select particles proportionally to their weights. However, SIR resampling has drawbacks such as large sample variance and difficulties in handling multi-modal distributions, which may require additional algorithmic adjustments.

Unlike SIR resampling, auxiliary resampling is based on auxiliary particle filter methodology. This approach better handles multi-modal distributions by incorporating predictive information about particles' future states, while effectively mitigating large sample variance issues. Implementation-wise, auxiliary resampling requires additional computation for proposal distributions and particle prediction steps, which increases computational load and memory requirements compared to basic SIR methods.

Finally, we introduce regular resampling - a relatively simple method based on equal probability sampling concepts. This approach avoids large sample variance problems and requires minimal computational resources as it systematically divides the cumulative weight space into equal segments. Code implementation typically involves creating uniform intervals and selecting one particle from each segment. However, since it ignores importance weight influences, regular resampling may perform poorly when dealing with multi-modal distributions or complex state spaces.

In summary, different resampling methods each have distinct advantages and limitations. Practical implementation should consider specific application requirements, computational constraints, and distribution characteristics when selecting the appropriate resampling strategy for particle filter applications.