Summary
Particle filters provide a general and flexible approach to numerically approximate the nonlinear filtering solution, a conditional probability distribution, of continuous-time partially observable stochastic signal processes. In this paper we consider the case of a continuous-time signal and discrete-time observation process. Standard implementations of particle methods suffer from particle degeneracy (also known as impoverishment or collapse), where after repeated application of Bayes' formula for updating the weights of the particles, few particles have any significant weight. The most common remedy in practice is to apply heuristic resampling methods. Control and flow-based particle methods have been introduced in the past decade as alternatives for handling the issue of degeneracy. The former method (control-based) modifies the dynamics between observations, whereas the latter (flow-based) introduces a pseudo-time for particle flow at the instance of observation. Both have advantages and disadvantages. In this paper, we develop theory and algorithms for a method that attempts to blend the advantages of both control and flow-based approaches. Motivated by chaotic systems from the geophysical sciences, we benchmark the hybrid method against standard, control, and flow-based particle methods on the atmospheric models of Lorenz.
