Unified framework for sampling/importance resampling algorithms

Sequential Monte Carlo (SMC) methods, i.e. particle filters, have been extensively studied and applied to various nonlinear Bayesian filtering problems throughout the last decade. The sampling/importance resampling (SIR) algorithm is one of the most commonly applied SMC methods and various proposals have been made for improving the performance of SIR algorithms. In this paper, we combine the work of various authors to provide a unified SIR framework which is then shown to cover some well known methods, such as the auxiliary particle filter. The description of the generalised framework is given from a measure theoretic point of view and the significance of resampling as a separate step of the algorithm is suppressed. Instead, resampling is regarded as an integral part of the random sample generation in importance sampling integration. By allowing a stratified sampling scheme, the generalised SIR framework is also shown to cover the sequential importance sampling algorithm which is generally not considered to be a SIR algorithm because of the absence of the resampling step. The general framework is illustrated by showing how it can be used for improving a SIR algorithm that uses extended Kalman filter equations for defining the importance distribution. The resulting algorithm is applied to a range-only tracking application and it is compared with some other choices of importance distribution.