On the performance of particle filters with adaptive number of particles

We investigate the performance of a class of particle filters (PFs) that can automatically tune their computational complexity evaluating online certain predictive statistics which are invariant for a broad class of state-space models. To be specific, we a family of block-adaptive PFs based on the methodology of Elvira et al. (IEEE Trans Signal Process 65(7):1781– 1794, 2017). In this class of algorithms, the number of Monte Carlo samples (known as particles) is adjusted periodically, and prove that the theoretical error bounds of the PF actually adapt to the updates in the number of particles. The evaluation the predictive statistics that lies at the core of the methodology is done by generating fictitious observations, i.e., particles the observation space. We study, both analytically and numerically, the impact of the number K of these particles on the of the algorithm. In particular, we prove that if the predictive statistics with K fictitious observations converged , then the particle approximation of the filtering distribution would match the first K elements in a series of moments the true filter. This result can be understood as a converse to some convergence theorems for PFs. From this analysis, we an alternative predictive statistic that can be computed (for some models) without sampling any fictitious observations all. Finally, we conduct an extensive simulation study that illustrates the theoretical results and provides further insights the complexity, performance and behavior of the new class of algorithms.