一般性粒子滤波算法收敛特性

Convergence Property of a Generic Particle Filter Algorithm

粒子滤波算法在处理最优滤波问题时受到了广泛的重视,对此类算法的收敛性研究是该领域研究的热点问题.首先介绍了一种变换的一般性粒子滤波算法,与一般性粒子滤波算法不同,在每次执行重要性采样步骤后,新算法需要判别是否需要重新执行重采样步骤和重要性采样步骤.随后对新算法的几乎必然收敛性进行了分析,并将对新算法的收敛性讨论推广到一般性粒子滤波算法中.研究了当感兴趣函数在扩展状态后验联合分布下四阶距存在并且递归次数有限时,由一般性粒子滤波算法得出的估计几乎收敛于最优估计的充分条件.最后,通过一组仿真实验来说明一般性粒子滤波算法的几乎必然收敛性.

Particle filters are widely utilized in the optimal filtering problems. These methods approximate the posteriori distribution of the state （or the posteriori joint distribution of the extened state） by a population of weighted particles which evolve randomly according to the dynamic system model and the measurements. Despite many theoretical advance which have been reported in the last decade,the study of the convergence property of particle filters is still an open question. In this paper,the almost sure convergence of the generic particle filter （GPF） is discussed in a circuitous way. First,a modified-generic particle filter （M-GPF） is constructed. Different from the GPF,the M-GPF will determine whether it is necessary to rerun both the resampling step and the importance sampling （IS） step according to a conditional criterion after performing the IS step at each time. Then the almost sure convergence of the M-GPF will be concerned. Later,when the recursive time is finite and the interesting function is 4th power integrable with respect to the posteriori joint distribution of the extended state,the sufficient condition for the GPF estimation converges almost surely to the optimal estimation is discussed. Finally,a novel simulation experiment will be presented to illustrate the almost sure convergence of the GPF.