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目的:重采样方法是粒子滤波设计的重要环节,也是避免或克服“权值退化”和“多样性匮乏”这一对粒子滤波难点问题的关键。当前研究领域已有几十余种重采样方法,然而尚缺乏一个基础性的重采样设计原则以及对这些方法的综合性能对比。针对于此,本文提出重采样“同分布”设计原则,并在此基础上,提出一种能够最大程度满足同分布原则的最优重采样方法。本文希望所提出的重采样同分布原则以及新方法有利于进一步的新方法设计或已有方法的工程选用。创新点:理论上严格定义了同分布原则作为重采样方法设计的普遍性原则,给出三种同分布测度方法;提出了一种最小采样方差(MSV:minimum sampling variance)最优重采样方法,在满足渐近无偏性的前提下获得最小采样方差。方法:给出三种“重采样同分布”测度方法:Kullback-Leibler偏差,Kolmogorov-Smirnov统计和采样方差(sampling variance)。所提出的最小采样方差重采样放宽了无偏性条件,仅满足渐近无偏,但获得了最小采样方差(参见定理2-4论证以及仿真性能对比)。结论:重采样前后粒子的概率分布应该统计上一致(即“同分布”)是重采样方法设计的一个重要原则。明确这一基本原则有利于规范化重采样新方法的设计与工程选用。所提出的MSV重采样新方法渐近无偏,并具有最小采样方差的优异理论特性,即最优地满足同分布原则。算法性能分析表明:大多数无偏或者渐近无偏重采样方法在滤波精度上差异较小,但是在采样方差、计算效率方面差异较大。另一方面,基于一些特殊规则或者问题模型设计的重采样方法可能具有特别优势。
Objective: The resampling method is an important part of the particle filter design, and it is also the key to avoid or overcome the particle filter difficulty problem of “weight degeneration” and “lack of diversity”. There are dozens of resampling methods in the current research area, however, there is a lack of a basic principle of resampling design and comprehensive performance comparison of these methods. In view of this, this paper presents the resampling “same distribution ” design principle, and on this basis, proposes an optimal resampling method that can best meet the same distribution principle. This paper hopes that the proposed resampling and distribution principle and the new method are beneficial to the further new method design or the engineering selection of the existing methods. Innovative point: In theory, the same distribution principle is strictly defined as the universal principle of resampling method design, and three methods for measuring the same distribution are given. An optimal resampling method with minimum sampling variance (MSV) is proposed, Get the minimum sampling variance while satisfying the asymptotic unbiasedness. Methods: Three methods of “resampling homogeneity” are given: Kullback-Leibler bias, Kolmogorov-Smirnov statistics and sampling variance. The proposed minimum-sampling-variance resampling relaxes the unbiased conditions and satisfies only asymptotic unbiasedness, but achieves the minimum sampling variance (see Theorem 2-4 for Arithmetic and Comparison of Simulation Performance). Conclusion: The probability distribution of particles before and after resampling should be statistically consistent (ie, “same distribution”) is an important principle of resampling method design. To clarify this basic principle is conducive to standardizing the design of new methods of sampling and engineering selection. The proposed new MSV resampling method is asymptotically unbiased and has the excellent theoretical property of minimum sampling variance, that is, the optimal distribution of MSV is satisfied. The performance analysis of the algorithm shows that most of the unbiased or asymptotic unbiased sampling methods have less difference in filtering accuracy, but differ greatly in sampling variance and computational efficiency. On the other hand, resampling methods based on some special rules or problem models may have particular advantages.