The particle filter (PF) is an important tool used for estimating states in a variety of nonlinear systems, such as positioning, mapping, and diagnostics. As it is so widely used, it is important that all the properties of the PF are well understood so that practitioners can spend their efforts on the issues facing their specific application instead of on the PF.

The PF has been in use for a long time now, and therefore much of its behavior has been figured out. At the same time, many of the core beliefs regarding the filter, such as the estimate improving when more particles are added, or marginalization always improving the estimate, are primarily based on experience rather than direct theory. While the topics have been studied, the theoretical results generally only apply as the number of particles tends to infinity and often require further assumptions of the systems being studied. This leaves room open for unexpected behaviors. In this work, analysis and explanation are given for two previously unexpected behaviors surrounding the PF.

The first behavior studied is a counter-intuitive phenomenon where, the mean squared error (MSE) of a PF can actually increase as more particles are added. An explanation for this is provided in the form of a new property, here called ‘projected instability’. It is based on a process in which each process noise is selected to minimize the next measurement error. Extensive simulations are used to show that the phenomenon can only occur when a projected instability is present. Further analysis is provided, both to explain why this behavior can occur, and to show what other conditions have an impact on the above phenomenon occurring.

The second result concerns the marginalized particle filter (MPF), also known as the Rao-Blackwellized particle filter (RBPF). The MPF utilizes conditionally linear substructures in the problem formulation, which allows for solving part of the estimation problem using particles, while the other part is estimated using a Kalman filter (KF) conditioned on each particle. The common understanding is that, the performance of an MPF is generally better than that of a PF with the same number of particles. What is shown here, is instead a broad subsection of problems for which the MPF gives identical quality results as the PF when applied. In fact, it is shown that after some time, the behavior of the filters is identical. Simple conditions are provided to determine when this is the case.