Particle filter

Particle filters or Sequential Monte Carlo (SMC) methods are a set of genetic-type particle Monte Carlo methodologies to solve . The term "particle filters" was first coined in 1996 by Del Moral in reference to mean field interacting particle methods used in fluid mechanics since the beginning of the 1960s. The terminology "sequential Monte Carlo" was proposed by Liu and Chen in 1998.

From the statistical and probabilistic point of view, particle filters can be interpreted as mean field particle interpretations of Feynman-Kac probability measures. These particle integration techniques were developed in molecular chemistry and computational physics by Theodore E. Harris and Herman Kahn in 1951, Marshall. N. Rosenbluth and Arianna. W. Rosenbluth in 1955 and more recently by Jack H. Hetherington in 1984. In computational physics, these Feynman-Kac type path particle integration methods are also used in Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods. Feynman-Kac interacting particle methods are also strongly related to mutation-selection genetic algorithms currently used in evolutionary computing to solve complex optimization problems.

The particle filter methodology is used to solve Hidden Markov Chain (HMM) and nonlinear filtering problems arising in signal processing and Bayesian statistical inference. The consists of estimating the internal states in dynamical systems when partial observations are made, and random perturbations are present in the sensors as well as in the dynamical system. The objective is to compute the conditional probability (a.k.a. posterior distributions) of the states of some Markov process, given some noisy and partial observations. With the notable exception of linear-Gaussian signal-observation models (Kalman filter) or wider classes of models (Benes filter) Mireille Chaleyat-Maurel and Dominique Michel proved in 1984 that the sequence of posterior distributions of the random states of the signal given the observations (a.k.a. optimal filter) have no finitely recursive recursion. Various numerical methods based on fixed grid approximations, Markov Chain Monte Carlo techniques (MCMC), conventional linearization, extended Kalman filters, or determining the best linear system (in expect cost-error sense) have never really coped with large scale systems, unstable processes or when the nonlinearities are not sufficiently smooth.

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