Sequential Monte Carlo methods in filter theory

Paul Fearnhead

Department of Statistics, 1, South Parks Road, Oxford, OX1 3TG.
email: p.fearnhead@lancs.ac.uk.

Summary

The need for accurate monitoring and analysis of sequential data arises in many scientific, industrial and financial problems. Although the Kalman filter (Kalman and Bucy, 1961) is effective for linear-Gaussian models, new methods of filtering are required for the general case.
Nonlinear and non-Gaussian filters are reviewed, with particular emphasis being placed on the particle filter, a recently developed filter, which uses sequential Monte Carlo methods. The particle filter is seen to cover a number of independently proposed, but related, filters, dating back to the SIR filter of Gordon, Salmond and Smith (1993).
All these particle filters approach the filtering problem from a sampling perspective, with the aim being to generate a random sample from the true posterior distribution. In this thesis filtering is viewed as a Monte Carlo integration problem. This perspective is used to suggest a number of new improvements for the particle filter. Along with these improvements, guidelines for the efficient implementation of the particle filter are given. It is also shown that the application of the particle filter to certain types of problems naturally leads to a filter similar to, but more efficient than, the random sampling algorithm of Akashi and Kumamoto (1977).
In order to demonstrate these improvements, the new particle filters are tested on various examples. These examples include the much studied bearings-only tracking problem, and a change-point detection problem based on oil well data. It is shown that significant increases in efficiency can be obtained by using the suggested improvements, and that the improved particle filters give promising results.
The application of particle filters to problems with fixed parameters is also considered. These are problems on which sequential Monte Carlo methods often struggle. Two simple examples will be studied, on which a number of different methods are tried. The results obtained will provide guidance for the construction of efficient methods for analysing more complicated fixed parameter problems.

Keywords

Importance resampling, Sequential filter, Markov Chain Monte Carlo, Rejection sampling, SIR filter, MHIR filter, ASIR fitler, Particle filter, Bearings-only tracking, Non-linear, State-space modelling, Change-point detection.

[postscript]

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