Discrete Time: Filtering Algorithms

  • Kody Law
  • Andrew Stuart
  • Konstantinos Zygalakis
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 62)

Abstract

In this chapter, we describe various algorithms for the filtering problem. Recall from Section  2.4 that filtering refers to the sequential update of the probability distribution on the state given the data, as data is acquired, and that \(Y _{j} =\{ y_{\ell}\}_{\ell=1}^{j}\) denotes the data accumulated up to time j. The filtering update from time j to time j + 1 may be broken into two steps: prediction , which is based on the equation for the state evolution, using the Markov kernel for the stochastic or deterministic dynamical system that maps \(\mathbb{P}(v_{j}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j})\); and analysis , which incorporates data via Bayes’s formula and maps \(\mathbb{P}(v_{j+1}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j+1})\). All but one of the algorithms we study (the optimal proposal version of the particle filter) will also reflect these two steps.

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Supplementary material

332018_1_En_4_MOESM1_ESM.zip (15 kb)
matlab files (ZIP 15 KB)

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kody Law
    • 1
  • Andrew Stuart
    • 2
  • Konstantinos Zygalakis
    • 3
  1. 1.Oak Ridge National LaboratoryComputer Science and Mathematics DivisionOak RidgeUSA
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Division of Mathematical SciencesUniversity of SouthamptonSouthamptonUK

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