Direct, prediction- and smoothing-based Kalman and particle filter algorithms

François Desbouvries*, Yohan Petetin, Boujemaa Ait-El-Fquih

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


We address the recursive computation of the filtering probability density function (pdf) pnn in a hidden Markov chain (HMC) model. We first observe that the classical path pn-1n-1→pn n-1→pnn is not the only possible one that enables to compute pnn recursively, and we explore the direct, prediction-based (P-based) and smoothing-based (S-based) recursive loops for computing pn n. We next propose a common methodology for computing these equations in practice. Since each path can be decomposed into an updating step and a propagation step, in the linear Gaussian case these two steps are implemented by Gaussian transforms, and in the general case by elementary simulation techniques. By proceeding this way we routinely obtain in parallel, for each filtering path, one set of Kalman filter (KF) equations and one generic sequential Monte Carlo (SMC) algorithm. Finally we classify in a common framework four KF (two of which are original), which themselves can be associated to four generic SMC algorithms (two of which are original). We finally compare our algorithms via simulations. S-based filters behave better than P-based ones, and within each class of filters better results are obtained when updating precedes propagation.

Original languageEnglish (US)
Pages (from-to)2064-2077
Number of pages14
JournalSignal Processing
Issue number8
StatePublished - Aug 2011
Externally publishedYes


  • Kalman filters
  • Particle filtering
  • Sampling importance resampling
  • Sequential Monte Carlo
  • Sequential importance sampling

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Software
  • Signal Processing
  • Computer Vision and Pattern Recognition
  • Electrical and Electronic Engineering


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