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Advances in Meteorology
Volume 2016, Article ID 9372786, 18 pages
Research Article

From the Kalman Filter to the Particle Filter: A Geometrical Perspective of the Curse of Dimensionality

1CNRM, UMR 3589, 42 Av. Coriolis, 31057 Toulouse, France
2INPT-ENM, 42 Av. Coriolis, 31057 Toulouse, France

Received 28 June 2016; Accepted 20 October 2016

Academic Editor: James Cleverly

Copyright © 2016 Olivier Pannekoucke et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this contribution is to provide a description of the difference between Kalman filter and particle filter when the state space is of high dimension. In the Gaussian framework, KF and PF give the same theoretical result. However, in high dimension and using finite sampling for the Gaussian distribution, the PF is not able to reproduce the solution produced by the KF. This discrepancy is highlighted from the convergence property of the Gaussian law toward a hypersphere: in high dimension, any finite sample of a Gaussian law lies within a hypersphere centered in the mean of the Gaussian law and of radius square-root of the trace of the covariance matrix. This concentration of probability suggests the use of norm as a criterium that discriminates whether a forecast sample can be compatible or not with a given analysis state. The contribution illustrates important characteristics that have to be considered for the high dimension but does not introduce a new approach to face the curse of dimensionality.