Adapting the Number of Particles in Sequential Monte Carlo Methods Through an Online Scheme for Convergence Assessment

Particle filters are broadly used to approximate posterior distributions of hidden states in state-space models by means of sets of weighted particles. While the convergence of the filter is guaranteed when the number of particles tends to infinity, the quality of the approximation is usually unknown but strongly dependent on the number of particles. In this paper, we propose a novel method for assessing the convergence of particle filters in an online manner, as well as a simple scheme for the online adaptation of the number of particles based on the convergence assessment. The method is based on a sequential comparison between the actual observations and their predictive probability distributions approximated by the filter. We provide a rigorous theoretical analysis of the proposed methodology and, as an example of its practical use, we present simulations of a simple algorithm for the dynamic and online adaptation of the number of particles during the operation of a particle filter on a stochastic version of the Lorenz 63 system.

[1]  Nicholas G. Polson,et al.  Particle Filtering , 2006 .

[2]  P. Fearnhead,et al.  Improved particle filter for nonlinear problems , 1999 .

[3]  Ondřej Straka,et al.  PARTICLE FILTER ADAPTATION BASED ON EFFICIENT SAMPLE SIZE , 2006 .

[4]  Dieter Fox,et al.  Adapting the Sample Size in Particle Filters Through KLD-Sampling , 2003, Int. J. Robotics Res..

[5]  Eric Moulines,et al.  Adaptive methods for sequential importance sampling with application to state space models , 2008, 2008 16th European Signal Processing Conference.

[6]  D. Crisan,et al.  Fundamentals of Stochastic Filtering , 2008 .

[7]  P. Moral,et al.  Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering , 2000 .

[8]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[9]  Joaquín Míguez,et al.  A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models , 2012, Stat. Comput..

[10]  Jun S. Liu,et al.  Rejection Control and Sequential Importance Sampling , 1998 .

[11]  Nadia Oudjane,et al.  A sequential particle algorithm that keeps the particle system alive , 2005, 2005 13th European Signal Processing Conference.

[12]  R. Plackett,et al.  Karl Pearson and the Chi-squared Test , 1983 .

[13]  Rong Chen,et al.  Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering , 2000, IEEE Trans. Inf. Theory.

[14]  Nicolas Chopin,et al.  SMC2: an efficient algorithm for sequential analysis of state space models , 2011, 1101.1528.

[15]  P. Moral,et al.  On Adaptive Sequential Monte Carlo Methods , 2008 .

[16]  Anthony Lee,et al.  The Alive Particle Filter , 2013 .

[17]  Anthony Lee,et al.  On the role of interaction in sequential Monte Carlo algorithms , 2013, 1309.2918.

[18]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[19]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[20]  Nando de Freitas,et al.  An Introduction to Sequential Monte Carlo Methods , 2001, Sequential Monte Carlo Methods in Practice.

[21]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[22]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[23]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[24]  Petar M. Djuric,et al.  Assessment of Nonlinear Dynamic Models by Kolmogorov–Smirnov Statistics , 2010, IEEE Transactions on Signal Processing.

[25]  J. Míguez,et al.  Nested particle filters for online parameter estimation in discrete-time state-space Markov models , 2013, Bernoulli.

[26]  A. Beskos,et al.  On the stability of sequential Monte Carlo methods in high dimensions , 2011, 1103.3965.

[27]  Paul Krause,et al.  Dimensional reduction for a Bayesian filter. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Simon J. Godsill,et al.  An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo , 2007, Proceedings of the IEEE.

[29]  Petar M. Djuric,et al.  Resampling Methods for Particle Filtering , 2015 .

[30]  Dan Crisan,et al.  Particle Filters - A Theoretical Perspective , 2001, Sequential Monte Carlo Methods in Practice.

[31]  D. Crisan,et al.  Uniform approximations of discrete-time filters , 2008, Advances in Applied Probability.

[32]  Anthony Lee,et al.  ‘Variance estimation in the particle filter’ , 2015, Biometrika.

[33]  Yee Whye Teh,et al.  Asynchronous Anytime Sequential Monte Carlo , 2014, NIPS.

[34]  Anthony Lee,et al.  Variance estimation and allocation in the particle filter , 2015 .

[35]  Alvaro Soto,et al.  Self Adaptive Particle Filter , 2005, IJCAI.

[36]  Dan Crisan,et al.  Particle-kernel estimation of the filter density in state-space models , 2011, 1111.5866.

[37]  Xiao-Li Hu,et al.  A Basic Convergence Result for Particle Filtering , 2008, IEEE Transactions on Signal Processing.

[38]  Petar M. Djuric,et al.  Resampling Methods for Particle Filtering: Classification, implementation, and strategies , 2015, IEEE Signal Processing Magazine.

[39]  P. Moral,et al.  The Alive Particle Filter and Its Use in Particle Markov Chain Monte Carlo , 2015 .

[40]  Edward L. Ionides,et al.  Adaptive particle allocation in iterated sequential Monte Carlo via approximating meta-models , 2016, Stat. Comput..

[41]  A. Doucet,et al.  Particle Markov chain Monte Carlo methods , 2010 .

[42]  Alex Bateman,et al.  An introduction to hidden Markov models. , 2007, Current protocols in bioinformatics.

[43]  Petar M. Djuric,et al.  On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization , 2011, Statistics and Computing.