Numerical experiments for nonlinear filters with exact particle flow induced by log-homotopy

We show numerical experiments on a new theory of exact particle flow for nonlinear filters. This generalizes our theory of particle flow that was already many orders of magnitude faster than standard particle filters and which is several orders of magnitude more accurate than the extended Kalman filter for difficult nonlinear problems. The new theory generalizes our recent log-homotopy particle flow filters in three ways: (1) the particle flow corresponds to the exact flow of the conditional probability density; (2) roughly speaking, the old theory was based on incompressible flow (like subsonic flight in air), whereas the new theory allows compressible flow (like supersonic flight in air); (3) the old theory suffers from obstruction of particle flow as well as singularities in the equations for flow, whereas the new theory has no obstructions and no singularities. Moreover, our basic filter theory is a radical departure from all other particle filters in three ways: (a) we do not use any proposal density; (b) we never resample; and (c) we compute Bayes' rule by particle flow rather than as a point wise multiplication.

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

[2]  Simon J. Godsill,et al.  Improvement Strategies for Monte Carlo Particle Filters , 2001, Sequential Monte Carlo Methods in Practice.

[3]  Arnaud Doucet,et al.  Particle Filtering and Smoothing: Fifteen years later , 2008 .

[4]  F. LeGland,et al.  A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals , 2003 .

[5]  Fred Daum,et al.  Nonlinear filters with log-homotopy , 2007, SPIE Optical Engineering + Applications.

[6]  D. Mayne,et al.  Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering† , 1969 .

[7]  Yaakov Oshman,et al.  Adaptive Particle Filtering for Spacecraft Attitude Estimation from Vector Observations , 2009 .

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

[9]  Fred Daum,et al.  Particle flow for nonlinear filters with log-homotopy , 2008, SPIE Defense + Commercial Sensing.

[10]  P. Bickel,et al.  Sharp failure rates for the bootstrap particle filter in high dimensions , 2008, 0805.3287.

[11]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[12]  F. Daum.  Nonlinear filters: beyond the Kalman filter , 2005, IEEE Aerospace and Electronic Systems Magazine.

[13]  P. Bickel,et al.  Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems , 2008, 0805.3034.

[14]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[15]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[16]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[17]  Fred Daum,et al.  Nonlinear filters with particle flow , 2009, Optical Engineering + Applications.