Nonlinear filtering with transfer operator

This paper presents a new nonlinear filtering algorithm that is shown to outperform state-of-the-art particle filters with resampling. Starting from the Itô stochastic differential equation, the proposed algorithm harnesses Karhunen-Loéve expansion to derive an approximate non-autonomous dynamical system, for which transfer operator based density computation can be performed in exact arithmetic. It is proved that the algorithm is asymptotically consistent in mean-square sense. Numerical results demonstrate that explicitly accounting prior dynamics entail significant performance improvement for nonlinear non-Gaussian estimation problems with infrequent measurement updates, as compared to the performance of particle filters.

[1]  Abhishek Halder,et al.  Model validation: A probabilistic formulation , 2011, IEEE Conference on Decision and Control and European Control Conference.

[2]  Abhishek Halder,et al.  Uncertainty quantification for stochastic nonlinear systems using Perron-Frobenius operator and Karhunen-Loève expansion , 2012, 2012 IEEE International Conference on Control Applications.

[3]  J. Huang,et al.  Curse of dimensionality and particle filters , 2003, 2003 IEEE Aerospace Conference Proceedings (Cat. No.03TH8652).

[4]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[5]  H. Risken.  Fokker-Planck Equation , 1984 .

[6]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[7]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[8]  I. Michael Ross,et al.  Optimal Nonlinear Feedback Guidance for Reentry Vehicles , 2006 .

[9]  Abhishek Halder,et al.  Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville Equation , 2011 .

[10]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[11]  F. Daum.  Exact finite dimensional nonlinear filters , 1985, 1985 24th IEEE Conference on Decision and Control.

[12]  W. Gilks,et al.  Following a moving target—Monte Carlo inference for dynamic Bayesian models , 2001 .

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

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

[15]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[16]  Abhishek Halder,et al.  Further results on probabilistic model validation in Wasserstein metric , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[17]  C. Villani.  Topics in Optimal Transportation , 2003 .

[18]  G. Casella,et al.  Rao-Blackwellisation of sampling schemes , 1996 .

[19]  M. Grigoriu.  Stochastic Calculus: Applications in Science and Engineering , 2002 .

[20]  Mircea Grigoriu,et al.  White Noise Processes , 1987 .

[21]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

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

[23]  A direct computation of the bene[sbreve] filter conditional density , 1995 .

[24]  N. Oudjane,et al.  Progressive correction for regularized particle filters , 2000, Proceedings of the Third International Conference on Information Fusion.

[25]  C. Givens,et al.  A class of Wasserstein metrics for probability distributions. , 1984 .

[26]  R. Bhattacharya,et al.  Hypersonic State Estimation Using the Frobenius-Perron Operator , 2011 .

[27]  V. Benes.  Exact finite-dimensional filters for certain diffusions with nonlinear drift , 1981 .

[28]  N. X. Vinh,et al.  Optimum reentry trajectories of a lifting vehicle , 1980 .

[29]  P. Bickel,et al.  Obstacles to High-Dimensional Particle Filtering , 2008 .

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

[31]  C. S. Manohar,et al.  Monte Carlo filters for identification of nonlinear structural dynamical systems , 2006 .

[32]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[33]  J. Rogers.  Chaos , 1876 .

[34]  Fred Daum,et al.  Non-particle filters , 2006, SPIE Defense + Commercial Sensing.