Analysis of parallelizable resampling algorithms for particle filtering

Particle filtering methods are powerful tools for online estimation and tracking in nonlinear and non-Gaussian dynamical systems. They commonly consist of three steps: (1) drawing samples in the state-space of the system, (2) computing proper importance weights of each sample and (3) resampling. Steps 1 and 2 can be carried out concurrently for each sample, but standard resampling techniques require strong interaction. This is an important limitation, because one of the potential advantages of particle filtering is the possibility to perform very fast online signal processing using parallel computing devices. It is only very recently that resampling techniques specifically designed for parallel computation have been proposed, but little is known about the properties of such algorithms and how they compare to standard methods. In this paper, we investigate two classes of such techniques, distributed resampling with non-proportional allocation (DRNA) and local selection (LS). Namely, we analyze the effect of DRNA and LS on the sample variance of the importance weights; the distortion, due to the resampling step, of the discrete probability measure given by the particle filter; and the variance of estimators after resampling. Finally, we carry out computer simulations to support the analytical results and to illustrate the actual performance of DRNA and LS. Two typical problems are considered: vehicle navigation and tracking the dynamic variables of the chaotic Lorenz system driven by white noise.

[1]  Mónica F. Bugallo,et al.  A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics , 2004, EURASIP J. Adv. Signal Process..

[2]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

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

[4]  Petar M. Djuric,et al.  Gaussian particle filtering , 2003, IEEE Trans. Signal Process..

[5]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[6]  Xiaodong Wang,et al.  Joint multiple target tracking and classification in collaborative sensor networks , 2004, ISIT.

[7]  Vicente Pérez-Muñuzuri,et al.  Transition to Chaotic phase Synchronization through Random phase jumps , 2000, Int. J. Bifurc. Chaos.

[8]  Eric Moulines,et al.  Comparison of resampling schemes for particle filtering , 2005, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005..

[9]  Xiaodong Wang,et al.  Sampling-based soft equalization for frequency-selective MIMO channels , 2005, IEEE Transactions on Communications.

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

[11]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[12]  Petar M. Djuric,et al.  Gaussian sum particle filtering , 2003, IEEE Trans. Signal Process..

[13]  Fredrik Gustafsson,et al.  Particle filters for positioning, navigation, and tracking , 2002, IEEE Trans. Signal Process..

[14]  Jun S. Liu,et al.  Sequential Imputations and Bayesian Missing Data Problems , 1994 .

[15]  Petar M. Djuric,et al.  Blind equalization of frequency-selective channels by sequential importance sampling , 2004, IEEE Transactions on Signal Processing.

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

[17]  Grebogi,et al.  Exploiting the natural redundancy of chaotic signals in communication systems , 2000, Physical review letters.

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

[19]  Moon Gi Kang,et al.  Super-resolution image reconstruction , 2010, 2010 International Conference on Computer Application and System Modeling (ICCASM 2010).

[20]  Antonio Artés-Rodríguez,et al.  Particle Filtering Algorithms for Tracking a Maneuvering Target Using a Network of Wireless Dynamic Sensors , 2006, EURASIP J. Adv. Signal Process..

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[23]  Petar M. Djuric,et al.  Resampling algorithms and architectures for distributed particle filters , 2005, IEEE Transactions on Signal Processing.

[24]  Neil J. Gordon,et al.  Editors: Sequential Monte Carlo Methods in Practice , 2001 .

[25]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[26]  P. Fearnhead,et al.  An improved particle filter for non-linear problems , 1999 .

[27]  Petar M. Djuric,et al.  Resampling Algorithms for Particle Filters: A Computational Complexity Perspective , 2004, EURASIP J. Adv. Signal Process..

[28]  P. J. van Leeuwen,et al.  A variance-minimizing filter for large-scale applications , 2003 .

[29]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[30]  Joaquín Míguez,et al.  On a recursive method for the estimation of unknown parameters of partially observed chaotic systems , 2006 .

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

[32]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[33]  Mónica F. Bugallo,et al.  A sequential Monte Carlo method for adaptive blind timing estimation and data detection , 2005, IEEE Transactions on Signal Processing.

[34]  P. Djurić,et al.  Particle filtering , 2003, IEEE Signal Process. Mag..

[35]  J. Yorke,et al.  Chaos: An Introduction to Dynamical Systems , 1997 .

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

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