Obstacles to High-Dimensional Particle Filtering

Abstract Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and ...

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

[2]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[3]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[4]  R. Durrett.  Probability: Theory and Examples , 1993 .

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

[6]  Philippe Courtier,et al.  Unified Notation for Data Assimilation : Operational, Sequential and Variational , 1997 .

[7]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

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

[9]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[10]  G. Evensen,et al.  An ensemble Kalman smoother for nonlinear dynamics , 2000 .

[11]  T. Hamill.  Interpretation of Rank Histograms for Verifying Ensemble Forecasts , 2001 .

[12]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[13]  D. Pham.  Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems , 2001 .

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

[15]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[16]  J. Whitaker,et al.  Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter , 2001 .

[17]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[18]  D. McLaughlin,et al.  Hydrologic Data Assimilation with the Ensemble Kalman Filter , 2002 .

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

[20]  Freda Kemp,et al.  An Introduction to Sequential Monte Carlo Methods , 2003 .

[21]  C. Snyder,et al.  Assimilation of Simulated Doppler Radar Observations with an Ensemble Kalman Filter , 2003 .

[22]  Chris Snyder,et al.  Toward a nonlinear ensemble filter for high‐dimensional systems , 2003 .

[23]  G. Eyink,et al.  Ensemble Filtering for Nonlinear Dynamics , 2003 .

[24]  Xue Wei,et al.  Reanalysis without Radiosondes Using Ensemble Data Assimilation , 2004 .

[25]  Christian L. Keppenne,et al.  Ensemble Kalman filter assimilation of temperature and altimeter data with bias correction and application to seasonal prediction , 2005 .

[26]  Kuolin Hsu,et al.  Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter , 2005 .

[27]  E. Lorenz.  Predictability of Weather and Climate: Predictability – a problem partly solved , 2006 .

[28]  D. McLaughlin,et al.  Assessing the Performance of the Ensemble Kalman Filter for Land Surface Data Assimilation , 2006 .

[29]  Ionel Michael Navon,et al.  A Note on the Particle Filter with Posterior Gaussian Resampling , 2006 .

[30]  D. Nychka.  Data Assimilation” , 2006 .

[31]  T. M. Chin,et al.  An Ensemble-Based Smoother with Retrospectively Updated Weights for Highly Nonlinear Systems , 2007 .

[32]  T. Higuchi,et al.  Merging particle filter for sequential data assimilation , 2007 .

[33]  Brian R. Hunt,et al.  A non‐Gaussian Ensemble Filter for Assimilating Infrequent Noisy Observations , 2007 .

[34]  Keston W. Smith.  Cluster ensemble Kalman filter , 2007 .

[35]  T. Bengtsson,et al.  Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants , 2007 .

[36]  P. Bickel,et al.  Regularized estimation of large covariance matrices , 2008, 0803.1909.

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

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