A Deterministic Filter for non-Gaussian Bayesian Estimation

We present a fully deterministic method to compute sequential updates for stochastic state estimates of dynamic models from noisy measurements. It does not need any assumptions about the type of distribution for either data or measurement — in particular it does not have to assume any of them as Gaussian. It is based on a polynomial chaos expansion (PCE) of the stochastic variables of the model. We use a minimum variance estimator that combines an a priori state estimate and noisy measurements in a Bayesian way. For computational purposes, the update equation is projected onto a finite-dimensional PCE-subspace. The resulting Kalman-type update formula for the PCE coefficients can be efficiently computed solely within the PCE. As it does not rely on sampling, the method is deterministic, robust, and fast. In this paper we discuss the theory and practical implementation of the method. The original Kalman filter is shown to be a low-order special case. In a first experiment, we perform a bi-modal identification using noisy measurements. Additionally, we provide numerical experiments by applying it to the well known Lorenz-84 model and compare it to a related method, the ensemble Kalman filter.

[1]  Habib N. Najm,et al.  Stochastic spectral methods for efficient Bayesian solution of inverse problems , 2005, J. Comput. Phys..

[2]  Emmanuel D. Blanchard,et al.  Polynomial Chaos Approaches to Parameter Estimation and Control Design for Mechanical Systems with Uncertain Parameters , 2010 .

[3]  Craig H. Bishop,et al.  Adaptive sampling with the ensemble transform Kalman filter , 2001 .

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

[5]  C. Fox,et al.  Markov chain Monte Carlo Using an Approximation , 2005 .

[6]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

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

[8]  G. Evensen.  The ensemble Kalman filter for combined state and parameter estimation , 2009, IEEE Control Systems.

[9]  Edward N. Lorenz,et al.  Irregularity: a fundamental property of the atmosphere* , 1984 .

[10]  Steven Finette,et al.  Polynomial Chaos Quantification of the Growth of Uncertainty Investigated with a Lorenz Model , 2010 .

[11]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

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

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

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

[15]  Hermann G. Matthies,et al.  Uncertainty updating in the description of heterogeneous materials , 2010 .

[16]  Neri Merhav,et al.  Hidden Markov processes , 2002, IEEE Trans. Inf. Theory.

[17]  I. Segal,et al.  Integrals and operators , 1968 .

[18]  Roger G. Ghanem,et al.  Identification of Bayesian posteriors for coefficients of chaos expansions , 2010, J. Comput. Phys..

[19]  G. Evensen.  Using the Extended Kalman Filter with a Multilayer Quasi-Geostrophic Ocean Model , 1992 .

[20]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

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

[22]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[23]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[24]  Dongbin Xiu,et al.  A generalized polynomial chaos based ensemble Kalman filter with high accuracy , 2009, J. Comput. Phys..

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

[26]  J. Whitaker,et al.  Ensemble Square Root Filters , 2003, Statistical Methods for Climate Scientists.

[27]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[28]  Y. Marzouk,et al.  A stochastic collocation approach to Bayesian inference in inverse problems , 2009 .

[29]  J. Whitaker,et al.  Ensemble Data Assimilation without Perturbed Observations , 2002 .

[30]  Panos G. Georgopoulos,et al.  Uncertainty reduction and characterization for complex environmental fate and transport models: An empirical Bayesian framework incorporating the stochastic response surface method , 2003 .

[31]  N. Wiener.  The Homogeneous Chaos , 1938 .

[32]  Dirk P. Kroese,et al.  Kernel density estimation via diffusion , 2010, 1011.2602.

[33]  Georg A. Gottwald,et al.  Ensemble propagation and continuous matrix factorization algorithms , 2009 .

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

[35]  B. M. Golam Kibria.  Bayes Linear Statistics: Theory and Methods , 2008 .

[36]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[37]  Hermann G. Matthies,et al.  Uncertainty Quantification with Stochastic Finite Elements , 2007 .

[38]  S. Janson.  Gaussian Hilbert Spaces , 1997 .

[39]  Gene H. Golub,et al.  Matrix computations , 1983 .

[40]  R. M. Loynes,et al.  Studies In The Theory Of Random Processes , 1966 .

[41]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[42]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[43]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[44]  Benjamin L. Pence,et al.  A maximum likelihood approach to recursive polynomial chaos parameter estimation , 2010, Proceedings of the 2010 American Control Conference.

[45]  H. Matthies.  Stochastic finite elements: Computational approaches to stochastic partial differential equations , 2008 .

[46]  Edward N. Lorenz.  A look at some details of the growth of initial uncertainties , 2005 .

[47]  Nicholas Zabaras,et al.  An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method , 2009 .