Sequential Parameter Estimation of Time-Varying Non-Gaussian Autoregressive Processes
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Sequential Parameter Estimation of Time-Varying Non-Gaussian Autoregressive Processes Petar M. Djuri´c Department of Electrical and Computer Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794, USA Email: [email protected]
Jayesh H. Kotecha Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA Email: [email protected]
Fabien Esteve ENSEEIHT/T´eSA 2, rue Charles Camichel, BP 7122, 31071 Toulouse Cedex 7, France Email: [email protected]
Etienne Perret ENSEEIHT/T´eSA 2, rue Charles Camichel, BP 7122, 31071 Toulouse Cedex 7, France Email: [email protected] Received 1 August 2001 and in revised form 14 March 2002 Parameter estimation of time-varying non-Gaussian autoregressive processes can be a highly nonlinear problem. The problem gets even more difficult if the functional form of the time variation of the process parameters is unknown. In this paper, we address parameter estimation of such processes by particle filtering, where posterior densities are approximated by sets of samples (particles) and particle weights. These sets are updated as new measurements become available using the principle of sequential importance sampling. From the samples and their weights we can compute a wide variety of estimates of the unknowns. In absence of exact modeling of the time variation of the process parameters, we exploit the concept of forgetting factors so that recent measurements affect current estimates more than older measurements. We investigate the performance of the proposed approach on autoregressive processes whose parameters change abruptly at unknown instants and with driving noises, which are Gaussian mixtures or Laplacian processes. Keywords and phrases: particle filtering, sequential importance sampling, forgetting factors, Gaussian mixtures.
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INTRODUCTION
In on-line signal processing, a typical objective is to process incoming data sequentially in time and extract information from them. Applications vary and include system identification [1], equalization [2, 3], echo cancelation [4], blind source separation [5], beamforming [6, 7], blind deconvolution [8], time-varying spectrum estimation [6], adaptive detection [9], and digital enhancement of speech and audio signals [10]. These applications find practical use in communications, radar, sonar, geophysical explorations, astrophysics, biomedical signal processing, and financial time series analysis. The task of on-line signal processing usually amounts to estimation of unknowns and tracking them as they change with time. A widely adopted approach to addressing this
problem is the Kalman filter, which is optimal in the cases when the signal models are linear and the noises are additive and Gaussian [1]. The framework of the Kalman filter allows for derivation of all the recursive least squares (RLS) adaptive filters [11]. When nonlinearities have to be tackled, the extended Kalman filter becomes the tool for estimating the unknowns of interest [6, 12, 13]. I
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