A spectral approach to estimation and smoothing of continuous spatial processes |
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| Authors: | Z T Yucel R H Shumway |
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| Institution: | (1) Division of Statistics, University of California, 95616 Davis, CA, USA |
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| Abstract: | This paper is concerned with developing computational methods and approximations for maximum likelihood estimation and minimum mean square error smoothing of irregularly observed two-dimensional stationary spatial processes. The approximations are based on various Fourier expansions of the covariance function of the spatial process, expressed in terms of the inverse discrete Fourier transform of the spectral density function of the underlying spatial process. We assume that the underlying spatial process is governed by elliptic stochastic partial differential equations (SPDE's) driven by a Gaussian white noise process. SPDE's have often been used to model the underlying physical phenomenon and the elliptic SPDE's are generally associated with steady-state problems.A central problem in estimation of underlying model parameters is to identify the covariance function of the process. The cumbersome exact analytical calculation of the covariance function by inverting the spectral density function of the process, has commonly been used in the literature. The present work develops various Fourier approximations for the covariance function of the underlying process which are in easily computable form and allow easy application of Newton-type algorithms for maximum likelihood estimation of the model parameters. This work also develops an iterative search algorithm which combines the Gauss-Newton algorithm and a type of generalized expectation-maximization (EM) algorithm, namely expectation-conditional maximization (ECM) algorithm, for maximum likelihood estimation of the parameters.We analyze the accuracy of the covariance function approximations for the spatial autoregressive-moving average (ARMA) models analyzed in Vecchia (1988) and illustrate the performance of our iterative search algorithm in obtaining the maximum likelihood estimation of the model parameters on simulated and actual data. |
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| Keywords: | Gaussian process spatial correlation anisotropy Fourier transform Gauss-Newton ECM measurement error signal extraction irregular data |
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