Maximum Likelihood Estimation of Spatial Covariance Parameters

In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.

[1]  K. Mardia,et al.  Maximum likelihood estimation of models for residual covariance in spatial regression , 1984 .

[2]  Eulogio Pardo-Igúzquiza,et al.  MLREML: a computer program for the inference of spatial covariance parameters by maximum likelihood and restricted maximum liklihood , 1997 .

[3]  Georges Bastin,et al.  Variogram identification by the mean-squared interpolation error method with application to hydrologic fields , 1985 .

[4]  P. Kitanidis,et al.  Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton method , 1985 .

[5]  M. R. Osborne,et al.  Estimation of covariance parameters in kriging via restricted maximum likelihood , 1991 .

[6]  L. Gelhar.  Stochastic Subsurface Hydrology , 1992 .

[7]  Peter K. Kitanidis,et al.  Analysis of the Spatial Structure of Properties of Selected Aquifers , 1985 .

[8]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[9]  Michel Gevers,et al.  Identification and optimal estimation of random fields from scattered point-wise data , 1985, Autom..

[10]  B. Ripley.  Statistical inference for spatial processes , 1990 .

[11]  T. Brubaker,et al.  Nonlinear Parameter Estimation , 1979 .

[12]  Peter K. Kitanidis,et al.  PARAMETRIC ESTIMATION OF COVARIANCES OF REGIONALIZED VARIABLES , 1987 .

[13]  Eulogio Pardo-Igúzquiza,et al.  AMLE3D: a computer program for the inference of spatial covariance parameters by approximate maximum likelihood estimation , 1997 .

[14]  S. P. Neuman,et al.  estimation of spatial covariance structures by adjoint state maximum likelihood cross validation: 1. Theory , 1989 .

[15]  Dale L. Zimmerman,et al.  Computationally efficient restricted maximum likelihood estimation of generalized covariance functions , 1989 .

[16]  A. V. Vecchia.  Estimation and model identification for continuous spatial processes , 1988 .

[17]  Brian D. Ripley,et al.  Problems with likelihood estimation of covariance functions of spatial Gaussian processes , 1987 .

[18]  Peter K. Kitanidis,et al.  Statistical estimation of polynomial generalized covariance functions and hydrologic applications , 1983 .

[19]  P. J. Green,et al.  Probability and Statistical Inference , 1978 .

[20]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[21]  Kanti V. Mardia,et al.  On multimodality of the likelihood in the spatial linear model , 1989 .

[22]  Peter K. Kitanidis,et al.  Orthonormal residuals in geostatistics: Model criticism and parameter estimation , 1991 .

[23]  R. Fisher.  001: On an Absolute Criterion for Fitting Frequency Curves. , 1912 .

[24]  H. Akaike.  A new look at the statistical model identification , 1974 .