Abstract
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.
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REFERENCES
Akaike, H. 1974, A new look at the statistical model identification: IEEE Trans. on Automatic Control. v. AC-19,no. 6, p. 716–723.
Bastin, G., and Gevers, M., 1985, Identification and optimal estimation of random fields from scattered point-wise data: Automatica, v. 21,no. 2, p. 139–155.
Bard, Y., 1974, Nonlinear parameter estimation: Academic Press, New York, 341 p.
Dietrich, C. R., and Osborne, M. R., 1991, Estimation of covariance parameters in Kriging via restricted maximum likelihood: Math. Geology, v. 23,no. 7, p. 655–672.
Fisher, R. A., 1912, On an absolute criterion for fitting frequency curves: Messeng. Math., v. 41, p. 155–160.
Gelhar, L. W., 1993, Stochastic subsurface hydrology: Prentice Hall, Englewood Cliffs, New Jersey, 390 p.
Harville, H., 1977, Maximum likelihood approaches to variance component estimation and to related problems: Jour. Am. Stat. Assoc., v. 72,no. 358, p. 320–388.
Hoeksema, R. J., and Kitanidis, P. K., 1985. Analysis of the spatial structure of properties of selected aquifers: Water Resources Res., v. 21,no. 4, p. 563–572.
Kalbfleisch, 1979, Probability and statistical inference II: Springer Verlag, New York, 316 p.
Kitanidis, P. K., 1983, Statistical estimation of polynomial generalized covariance functions and hydrologic applications: Water Resources Res., v. 19,no. 2, p. 909–921.
Kitanidis, P. K., 1987, Parametric estimation of covariances of regionalized variables: Water Resources Bull., v. 23,no. 4, p. 671–680.
Kitanidis, P. K., 1991, Orthogonal residuals in geostatistics: model criticism and parameter estimation: Math. Geology, v. 23,no. 5, p. 741–758.
Kitanidis, P. K., and Lane, R. W., 1985, Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton method: Jour. Hydrology, v. 79,nos. 1–2, p. 53–71.
Lebel, T., and Bastin, G., 1985, Variogram identification by mean squared interpolation error method with applications to hydrologic fields: Jour. Hydrology, v. 77,nos. 1–4, p. 31–56.
Mardia, K. V., and Marshall, R. J., 1984, Maximum likelihood estimation of models for residual covariance in spatial regression: Biometrika, v. 71,no. 1, p. 135–146.
Mardia, K. V., and Watkins, 1989, On multimodality of the likelihood in the spatial linear model: Biometrika, v. 76,no. 2, p. 289–295.
Nelder, J. A., and Mead, R., 1965, A simplex method for function minimization: Computer Jour., v. 7, p. 308–313.
Pardo-Igúzquiza, E., 1997, MLREML: a computer program for the inference of spatial covariance parameters by maximum likelihood and restricted maximum likelihood: Computers & Geosciences, v. 23,no. 2, pp. 153–162.
Pardo-Iguzquiza, E., and Dowd, P. A., 1997, AMLE3D: a computer program for the statistical inference of covariance parameters by approximate maximum likelihood estimation. Computers & Geosciences, in press.
Ripley, B. D., 1988, Statistical inference for spatial processes: Cambridge Univ. Press, Cambridge, 148 p.
Ripley, B. D., 1992, Stochastic models fo the distribution of rock types in petroleum reservoirs, in Walden, A. T., and Guttorp, P., eds., Statistics in the Environmental and Earth Sciences: John Wiley & Sons, New York, 306 p.
Samper, F. J., and Neuman, S. P., 1989, Estimation of spatial covariance structures by adjoint state maximum likelihood cross-validation 1, Theory: Water Resources Res., v. 25,no. 3, p. 351–362.
Vecchia, A. V., 1988, Estimation and model identification for continuous spatial processes: Jour. Roy. Stat. Soc., Ser. B, v. 50,no. 2, p. 297–312.
Warnes, J. J., and Ripley, B. D., 1987, Problems with the likelihood estimation of covariance functions of spatial Gaussian processes: Biometrika, v. 74,no. 3, p. 640–642.
Zimmerman, D. L., 1989, Computationally efficient restricted maximum likelihood estimation of generalized covariance functions: Math. Geology, v. 21,no. 7, p. 655–672.
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Pardo-Igúzquiza, E. Maximum Likelihood Estimation of Spatial Covariance Parameters. Mathematical Geology 30, 95–108 (1998). https://doi.org/10.1023/A:1021765405952
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DOI: https://doi.org/10.1023/A:1021765405952