Hybrid Estimation of Semivariogram Parameters
Tóm tắt
Two widely used methods of semivariogram estimation are weighted least squares estimation and maximum likelihood estimation. The former have certain computational advantages, whereas the latter are more statistically efficient. We introduce and study a “hybrid” semivariogram estimation procedure that combines weighted least squares estimation of the range parameter with maximum likelihood estimation of the sill (and nugget) assuming known range, in such a way that the sill-to-range ratio in an exponential semivariogram is estimated consistently under an infill asymptotic regime. We show empirically that such a procedure is nearly as efficient computationally, and more efficient statistically for some parameters, than weighted least squares estimation of all of the semivariogram’s parameters. Furthermore, we demonstrate that standard plug-in (or empirical) spatial predictors and prediction error variances, obtained by replacing the unknown semivariogram parameters with estimates in expressions for the ordinary kriging predictor and kriging variance, respectively, perform better when hybrid estimates are plugged in than when weighted least squares estimates are plugged in. In view of these results and the simplicity of computing the hybrid estimates from weighted least squares estimates, we suggest that software that currently estimates the semivariogram by weighted least squares methods be amended to include hybrid estimation as an option.
Tài liệu tham khảo
Clark, I., 1979, Practical geostatistics: Applied Science Publishers: Essex, England, 129 p.
Cressie, N., 1985, Fitting variogram models by weighted least squares: Math. Geol., v. 17, no. 5, p. 563–586.
Curriero, F. C., and Lele, S., 1999, A composite likelihood approach to semivariogram estimation: Journ. of Agricultural, Biological, and Environmental Stat., v. 4, no. 1, p. 9–28.
Dubin, R. A., 1994, Estimating correlograms: A Monte Carlo study: Geographical Systems, v. 1, p. 189–202.
Ecker, M. D., and Gelfand, A. E., 1997, Bayesian variogram modeling for an isotropic spatial process: Journal of Agricultural, Biological, and Environmental Stat., v. 2, no. 4, p. 347–369.
Ecker, M. D., and Gelfand, A. E., 1999, Bayesian modeling and inference for geometrically anisotropic spatial data: Math. Geol., v. 31, no. 1, p. 67–83.
Genton, M. G., 1998, Variogram fitting by generalized least squares using an explicit formula for the covariance structure: Math. Geol., v. 30, no. 4, p. 323–345.
Handcock, M. S., and Stein, M. L., 1993, A Bayesian analysis of kriging: Technometrics, v. 35, no. 4, p. 403–410.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Kitanidis, P. K., 1983, Statistical estimation of polynomial generalized covariance functions and hydrologic applications: Water Resources Res., v. 19, no. 4, p. 909–921.
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.
Stein, M. L., 1988, Asymptotically efficient spatial interpolation with a misspecified covariance function: Annals of Stat., v. 16, no. 1, p. 55–63.
Stein, M. L., 1990, Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure: Annals of Stat., v. 18, no. 2, p. 850–872.
Stein, M. L., and Handcock, M. S., 1989, Some aymptotic properties of kriging when the covariance function is misspecified: Math. Geol., v. 21, no. 2, p. 171–190.
Stein, M. L., 1999, Interpolation of spatial data: Springer, New York, 247 p.
Ying, Z., 1991, Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process: Journal of Multivariate Analysis, v. 36, no. 2, p. 280–296.
Zhang, H., 2004, Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics: Jour. of the Am. Stat. Association, v. 99, no. 465, p. 250–261.
Zimmerman, D. L., 1989, Computationally efficient restricted maximum likelihood estimation of generalized covariance functions: Math. Geol., v. 21, no. 7, p. 655–672.
Zimmerman, D. L., and Zimmerman, M. B., 1991, A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors: Technometrics, v. 33, no. 1, p. 77–91.