An evaluation of a non-parametric method of estimating semi-variograms of isotropic spatial processes

Semi-variograms are useful for describing the correlation structure of spatial random variables. Valid semi-variograms must be conditionally negative definite. To ensure this restriction when estimating these functions, a valid parametric model is typically fitted to a sample semi-variogram. Recently, a method of fitting valid semi-variograms without having to choose a parametric family has been described in the literature. The method is based on the spectral representation of positive definite functions. In this paper, the method is evaluated using simulated data. The fits obtained using the non-parametric method are compared with fits obtained by fitting four parametric models (exponential, Gaussian, rational quadratic and power) to simulated data using non-linear least squares. The comparisons are based on the integrated squared errors of the resulting fits. The nonparametric estimator always resulted in fits that were as good as those obtained using the parametric models. The non-parametric method is faster, easier to use and more objective than the parametric methods. Some examples are presented.