Inference of spatial indicator covariance parameters by maximum likelihood using MLREML

Abstract

This paper discusses an application of a previously published code to show how maximum likelihood estimation may be used for the inference of spatial indicator covariance parameters. Although the maximum likelihood equations are derived on the assumption of a multivariate normal distribution for the experimental data, when the data are neither Gaussian nor can be transformed to normal scores (as is clearly the case with indicator data) maximum likelihood still provides a weighted least squares criterion of fitting spatial covariance models. The estimator is still consistent, asymptotically unbiased and the estimates are asymptotically normally distributed. Maximum likelihood also provides the uncertainty of the estimates by assessing the standard errors of the estimates. The method is recommended for cases in which the number of experimental data is relatively small (several dozens) and irregularly distributed. In such situations, the experimental indicator variogram, calculated by classical non-parametric methods, is so noisy that it is extremely difficult to fit it to a model. This difficulty increases as the threshold moves further away from the median of the data. A simulated case study is used to demonstrate the application of the method and illustrate the results. The program MLREML (previously published in Computers and Geosciences) has been used to perform maximum likelihood estimation of spatial indicator covariances.