A comparison of approaches for valid variogram achievement

Abstract

Variogram estimation is a major issue for statistical inference of spatially correlated random variables. Most natural empirical estimators of the variogram cannot be used for this purpose, as they do not achieve the conditional negative-definite property. Typically, this problem's resolution is split into three stages: empirical variogram estimation; valid model selection; and model fitting. To accomplish these tasks, there are several different approaches strongly defended by their authors. Our work's main purpose was to identify these approaches and compare them based on a numerical study, covering different kind of spatial dependence situations. The comparisons are based on the integrated squared errors of the resulting valid estimators. Additionally, we propose an easily implementable empirical method to compare the main features of the estimated variogram function.

Estimation of the spatial distribution through the kernel indicator variogram

“…The kernel approach does not reduce the number of indicator variograms to be approximated, equaling that of thresholds, although its application provides consistent estimators, which are smoother than the sample indicator variograms. This property helps characterize the main features of the spatial dependence in the non‐indicator setting, as analyzed in Menezes et al (), so it is expected to follow for the indicator setting.…”

BLP approach for the estimation of variogram parameters

“…This is an improvement of the algorithm structure for the estimation of variogram parameters. Due to the structural features of their own formula, commonly used estimators may fail the two conditions for choosing a valid variogram model (Menezes et al 2005).…”

“…However, all these estimators may fail the conditionally positive-definite property which may lead to absurd negative values for the mean square prediction errors, as proved by Cressie (1993). Menezes et al (2005) compared some distinct approaches to achieve variogram estimators fullfilling previous property, in such a way that these estimators can be used for inference and prediction. Suppose that the conditionally positive-definite property is satisfied by these estimators, the resulting experimental variogram could originate misleading estimates of the variogram parameters as it depends on the lag distances to which the model is fitted (Pardo-Igúzquiza and Dowd 2001;Kerry and Oliver 2007).…”

Assessing the effect of clustered and biased multi‐stage sampling

“…They adapt the Nadaraya-Watson regression estimation to the context of spatial data and propose an asymptotically optimal bandwidth parameter. In Menezes et al (2005), the performance of the NW kernel estimator and the Matheron one are compared, under different spatial correlation models; the results suggest the usual superiority of the former estimator.…”