Geostatistical Data Versus Point Process Data: Analysis of Second-Order Characteristics

Abstract

This paper considers data z ₁,... ,z _n assumed to stem from a realization of a spatial process Z and collected at sites s ₁,..., s _n The random field and the marked point process are two kinds of spatial processes. The former is defined in every point of the area of interest and the sample positions can be determined by the scientist himself. For the latter the locations are given by a stochastic point process. In general it is not possible to extend a given marked point process to a random field because of the interactions among the locations and the marks of the point process. However, such an extension is possible in the so called random field model which is therefore of particular interest in data analysis as a reference model.

Second-order characteristics describe the association between the random variables Z(s ₁) and Z(s ₂) located at the locations s ₁ and S ₂- Quantities like pair correlation, mark correlation and mark variogram functions are useful in order to assess the second-order characteristics of marked point processes, while covariance/correlation functions and the variogram are commonly used for the random fields.

The goal of this paper is to analyze the practical implications of all the above mentioned characteristics using examples from ecology and, in general, from environmental science fields. Comparisons between statistics in the geostatistical and the point process context are developed.