Geostatistical data versus point process data: analysis of
second-order characteristics
Jorge Mateu(1) & Paulo J. Ribeiro Jr.(2)
Abstract
This paper considers data $z_1,\ldots,z_n$ assumed to
stem from a realization of a spatial process $Z$ and collected at sites
$s_1,\ldots,s_n$. 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.
Keywords: second-order characteristics, mark correlation function, geostatistics, marked point process.
(1) Universitat Jaume I.
Address: Department of Mathematics, Universitat Jaume I, Campus Penyeta Roja,
12071. Castellon, Spain.
e-mail: mateu@mat.uji.es
(2) UFPR/ CAPES/ Lancaster University.
Address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1-4YF
e-mail: paulojus@est.ufpr.br
http://www.maths.lancs.ac.uk/~ribeiro/
Last modified: Tue Oct 24 13:09:26 BST 2000