Nonparametric measures of association between a spatial point process and a random set, with geological applications

In mining exploration it is often desired to predict the occurrence of ore deposits given other geological information, such as the locations of faults. More generally it is of interest to measure the spatial association between two spatial patterns observed in the same survey region. Berman developed parametric methods for conditional inference about a point process X given another spatial process Y. This paper proposes an alternative, nonparametric, approach using distance methods, analogous to the use of the summary functions F, G and J for univariate point patterns. Our methods apply to a bivariate spatial process (X, Y) consisting of a point process X and a random set Y. In particular we develop a bivariate analogue of the J-function of van Lieshout and Baddeley which shows promise as a summary statistic and turns out to be closely related to Berman's analysis. Properties of the bivariate J-function include a multiplicative identity under independent superposition, which has no analogue in the univariate case. Two geological examples are investigated.