A cautionary example on the use of second-order methods for analyzing point patterns

A popular approach to the analysis of mapped planar point patterns is through secondorder methods, where the object of attention is the variance of the number of points falling in a test set of a given size and shape, or the behaviour of all distances between pairs of points in the pattern. Standard references for the analysis of spatial data are works by Ripley (1981, Ch. 8) and Diggle (1983). Many of the methods for point-pattern analysis given in the literature are second-order in nature. The analysis of mapped point patterns is important in a variety of biological applications, ranging from ecology to physiology, and some examples are given in these books by Ripley and by Diggle. Suppose that an observed point pattern can be considered as a realization of a random point-process model which is stationary and isotropic. Of course this is a strong assumption which may not be justified in practice, but it is nevertheless the assumption under which much of the existing point-process methodology has been developed. The property of the underlying random process elicited, directly or indirectly, by second-order methods is the second-moment cumulative function K(t), which satisfies the following properties under suitable regularity conditions (see Ripley, 1977, p. 150): (i) if X is the intensity of the process, then XK(t) is the expected number of further points within distance t of a typical point of the process; (ii) specifying K(t) for all t is equivalent to specifying the variance of the number of points falling in any given set. The function K(t) defined by Ripley (1977) is an edge-corrected version of the empirical distribution function of all interpoint distances in the observed pattern, and provides an approximately unbiased estimator of K(t). The estimate K(t) can be used to construct tests of the hypothesis that an observed pattern is consistent with the 'completely random' Poisson point-process model, and, more importantly, to quantify the apparent deviation