Rings, circles, and null-models for point pattern analysis in ecology

A large number of methods for the analysis of point pattern data have been developed in a wide range of scientific fields. First-order statistics describe large-scale variation in the intensity of points in a study region, whereas second-order characteristics are summary statistics of all point-to-point distances in a mapped area and offer the potential for detecting both different types and scales of patterns. Second-order analysis based on Ripley's K-function is increasingly used in ecology to characterize spatial patterns and to develop hypothesis on underlying processes; however, the full range of available methods has seldomly been applied by ecologists. The aim of this paper is to provide guidance to ecologists with limited experience in second-order analysis to help in the choice of appropriate methods and to point to practical difficulties and pitfalls. We review (1) methods for analytical and numerical implementation of two complementary second-order statistics, Ripley's K and the O-ring statistic, (2) methods for edge correction, (3) methods to account for first-order effects (i.e. heterogeneity) of univariate patterns, and (4) a variety of useful standard and non-standard null models for univariate and bivariate patterns. For illustrative purpose, we analyze examples that deal with non-homogeneous univariate point patterns. We demonstrate that large-scale heterogeneity of a point-pattern biases Ripley's K-function at smaller scales. This bias is difficult to detect without explicitly testing for homogeneity, but we show that it can be removed when applying methods that account for first-order effects. We synthesize our review in a number of step-by-step recommendations that guide the reader through the selection of appropriate methods and we provide a software program that implements most of the methods reviewed and developed here.