A Statistical Test for Ripley’s Function Rejection of Poisson Null Hypothesis

Ripley’s function is the classical tool to characterize the spatial structure of point patterns. It is widely used in vegetation studies. Testing its values against a null hypothesis usually relies on Monte-Carlo simulations since little is known about its distribution. We introduce a statistical test against complete spatial randomness (CSR). The test returns the value to reject the null hypothesis of independence between point locations. It is more rigorous and faster than classical Monte-Carlo simulations. We show how to apply it to a tropical forest plot. The necessary R code is provided. 1. Introduction The commonest tool used to characterize the spatial structure of a point set is Ripley's statistic [1, 2]. It has been widely used in ecology and other scientific fields and is well referenced in handbooks [3–7]. Classically, an observed set of points is tested against a homogeneous Poisson point process taken as a null model. Since little is known about the distribution of the function, the test of rejection of the null hypothesis relies on Monte-Carlo simulations. Large point patterns are difficult to deal with because computation time is roughly proportional to the square of the number of points (to calculate the distances between all pairs of points) multiplied by the number of simulations. Moreover, the test is valid for one distance but using it repeatedly for all distances where the function is calculated dramatically increases the risk to reject the null hypothesis by error [8]. Duranton and Overman [9] provided a heuristic global test followed by Marcon and Puech [10]. Loosmore and Ford proposed a goodness-of-fit test to eliminate this error, but still rely on Monte-Carlo simulations. We showed in [11] that a global test was able to return a classical value, that is to say, the probability to erroneously reject the null model, relying on exact values of the biases and variances of the statistics. We derived its asymptotic properties in a growing square window. We develop it in this paper so that it can be used in a rectangular window, as most applications require. We show that it can be applied to real point patterns, even with a little number of points and discuss in depth the way to employ it, so that it can be used by empirical researchers. We first present our motivating example: a tropical forest plot where we want to elucidate the spatial structure of two species of trees. We provide the mathematical framework supporting the test. We apply it to the dataset and present the results. In the Discussion, we review the literature of