📝 SummarXiv

Abstract

We introduce a new type of test for complete spatial randomness that applies to mapped point patterns in a rectangle or a cube of any dimension. This is the first test of its kind to be based on characteristic functions and utilizes a weighted $L_2$-distance between the empirical and uniform characteristic functions. The test shows surprising connections to Ripley's $K$-function and Zimmerman's $\bar{\omega}^2$ statistic. It is also simple to calculate and does not require adjusting for edge effects. An efficient algorithm is developed to find the asymptotic null distribution of the test statistic under the Cauchy weight function. This makes the test fast to compute. In simulations, our test shows varying sensitivity to different levels of spatial interaction depending on the scale parameter of the Cauchy weight function. Tests with different parameter values can be combined to create a Bonferroni-corrected omnibus test, which is more powerful than the popular $L$-test and the Clark-Evans test in most simulation settings of heterogeneity, aggregation and regularity, especially when the sample size is large. The simplicity of the empirical characteristic function makes it straightforward to extend our test to non-rectangular or sparsely sampled point patterns.