A Uniformly Bounded Correlation Function for Spatial Point Patterns

A point pattern is a dataset of coordinates, typically in 2D or 3D space. Point patterns are ubiquitous in diverse applications including Geographic Information Systems, Astronomy, Ecology, Biology and Medicine. Among the statistics used to quantify point patterns, most are based on Ripley's 𝐾-function, which measures the deviation of the observed pattern from a completely random arrangement of points. This approach is useful for constructing null hypothesis tests, but Ripley's 𝐾 and its variants are less suitable as quantitative effect sizes because their ranges and expected values generally depend on the scale or the size of the region in which the pattern is observed. To address this, we propose a new function that behaves like a correlation coefficient for point patterns: it is tightly bounded by -1 and 1, with a value of -1 corresponding to a maximally dispersed arrangement of points, 0 indicating complete spatial randomness, and 1 representing maximal clustering. These properties are independent of scale and observation window size assuming appropriate edge correction. Evaluating our function on simulated data, we show that it has comparable statistical calibration and power to 𝐾-based baselines. We hope that the ease of interpretation of our bounded function will facilitate the analysis of spatial data across multiple fields.