Global envelope tests for spatial processes

TL;DR

This paper introduces new global envelope tests for spatial processes that improve goodness-of-fit testing by controlling error rates over an entire interval of distances, providing p-values and better interpretability.

Contribution

The paper proposes two novel approaches for global envelope tests in spatial statistics, enabling a priori error control and p-value computation, enhancing existing methods.

Findings

New tests control type I error over distance intervals

Methods provide p-values for goodness-of-fit assessments

Illustrated with simulated and real data

Abstract

Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function $T(r)$ with its simulated counterparts from the null model. However, the type I error probability $\alpha$ is conventionally controlled for a fixed distance $r$ only, whereas the functions are inspected on an interval of distances $I$. In this study, we propose two approaches related to Barnard's Monte Carlo test for building global envelope tests on $I$:(1) ordering the empirical and simulated functions based on their $r$-wise ranks among each other, and (2) the construction of envelopes for a deviation test. These new tests allow the a priori selection of the global $\alpha$ and they yield $p$-values. We illustrate these tests using simulated and real point pattern data.