Envelope tests for spatial point patterns with and without simulation

• Main Authors: Thorsten Wiegand, Pavel Grabarnik, Dietrich Stoyan
• Format: Article
• Language: English
• Published: Wiley 2016-06-01
• Series: Ecosphere
• Subjects: deviation test; global envelopes; goodness‐of‐fit; Monte Carlo test; null model; pair correlation function
• Online Access: https://doi.org/10.1002/ecs2.1365

Abstract Model testing is a central step of spatial point pattern analysis, which allows ecologists to judge if their data agree with ecological hypotheses. We present a simple and elegant solution of a challenging problem: the construction of a goodness‐of‐fit envelope test with prescribed significance level α. Our new Analytical Global Envelope (AGE) test is not restricted to the narrow frame of complete spatial randomness testing and its envelopes can be determined by mathematical calculations. This allows us to investigate the influence of key settings of the AGE test on the width of the envelope strip. To circumvent some assumptions of the simulation‐free AGE test we present a corresponding Simulation‐Based Global Envelope (SBGE) test. The envelope strip of the AGE and the SBGE test encircles the range of a summary function such as the pair correlation function under the null model, and it has the desired property that the null hypothesis can be rejected with significance level α if the empirical summary function wanders outside the envelopes. The AGE test can be applied under the mild conditions that the values of the summary functions under the null model are (approximately) normally distributed and are (approximately) independent for different distance bins rj. The SBGE test requires only the independence assumption. The width of the strip of the AGE envelopes scales for a broad range of point processes with 1/n, where n is the number of points. This casts doubt about attempts of goodness‐of‐fit testing with low n (say <100). The AGE and SBGE test operate with wider envelope strips than the classical “pointwise” test. Therefore, the pointwise test has to be considered as too liberal. Furthermore, we show that the width of the AGE/SBGE strip increases approximately with ln(b), where b is the number of distance bins. For example, the AGE/SBGE envelopes are for b = 20 more than 50% wider than the corresponding pointwise envelopes. Our study opens up new avenues to the test problem in point pattern statistics and the new AGE and SBGE tests can be widely applied in ecology to improve the practice in null model testing.