A Monte Carlo-Adjusted Goodness-of-Fit Test for Parametric Models Describing Spatial Point Patterns

Abstract

Assessing the goodness-of-fit (GOF) for intricate parametric spatial point process models is important for many application fields. When the probability density of the statistic of the GOF test is intractable, a commonly used procedure is the Monte Carlo GOF test. Additionally, if the data comprise a single dataset, a popular version of the test plugs a parameter estimate in the hypothesized parametric model to generate data for the Monte Carlo GOF test. In this case, the test is invalid because the resulting empirical level does not reach the nominal level. In this article, we propose a method consisting of nested Monte Carlo simulations which has the following advantages: the bias of the resulting empirical level of the test is eliminated, hence the empirical levels can always reach the nominal level, and information about inhomogeneity of the data can be provided. We theoretically justify our testing procedure using Taylor expansions and demonstrate that it is correctly sized through various simulation studies. In our first data application, we discover, in agreement with Illian et al., that Phlebocarya filifolia plants near Perth, Australia, can follow a homogeneous Poisson clustered process that provides insight into the propagation mechanism of these plants. In our second data application, we find, in contrast to Diggle, that a pairwise interaction model provides a good fit to the micro-anatomy data of amacrine cells designed for analyzing the developmental growth of immature retina cells in rabbits. This article has supplementary material online.

Key Words:

• Adjusted level
• Distribution of p-values
• Level bias
• Nested Monte Carlo simulation
• Spatial point process

SUPPLEMENTARY MATERIALS

Supplements: R-code for goodness-of-fit (GOF) tests described in the article (GOF.R).

Phlebocarya filifolia plants data: Positions of 207 plants described in the article (Phlebocarya.txt).

Amacrine cells: Positions of 294 on/off cells in the retina of a rabbit (amacrineson.txt and amacrinesoff.txt).

Proof of Proposition 1 and pseudocode of the Monte Carlo-adjusted GOF test.

All files can be found in a single zip file (AGOF.zip).

ACKNOWLEDGMENTS

This publication is based in part on work supported by award no. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST) and by NSF grants DMS-1007504 and DMS-1106494. The authors thank the editor, the associate editor, and a referee for their helpful comments and suggestions. The first author thanks Dr. Naisyin Wang for suggesting the topic of this article. The authors also acknowledge the Texas A&M University Brazos HPC cluster that contributed to the research reported here.