Structured Space-Sphere Point Processes and K-Functions

Abstract

This paper concerns space-sphere point processes, that is, point processes on the product space of $\mathbb {R}^{d}$ (the d-dimensional Euclidean space) and $\mathbb {S}^{k}$ (the k-dimensional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere K-function which is a natural extension of the inhomogeneous K-function for point processes on $\mathbb {R}^{d}$ to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere K-function is shown to be proportional to the product of the inhomogeneous spatial and spherical K-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere K-function is illustrated for real and simulated datasets with varying dimensions d and k.

Appendix A

In Sections 6.1–6.3, we used the global rank envelope test presented in Myllymäki et al. (2017) to test for various point process models. In this appendix, we briefly explain the idea and use of such a test. A global rank envelope test compares a chosen test function for the observed data with the distribution of the test function under the null model; as this distribution is typically unknown it is approximated using a Monte Carlo approach. The comparison is based on a rank that only gives a weak ordering of the test functions. Thus, instead of a single p-value, the global rank envelope test provides an interval of p-values, where the end points specify the most liberal and conservative p-values of the test. A narrow p-interval is desirable as the test is inconclusive if the p-interval contains the chosen significance level. The width of the p-interval depends on the number of simulations, smoothness of the test functions and dimensionality. Myllymäki et al. (2017) recommended to use 2499 simulations for one-dimensional test functions and a significance level of 5%.

An advantage of the global rank envelope procedure is that it provides a graphical interpretation of the test in form of critical bounds (called a global rank envelope) for the test function. For example, if the observed test function is not completely inside the 95% global rank envelope, this corresponds to a rejection of the null hypothesis at a significance level of 5%. Furthermore, locations where the observed test function falls outside the global rank envelope reveal possible reasons for rejecting the null model.

In their supplementary material, Myllymäki et al. (2017) discussed two approaches for calculating test functions that rely on an estimate of the intensity. One approach is to reuse the intensity estimate for the observed point pattern in calculation of all the test functions, another is to reestimate the intensity for each simulation and then use this estimate when calculating the associated test function. For the L-function, which is a transformation of K₁, Myllymäki et al. (2017) concluded that the reestimation approach give the more powerful test. In this paper, we have therefore based all our global rank envelope tests on that approach.