Structured Space-Sphere Point Processes and K-Functions

This paper concerns space-sphere point processes, that is, point processes on the product space of ℝ d $\mathbb {R}^{d}$ (the d-dimensional Euclidean space) and S k $\mathbb {S}^{k}$ (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 ℝ d $\mathbb {R}^{d}$ 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.