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This paper concerns space-sphere point processes, that is, point processes onthe product space of $\mathbb R^d$ (the $d$-dimensional Euclidean space) and$\mathbb S^k$ (the $k$-dimen\-sional sphere). We consider specific classes ofmodels for space-sphere point processes, which are adaptations of existingmodels for either spherical or spatial point processes. For model checking orfitting, we present the space-sphere $K$-function which is a natural extensionof the inhomogeneous $K$-function for point processes on $\mathbb R^d$ to thecase of space-sphere point processes. Under the assumption that the intensityand pair correlation function both have a certain separable structure, thespace-sphere $K$-function is shown to be proportional to the product of theinhomogeneous spatial and spherical $K$-functions. For the presentedspace-sphere point process models, we discuss cases where such a separablestructure can be obtained. The usefulness of the space-sphere $K$-function isillustrated for real and simulated datasets with varying dimensions $d$ and$k$.

Structured space-sphere point processes and $K$-functions