Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.

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Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen-Loève expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen-Loève coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen-Loève expansions are used to approximate axially symmetric processes. For such approximations, bounds for the L 2-error are provided. Numerical experiments are conducted to illustrate our findings.

The Matérn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Matérn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Matérn model using chordal distances.

This paper concerns space-sphere point processes, that is, point processes on the product space of R d (the d-dimensional Euclidean space) and S k (the kdimensional 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 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.