High-dimensional problems for spatial point processes

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Nowadays, new technologies allow, on the one hand, the acquisition of an increasing mass of data and on the other hand the observation of more and more complex phenomena. Statistics and in particular the sub-branch of spatial statistics does not avoid these questions. The present research program intends to consider high-dimensional problems for one specific class of spatial models which is the class of (spatial) point processes.******Context***Point processes model random sets of points or events in interaction. Point patterns arise in a broad range of fields. When the observation domain, say S, corresponds to a subset of Rd (with the dimension d=2,3), such processes can model for instance galaxies in astrophysics, hundreds of trees species in forestry, sources of outbreak of a disease in epidemiology, ocular fixations from different individuals watching images or videos in vision, etc. Classical questions are about the modelling of the dependency between point patterns (are two trees species independent?) and/or to relate the distribution of points to extra information like the altitude map, soil nature, for forestry applications. Many statistical methodologies exist in the literature, however very few things are known when many point patterns are simultaneously observed and/or when the amount of extra information is important. How to extract information, to efficiently select covariates are the implicit questions. Very recently, when the dimension of S is large (think of a unit cube [0,1]d with d=50), point processes have appeared in computer experiments to construct random designs and when S is a discrete space they have emerged in machine learning, compressed sensing as an efficient tool for subsampling a possibly high-dimensional dataset. To illustrate one of of the questions an expected feature for a sample of points derived from a stochastic model is to "nicely" cover the unit cube, which can be achieved if the pattern exhibits some kind of regularity. But it is still an open question to have a simple model which satisfies also this kind of regularity when the same sample of points is projected on any subspace of the unit cube, a property that classical experimental designs like Latin hypercubes are able to handle.******Objective***The goal of this research program is to bring modern questions induced by the high-dimension feature to the relatively recent class of point processes models, which arises in an increasing number of applications. By the nature of these two research areas, this research program is modern and innovative. Problems will be investigated both from a theoretical point of view by providing the statistics community new methodologies and results to understand their limitations and from a practical/computational point of view by providing practitioners with a systematic implementation of the developed methodologies within the (free) R software.