Abstract:
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Determinantal point processes (DPPs) are models for repulsiveness between points in `space', where the two most studied cases of `space' is a finite set or the $d$-dimensional Euclidean space. Their moment properties and density expressions are known; they can easily and quickly be simulated; rather flexible parametric models can be constructed and likelihood or moment based inference procedures apply; and they are used in mathematical physics, combinatorics, random-matrix theory, machine learning and spatial statistics, see [1] and the references therein. This talk concerns DPPs defined on the $d$-dimensional unit sphere $\mathbb S^d$ where the one- and two-dimensional cases are the practically most relevant cases. The connection between Mercer and Schoenberg representations of the kernel for a DPP will be established, whereby we can construct new parametric models for kernels (covariance functions) which provide tractable and flexible DPP models on $\mathbb S^d$. References: [1] F. Lavancier, J. M{\o}ller and E. Rubak (2015). Determinantal point process models and statistical inference. To appear in Journal of Royal Statistical Society: Series B (Statistical Methodology).
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