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Going off grid: computationally efficient inference for log-Gaussian Cox processes. (English) Zbl 1452.62704
Summary: This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, whereas an approximation based on a counting process on a partition of the domain achieves only first-order convergence. The results improve upon the general theory of convergence for stochastic partial differential equation models introduced by F. Lindgren et al. [J. R. Stat. Soc., Ser. B, Stat. Methodol. 73, No. 4, 423–498 (2011; Zbl 1274.62360)]. The new method is demonstrated on a standard point pattern dataset, and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of A. Chakraborty et al. [“Point pattern modelling for degraded presence-only data over large regions”, Appl. Stat. 60, No. 5, 757–776 (2011; doi:10.1111/j.1467-9876.2011.00769.x)]. The second extension constructs a log-Gaussian Cox process on the world’s oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

62M30 Inference from spatial processes
60G15 Gaussian processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G60 Random fields
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
62-08 Computational methods for problems pertaining to statistics
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