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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.

MSC:
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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