Approximate Marginal Posterior for Log Gaussian Cox Processes

TL;DR

This paper introduces an efficient, scalable pseudo-marginal MCMC method for inference in log Gaussian Cox processes, reducing computational complexity and improving parameter estimation for complex spatial point pattern data.

Contribution

It develops a novel pseudo-marginal MCMC algorithm for scalable inference in log Gaussian Cox processes, addressing computational challenges of high-dimensional covariance matrices.

Findings

The method effectively estimates parameters in univariate and multivariate models.

Simulation studies demonstrate computational efficiency and accuracy.

Application to tree data reveals complex inter-species interactions.

Abstract

The log Gaussian Cox process is a flexible class of Cox processes, whose intensity surface is stochastic, for incorporating complex spatial and time structure of point patterns. The straightforward inference based on Markov chain Monte Carlo is computationally heavy because the computational cost of inverse or Cholesky decomposition of high dimensional covariance matrices of Gaussian latent variables is cubic order of their dimension. Furthermore, since hyperparameters for Gaussian latent variables have high correlations with sampled Gaussian latent processes themselves, standard Markov chain Monte Carlo strategies are inefficient. In this paper, we propose an efficient and scalable computational strategy for spatial log Gaussian Cox processes. The proposed algorithm is based on pseudo-marginal Markov chain Monte Carlo approach. Based on this approach, we propose estimation of approximate marginal posterior for parameters and comprehensive model validation strategies. We provide details for all of the above along with some simulation investigation for univariate and multivariate settings and analysis of a point pattern of tree data exhibiting positive and negative interaction between different species.