# Quick inference for log Gaussian Cox processes with non-stationary underlying random fields

## TL;DR

This paper introduces a fast three-step composite likelihood method for non-stationary log Gaussian Cox processes, enabling modeling of spatially varying point patterns with non-stationary mean and covariance functions.

## Contribution

It proposes a novel parametric and semiparametric modeling framework for non-stationary LGCPs, addressing limitations of traditional stationary assumptions and estimation methods.

## Key findings

- Effective modeling of fish aggregation and biological particle dispersal.
- Demonstrated computational efficiency of the three-step estimation procedure.
- Captured complex spatial heterogeneity in real datasets.

## Abstract

For point patterns observed in natura, spatial heterogeneity is more the rule than the exception. In numerous applications, this can be mathematically handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief, a LGCP is a Cox process driven by an underlying log Gaussian random field (log GRF). This allows the representation of point aggregation, point vacuum and intermediate situations, with more or less rapid transitions between these different states depending on the properties of GRF. Very often, the covariance function of the GRF is assumed to be stationary. In this article, we give two examples where the sizes (that is, the number of points) and the spatial extents of point clusters are allowed to vary in space. To tackle such features, we propose parametric and semiparametric models of non-stationary LGCPs where the non-stationarity is included in both the mean function and the covariance function of the GRF. Thus, in contrast to most other work on inhomogeneous LGCPs, second-order intensity-reweighted stationarity is not satisfied and the usual two step procedure for parameter estimation based on e.g. composite likelihood does not easily apply. Instead we propose a fast three step procedure based on composite likelihood. We apply our modelling and estimation framework to analyse datasets dealing with fish aggregation in a reservoir and with dispersal of biological particles.