A convolution type model for the intensity of spatial point processes applied to eye-movement data

TL;DR

This paper introduces a convolution-based semi-parametric model for estimating the intensity function of spatial point processes, specifically applied to eye-movement data, using Fourier series and penalized likelihood for estimation.

Contribution

It proposes a novel convolution-type model for intensity estimation in spatial point processes, with a Fourier series approach and penalized likelihood, tailored for eye-movement data.

Findings

Efficient estimation demonstrated on simulated data.

Successful application to eye-movement data.

Model captures spatial intensity variations effectively.

Abstract

Estimating the first-order intensity function in point pattern analysis is an important problem, and it has been approached so far from different perspectives: parametrically, semiparametrically or nonparametrically. Our approach is close to a semiparametric one. Motivated by eye-movement data, we introduce a convolution type model where the log-intensity is modelled as the convolution of a function $\beta(\cdot)$, to be estimated, and a single spatial covariate (the image an individual is looking at for eye-movement data). Based on a Fourier series expansion, we show that the proposed model \rev{can be viewed as a} log-linear model with an infinite number of coefficients, which correspond to the spectral decomposition of $\beta(\cdot)$. After truncation, we estimate these coefficients through a penalized Poisson likelihood. We illustrate the efficiency of the proposed methodology on simulated data and on eye-movement data.