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

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 logintensity is modelled as the convolution of a function β(•), 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 is related to the log-linear model with infinite number of coefficients, which correspond to the spectral decomposition of β(•). After truncation, we estimate these coefficients through a penalized Poisson likelihood and prove infill asymptotic results for a large class of spatial point processes. We illustrate the efficiency of the proposed methodology on simulated data and real data.