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Estimating the first-order intensity function in point pattern analysis is animportant problem, and it has been approached so far from differentperspectives: parametrically, semiparametrically or nonparametrically. Ourapproach is close to a semiparametric one. Motivated by eye-movement data, weintroduce a convolution type model where the log-intensity is modelled as theconvolution of a function $\beta(\cdot)$, to be estimated, and a single spatialcovariate (the image an individual is looking at for eye-movement data). Basedon a Fourier series expansion, we show that the proposed model \rev{can beviewed as a} log-linear model with an infinite number of coefficients, whichcorrespond to the spectral decomposition of $\beta(\cdot)$. After truncation,we estimate these coefficients through a penalized Poisson likelihood. Weillustrate the efficiency of the proposed methodology on simulated data and oneye-movement data.

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