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 modeled as the convolution of a function $\beta \left(\cdot \right)$, 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 can be viewed as a log-linear model with an infinite number of coefficients, which correspond to the spectral decomposition of $\beta \left(\cdot \right)$. 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.

估计点模式分析中的一阶强度函数是一个重要问题，到目前为止，它已经从不同的角度进行了处理：参数化、半参数化或非参数化。我们的方法接近于半参数方法。受眼球运动数据的启发，我们引入了一种卷积类型模型，其中对数强度被建模为函数的卷积$\beta \left(\cdot \right)$，待估计，以及单个空间协变量（个人正在查看的图像以获取眼球运动数据）。基于傅里叶级数展开，我们表明所提出的模型可以被视为具有无限个系数的对数线性模型，对应于$\beta \left(\cdot \right)$. 截断后，我们通过惩罚泊松似然估计这些系数。我们说明了所提出的方法在模拟数据和眼动数据上的效率。