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.
翻译:在点形分析中估算一阶强度函数是一个重要问题,迄今为止,从不同的角度,即对准、半对称或非对称的角度,对一阶强度函数进行了估计,这是一个重要的问题。我们的方法接近于半对数。受眼动数据驱动,我们引入了一种卷变型模型,即对数密度的模拟模型是:一个函数($\beta(cdot))的递增,将加以估计,而单一空间共变(一个人正在寻找的图像是眼睛移动数据)。根据四倍系列的扩展,我们显示,拟议的模型\rev{可以被视为一个有无限数系数的逻辑线性模型,这与$\beta(\cdott)美元的光谱分解位置相对应。在蒸发后,我们通过受抑制的Poisson可能性来估计这些系数。我们说明了关于模拟数据和眼睛移动数据的拟议方法的效率。