The availability of large spatial data geocoded at accurate locations has fueled a growing interest in spatial modeling and analysis of point processes. The proposed research is motivated by the intensity estimation problem for large spatial point patterns on complex domains, where many existing spatial point process models suffer from the problems of "leakage" and computation. We propose an efficient intensity estimation algorithm to estimate the spatially varying intensity function and to study the varying relationship between intensity and explanatory variables on complex domains. The method is built upon a graph regularization technique and hence can be flexibly applied to point patterns on complex domains such as regions with irregular boundaries and holes, or linear networks. An efficient proximal gradient optimization algorithm is proposed to handle large spatial point patterns. We also derive the asymptotic error bound for the proposed estimator. Numerical studies are conducted to illustrate the performance of the method. Finally, We apply the method to study and visualize the intensity patterns of the accidents on the Western Australia road network, and the spatial variations in the effects of income, lights condition, and population density on the Toronto homicides occurrences.

翻译：在准确地点提供大型空间数据地理编码,促使人们越来越关注点点过程的空间建模和分析。拟议研究的动机是复杂领域大型空间点模式的强度估计问题,许多现有空间点进程模型都存在“渗漏”和计算问题。我们建议一种高效的强度估计算法,以估计空间差异的强度功能,并研究复杂领域强度和解释变量之间的不同关系。这种方法以图解正规化技术为基础,因此可以灵活地应用到复杂领域,如有非正常边界和孔或线性网络的区域。建议一种高效的准氧化梯度优化算法,以处理大空间点模式。我们还从中推算出为拟议估计值设计的无症状错误。进行了数值研究,以说明方法的性能。最后,我们采用该方法来研究和直观西澳大利亚公路网事故的强度模式,以及多伦多杀人事件的收入、灯光状况和人口密度影响的空间变化。