  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.
