Semiparametric Multinomial Logistic Regression for Multivariate Point Pattern Data

We propose a new method for analysis of multivariate point pattern data observed in a heterogeneous environment and with complex intensity functions. We suggest semiparametric models for the intensity functions that depend on an unspecified factor common to all types of points. This is for example well suited for analyzing spatial covariate effects on events such as street crime activities that occur in a complex urban environment. A multinomial conditional composite likelihood function is introduced for estimation of intensity function regression parameters and the asymptotic joint distribution of the resulting estimators is derived under mild conditions. Crucially, the asymptotic covariance matrix depends on ratios of cross pair correlation functions of the multivariate point process. To make valid statistical inference without restrictive assumptions, we construct consistent nonparametric estimators for these ratios. Finally, we construct standardized residual plots, predictive probability plots, and semiparametric intensity plots to validate and to visualize the findings of the model. The effectiveness of the proposed methodology is demonstrated through extensive simulation studies and an application to analyzing the effects of socio-economic and demographical variables on occurrences of street crimes in Washington DC. Supplementary materials for this article are available online.