Semiparametric Multinomial Logistic Regression for Multivariate Point Pattern Data

Abstract

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.

Keywords:

• Conditional likelihood
• Cross pair correlation functions
• Multivariate point process

Supplementary Materials

The supplementary materials for this article contain further simulation studies and plots, proofs, and auxiliary results.

Additional information

Funding

Kristian Bjørn Hessellund and Rasmus Waagepetersen gratefully acknowledge The Danish Council for Independent Research—Natural Sciences, grant DFF-7014-00074 “Statistics for point processes in space and beyond,” and the “Centre for Stochastic Geometry and Advanced Bioimaging,” funded by grant 8721 from the Villum Foundation. Xu’s research is supported by NSF grant SES-1902195 and Yongtao Guan is supported by NSF grant DMS-1810591.