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Semi-parametric multinomial logistic regression for multivariate point pattern data

By Kristian Bjørn Hessellund, Ganggang Xu, Yongtao Guan and Rasmus Waagepetersen
CSGB Research Reports
No. 10, October 2019

We propose a new method for analysis of multivariate point pattern data observed in a heterogeneous environment and with complex intensity functions. We suggest semi-parametric 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 the cross pair correlation functions of the multivariate point process. To make valid statistical inference without restrictive assumptions, we construct consistent non-parametric estimators for cross pair correlation function ratios. Finally, we construct standardized residual plots and predictive probability plots to validate and to visualize findings of the model. The effectiveness of the proposed methodology is demonstrated through extensive simulation studies and an application to analyzing effects of socio-economic and demographical variables on occurrences of street crimes in Washington DC.

Keywords: Conditional likelihood, Cross pair correlation functions, multinomial logistic regression, multivariate point process, semi-parametric.

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