Nonparametric estimation of the pair correlation function of replicated inhomogeneous point processes

Ganggang Xu,Jingfei Zhang,Rasmus Waagepetersen,Abdollah Jalilian,Yongtao Guan,Chong Zhao

doi:10.1214/20-ejs1755

Abstract

We consider the nonparametric estimation of the isotropic pair correlation function (PCF) of inhomogeneous point processes when replicates are available. Based on carefully designed estimating equations, two types of nonparametric estimators, i.e., the local polynomial estimator and the orthogonal series estimator, are proposed and studied. The proposed estimators circumvent the problems caused by the need for estimating the unknown intensity function for kernel smoothed PCF estimators and they are free of edge correction terms. Asymptotic properties are investigated for both estimators and valid point-wise confidence bands are derived. Finite sample performances of the proposed estimators are demonstrated by simulation as well as an application to the Sina Weibo posting data.

Highlights

Statistical analysis, and in suggesting suitable models for the data
While the first scenario is more suitable for temporal processes where replicates are available such as in our Sina Weibo posting data, the second scenario may be more preferable for spatial point processes where the number of replicates is limited
We proposed two types of nonparametric estimators for the pair correlation function (PCF) of inhomogeneous point processes when replicates are available

Summary

Introduction

Statistical analysis, and in suggesting suitable models for the data. the popular second-order characteristics called Ripley’s K-function (Ripley, 1976) and Besag’s L-function Besag (1977) are in one-to-one correspondence with the PCF. There is extensive literature on estimating the PCF nonparametrically based on a single observed point pattern, ranging from the kernel estimators (Stoyan and Stoyan, 1994; Baddeley et al, 2000; Illian et al, 2008), the Bayesian estimator (Yue and Loh, 2010) and more recently the orthogonal series estimator (Jalilian et al, 2019) All of these methods require that the unknown intensity function λ(·) is replaced by an estimate λ(·). While the first scenario is more suitable for temporal processes where replicates are available such as in our Sina Weibo posting data, the second scenario may be more preferable for spatial point processes where the number of replicates is limited In both scenarios, we propose empirical estimators for the asymptotic variances of the estimated PCFs and construct valid confidence bands. Another advantage of the proposed estimators is that they do not require any edge correction terms due to the design of the estimating equations

Objectives

Methods

Conclusion