On the classification problem for Poisson Point Processes

TL;DR

This paper investigates binary classification for Poisson point processes in metric spaces, proposing two methods: nonparametric intensity estimation and a k-nearest neighbor rule, with theoretical consistency proofs and simulation comparisons.

Contribution

It introduces and proves the consistency of two classification approaches for Poisson processes, including a novel adaptation of k-NN with theoretical validation.

Findings

The intensity estimation approach is consistent.

The k-NN classifier satisfies the Besicovitch condition.

In simulations, k-NN outperforms intensity-based classification.

Abstract

We study the binary classification problem for Poisson point processes, which are allowed to take values in a general metric space. The problem is tackled in two different ways: estimating nonparametricaly the intensity functions of the processes (and then plugged into a deterministic formula which expresses the regression function in terms of the intensities), and performing the classical $k$ nearest neighbor rule by introducing a suitable distance between patterns of points. In the first approach we prove the consistency of the estimated intensity so that the rule turns out to be also consistent. For the $k$-NN classifier, we prove that the regression function fulfils the so called "Besicovitch condition", usually required for the consistency of the classical classification rules. The theoretical findings are illustrated on simulated data, where in one case the $k$-NN rule outperforms the first approach.