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Palm distributions of superposed point processes for statistical inference

Mario Beraha, Federico Camerlenghi, Lorenzo Ghilotti

Abstract

Palm distributions play a central role in the study of point processes and their associated summary statistics. In this paper, we characterize the Palm distributions of the superposition of independent point processes, establishing a simple mixture representation depending on the point processes' Palm distributions and moment measures. We explore two statistical applications enabled by our main result. First, we consider minimum contrast estimation for corrupted point processes. Second, we investigate the class of shot noise Cox processes and derive explicit expressions for their higher-order Palm distributions. In the finite case, we further obtain a tractable expression for the Janossy density, which plays the role of a likelihood function and thus can be used for new likelihood-based inference strategies. Extensions to the superposition of multiple point processes and to higher-order Palm distributions are also presented.

Palm distributions of superposed point processes for statistical inference

Abstract

Palm distributions play a central role in the study of point processes and their associated summary statistics. In this paper, we characterize the Palm distributions of the superposition of independent point processes, establishing a simple mixture representation depending on the point processes' Palm distributions and moment measures. We explore two statistical applications enabled by our main result. First, we consider minimum contrast estimation for corrupted point processes. Second, we investigate the class of shot noise Cox processes and derive explicit expressions for their higher-order Palm distributions. In the finite case, we further obtain a tractable expression for the Janossy density, which plays the role of a likelihood function and thus can be used for new likelihood-based inference strategies. Extensions to the superposition of multiple point processes and to higher-order Palm distributions are also presented.

Paper Structure

This paper contains 24 sections, 14 theorems, 141 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Superposition of point processes
  3. Background and notation for point processes
  4. Palm distributions of the superposition of independent processes
  5. Inference for corrupted processes via minimum contrast
  6. Summary statistics for superposed point processes
  7. Fitting a corrupted Matérn cluster process via minimum contrast estimation
  8. Shot noise Cox processes: Palm distributions and Janossy densities
  9. Discussion
  10. Mathematical background on point processes
  11. General results on Palm distributions of point processes
  12. Proofs of the main results and extensions
  13. Extensions of \ref{['teo:main']}
  14. Summary statistics for corrupted point processes
  15. Inhomogeneous summary statistics
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\Phi_1$ and $\Phi_2$ be two independent point processes on $\mathbb{X}$. Then, for any $x \in \mathbb{X}$, the Palm version $(\Phi_1 + \Phi_2)_{x}$ can be expressed as the following mixture:

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition S1: Palm distributions of mixtures of point processes
  • proof
  • Theorem S1: Janossy measures of finite point processes
  • proof
  • Corollary S1
  • proof
  • ...and 16 more