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Cox processes driven by transformed Gaussian processes on linear networks -- A review and new contributions

Jesper Møller, Jakob G. Rasmussen

TL;DR

Three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks are introduced, and for the first time statistical procedures and applications for parametric families of such models are considered.

Abstract

There is a lack of point process models on linear networks. For an arbitrary linear network, we consider new models for a Cox process with an isotropic pair correlation function obtained in various ways by transforming an isotropic Gaussian process which is used for driving the random intensity function of the Cox process. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

Cox processes driven by transformed Gaussian processes on linear networks -- A review and new contributions

TL;DR

Three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks are introduced, and for the first time statistical procedures and applications for parametric families of such models are considered.

Abstract

There is a lack of point process models on linear networks. For an arbitrary linear network, we consider new models for a Cox process with an isotropic pair correlation function obtained in various ways by transforming an isotropic Gaussian process which is used for driving the random intensity function of the Cox process. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.
Paper Structure (28 sections, 7 theorems, 55 equations, 7 figures, 5 tables)

This paper contains 28 sections, 7 theorems, 55 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Linear networks and metrics
  3. Setting
  4. The resistance metric
  5. Gaussian processes and isotropic covariance functions
  6. Isotropic covariance functions
  7. Simulation of GPs on linear networks
  8. Point processes and some of their characteristics
  9. Point process setting
  10. Moment properties
  11. $K$-function
  12. Estimation and model checking
  13. Cox processes driven by transformed Gaussian processes
  14. Cox processes on linear networks
  15. Models
  16. ...and 13 more sections

Key Result

Theorem 1

We have the following properties of $d_{\mathcal{G}}$ and $d_{\mathcal{R}}$.

Figures (7)

  • Figure 1: Left: The locations of street crimes in a part of Chicago. Right: The locations of spines on a dendrite tree HeidiMe. The circle marks the root of the tree, the black lines are a main branch, and the grey lines are side branches.
  • Figure 2: Correlations functions with inverse gamma Bernstein CDFs (black curves) with $(\tau,\phi)=(1,1)$ (left column) and $(\tau,\phi)=(5,5)$ (right column). The grey curves in each plot show 100 approximated correlation functions given by \ref{['e:approxc']} with $n=20$ (upper row), $n=50$ (middle row), and $n=200$ (lower row).
  • Figure 3: Simulation of zero mean GPs on the Chicago street network with an isotropic exponential covariance function $c(u,v)=r_0(d_{\mathcal{R}}(u,v))$. Top row: When $r_0(t)=\exp(-st)$ with parameter $s=0.1$ (left) or $s=0.01$ (right). Middle row: When $r_0$ has Bernstein CDF given by a gamma distribution (left) or inverse gamma distribution (right), where in both cases the mean of $s$ is 0.01. Bottom row: For plots 2--4, the corresponding densities for $F$ and correlation functions, where the curves in solid, dashed, and dotted correspond to plots 2--4, respectively.
  • Figure 4: Pair correlation functions for the Chicago crime dataset: Non-parametric estimate (solid curve) and curves estimated by minimum contrast for a LGCP (dashed curve), an ICP (dotted curve), and a PCPP (dashed-dotted curve) model.
  • Figure 5: Left column: Simulations of point patterns under the estimated LGCP (upper row), ICP (middle row) and PCPP (lower row) models for the Chicago crime dataset. The width of the lines shows the simulation of the underlying transformed Gaussian processes (the random intensity functions), while the points show the simulated point pattern. Right column: 95% global envelopes for the fitted LGCP (upper row), ICP (middle row), and PCPP (lower row) models for the Chicago crime dataset. The $p$-values of the global envelope test are shown above each plot.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Proposition 1
  • proof
  • Example 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 4 more