[Paper Review] A J-function for inhomogeneous point processes

M. N. M. van Lieshout|arXiv (Cornell University)|Aug 26, 2010

Point processes and geometric inequalities4 citations

TL;DR

This paper introduces a new inhomogeneous $J$-function for intensity-reweighted moment stationary point processes, generalizing the classical $J$-function to account for spatial and temporal inhomogeneity. By expressing the statistic via generating functionals and conditional intensities, it enables detection of spatial interactions—such as inhibition or clustering—under non-stationary intensity, with explicit computation for Poisson, log-Gaussian Cox, and thinned hard core processes, validated through minus sampling estimators and simulations.

ABSTRACT

We propose new summary statistics for intensity-reweighted moment stationary point processes that generalise the well known J-, empty space, and nearest-neighbour distance distribution functions, represent them in terms of generating functionals and conditional intensities, and relate them to the inhomogeneous reduced second moment function. Extensions to space time and marked point processes are briefly discussed.

Motivation & Objective

• To extend the classical $J$-function to inhomogeneous point processes where intensity varies across space or time.
• To develop a summary statistic that captures interaction structure (clustering or inhibition) while accounting for intensity non-stationarity.
• To provide a theoretical framework based on generating functionals and conditional intensities for the new $J_{\text{inhom}}$ function.
• To derive explicit expressions for $J_{\text{inhom}}$ under key models: Poisson, log-Gaussian Cox, and thinned hard core processes.
• To propose a minus sampling estimator for practical estimation and validate it through simulated examples.

Proposed method

• Define the inhomogeneous $J$-function $J_{\text{inhom}}(t)$ using $n$-point correlation functions and the generating functional of the point process.
• Represent $J_{\text{inhom}}$ in terms of the conditional intensity and intensity-reweighted moment stationarity.
• Express $J_{\text{inhom}}$ via the ratio of two generating functionals: $G_B^{!x}(1 - u_t^x)$ and $G(1 - u_t^x)$, where $u_t^x$ encodes spatial distance and intensity.
• Derive explicit forms of $J_{\text{inhom}}$ for three model classes: Poisson, log-Gaussian Cox, and thinned hard core processes.
• Propose a minus sampling estimator for $K_{\text{inhom}}$ to estimate the inhomogeneous $J$-function from observed data.
• Extend the framework to space-time and marked point processes by adapting the correlation structure and defining $J^{ST}_{\text{inhom}}$ and $J^{B}_{\text{inhom}}$ functions with appropriate translation invariance assumptions.

Experimental results

Research questions

• RQ1 How can the classical $J$-function be generalized to accommodate spatial and temporal inhomogeneity in point processes?
• RQ2 What is the theoretical representation of the inhomogeneous $J$-function in terms of generating functionals and conditional intensities?
• RQ3 How does the new $J_{\text{inhom}}$ function behave under different point process models, such as Poisson, log-Gaussian Cox, and thinned hard core processes?
• RQ4 Can a consistent and practical estimator for $J_{\text{inhom}}$ be constructed from observed data, and how does it perform in simulations?
• RQ5 How can the framework be extended to space-time and marked point processes while preserving interpretability and statistical consistency?

Key findings

• For the Poisson point process with inhomogeneous intensity, $J_{\text{inhom}}(t) = 1$, confirming the absence of interaction, as expected.
• For the log-Gaussian Cox process with positive correlation in the underlying Gaussian field, $J_{\text{inhom}}(t)$ lies below the Poisson reference, indicating clustering after accounting for inhomogeneity.
• For the thinned hard core process, $J_{\text{inhom}}(t)$ shows a flat initial segment below the Poisson reference up to $r \approx 0.2$, reflecting inhibition due to the hard core distance.
• The minus sampling estimator of $K_{\text{inhom}}$ correctly identifies deviations from the Poisson model, with estimated values below $\pi t^2$ for $t > 0.13$ in the log-Gaussian Cox case.
• In the thinned hard core process, the estimator confirms inhibition by showing values below the Poisson reference up to $t \approx 0.2$, consistent with the $J_{\text{inhom}}$-function behavior.
• The extension to space-time and marked point processes is feasible: $J^{ST}_{\text{inhom}}$ reduces to the $K^*$-approach of Gabriel and Diggle when truncated at $n=1$, and $J^{B}_{\text{inhom}}$ allows interaction analysis conditioned on mark sets.