[Paper Review] Spatial Cox processes in an infinite-dimensional framework

M. P. Frías, A. Torres-Signes+2 more|arXiv (Cornell University)|Nov 27, 2018

Spatial and Panel Data Analysis30 references4 citations

TL;DR

This paper introduces a novel class of spatial Cox processes driven by a Hilbert-valued log-intensity, enabling functional data analysis of spatial point patterns. It proposes a Whittle-type spectral estimation method based on the periodogram operator, proving strong consistency of the parametric estimator in a linear spatial autoregressive Hilbertian model, and applies the method to predict respiratory disease mortality across the Iberian Peninsula (1980–2015).

ABSTRACT

We introduce a new class of spatial Cox processes driven by a Hilbert--valued random log--intensity. We adopt a parametric framework in the spectral domain, to estimate its spatial functional correlation structure. Specifically, we consider a spectral functional, based on the periodogram operator, inspired on Whittle estimation methodology. Strong-consistency of the parametric estimator is proved in the linear case. We illustrate this property in a simulation study under a Gaussian first order Spatial Autoregressive Hilbertian scenario for the log--intensity model. Our method is applied to the spatial functional prediction of respiratory disease mortality in the Spanish Iberian Peninsula, in the period 1980--2015.

Motivation & Objective

• To develop a new class of spatial Cox processes driven by infinite-dimensional (Hilbert-valued) log-intensity random fields.
• To address the gap in parametric estimation methods for functional spatial point processes by introducing a spectral-domain approach.
• To establish strong consistency of a parametric estimator based on a periodogram operator in a linear spatial autoregressive Hilbertian framework.
• To apply the proposed methodology to real-world spatial functional data, specifically respiratory disease mortality in Spain (1980–2015).

Proposed method

• Models the log-intensity of a spatial Cox process as a Hilbert-space-valued stochastic process, enabling functional data analysis of spatial point patterns.
• Employs a parametric spectral framework based on the periodogram operator, inspired by Whittle’s estimation methodology for time series.
• Derives a functional estimating equation using Fourier coefficients and spectral density operators in the frequency domain.
• Applies Parserval’s identity and truncation techniques to analyze convergence properties of the spectral estimator.
• Uses the inverse spectral density operator and functional series expansions to derive asymptotic behavior of the estimator.
• Establishes strong consistency of the estimator by contradiction, leveraging bounds on spectral ratios and operator norms.

Experimental results

Research questions

• RQ1 How can spatial Cox processes be extended to infinite-dimensional (Hilbert-valued) log-intensity fields to model functional spatial point patterns?
• RQ2 Can a Whittle-type spectral estimation procedure be adapted to functional spatial data to achieve consistent parameter estimation?
• RQ3 What are the theoretical conditions under which a parametric estimator of the spatial functional correlation structure is strongly consistent?
• RQ4 How does the proposed method perform in finite-sample settings with realistic spatial dependence structures?
• RQ5 Can the method be effectively applied to real-world spatio-temporal health data, such as regional mortality patterns?

Key findings

• The proposed parametric estimator for the spatial functional correlation structure is strongly consistent under the linear spatial autoregressive Hilbertian model.
• The spectral estimation method based on the periodogram operator achieves convergence to the true parameter value almost surely as the functional sample size increases.
• A simulation study confirms the consistency of the estimator under a Gaussian first-order spatial autoregressive Hilbertian model for the log-intensity.
• The method successfully captures the spatial functional correlation structure in high-dimensional functional data, enabling accurate prediction of spatial point patterns.
• The approach is applied to predict respiratory disease mortality across the Iberian Peninsula from 1980 to 2015, demonstrating practical utility on real data.
• Theoretical convergence is established via bounds on spectral ratios and the use of truncated spectral series, with error terms vanishing as sample size grows.