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Fourier analysis of spatial point processes

Junho Yang, Yongtao Guan

TL;DR

Fourier analysis of spatial point processes provides a frequency-domain framework to estimate and infer second-order structure in irregular spatial patterns, leveraging the discrete Fourier transform (DFT) and its tapered variant. The authors prove that, under second-order stationarity and an α-mixing condition, the DFTs and periodograms are asymptotically jointly Gaussian with asymptotic uncorrelatedness across distant frequencies, enabling a central limit theorem for the integrated periodogram. They derive the asymptotic distribution for the kernel spectral density estimator and introduce a Whittle-type likelihood for parametric model fitting that remains meaningful under misspecification, with estimators converging to the best-fitting spectral-density parameter. Through simulations on Hawkes, Neyman-Scott, Log-Gaussian Cox, and determinantal point processes, the method demonstrates competitive finite-sample performance and favorable computation times relative to spatial-domain methods, while also offering guidelines for practical implementation and potential extensions to multivariate and inhomogeneous settings.

Abstract

In this article, we develop comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an $α$-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models.

Fourier analysis of spatial point processes

TL;DR

Fourier analysis of spatial point processes provides a frequency-domain framework to estimate and infer second-order structure in irregular spatial patterns, leveraging the discrete Fourier transform (DFT) and its tapered variant. The authors prove that, under second-order stationarity and an α-mixing condition, the DFTs and periodograms are asymptotically jointly Gaussian with asymptotic uncorrelatedness across distant frequencies, enabling a central limit theorem for the integrated periodogram. They derive the asymptotic distribution for the kernel spectral density estimator and introduce a Whittle-type likelihood for parametric model fitting that remains meaningful under misspecification, with estimators converging to the best-fitting spectral-density parameter. Through simulations on Hawkes, Neyman-Scott, Log-Gaussian Cox, and determinantal point processes, the method demonstrates competitive finite-sample performance and favorable computation times relative to spatial-domain methods, while also offering guidelines for practical implementation and potential extensions to multivariate and inhomogeneous settings.

Abstract

In this article, we develop comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an -mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models.
Paper Structure (49 sections, 35 theorems, 298 equations, 8 figures, 3 tables)

This paper contains 49 sections, 35 theorems, 298 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Spectral density functions for the second-order stationary point processes
  3. Preliminaries
  4. Spectral density function and its estimator
  5. Features of the spectral density functions and their estimators
  6. Asymptotic properties of the DFT and periodogram
  7. Asymptotic results
  8. Nonparametric kernel spectral density estimator
  9. Estimation of the spectral mean statistics
  10. Examples of spatial point process models
  11. Example I: Hawkes processes
  12. Example II: Neyman-Scott point processes
  13. Example III: Log-Gaussian Cox Processes
  14. Example IV: Determinantal point processes
  15. Frequency domain parameter estimation under possible model misspecification
  16. ...and 34 more sections

Key Result

Theorem 3.1

Let $X$ be a second-order stationary point process on $\mathbb{R}^d$. Suppose that Assumptions assum:A, assum:C (for $\ell=2$), and assum:E(i) hold. Let $\{\boldsymbol{\omega}_{1,n}\}$ and $\{\boldsymbol{\omega}_{2,n}\}$ be sequences on $\mathbb{R}^d$ such that $\{\boldsymbol{\omega}_{1,n}\}$, $\{\b If we further assume Assumption assum:C for $\ell=4$ holds and $\{\boldsymbol{\omega}_{1,n}\}$ and

Figures (8)

  • Figure 1: Top: Realizations of the four different stationary isotropic spatial point processes on the observation domain $[-20,20]^2$. Middle left: Plot of the pair correlation function $g(\boldsymbol{x})-1$ against $\|\boldsymbol{x}\| \in [0,\infty)$ for each model. Middle right: Plot of the spectral density function $f(\boldsymbol{\omega})$ in log-scale against $\|\boldsymbol{\omega}\| \in [0,\infty)$ for each model. Bottom: Plot of the periodogram $\widehat{I}_{h,n}(\boldsymbol{\omega})$.
  • Figure 2: The true spectral density function ($f(\boldsymbol{\omega})$; solid line) as in (\ref{['eq:fLGCP']}) and the two best fitting TCP spectral densities for $A = 10$ evaluated on $D_{2\pi}$ (dashed line) and on $D_{5\pi}$ (dotted line). All three spectral densities are plotted in log-scale against $\|\boldsymbol{\omega}\| \in [0,\infty)$. The horizontal line indicates the asymptote of $f$ which takes a value $(2\pi)^{-2} \lambda^{(true)}$.
  • Figure C.1: Example of disjoint partitions of $D_n \backslash (D_n-\boldsymbol{t})$ when $d=2$.
  • Figure H.1: Top: Kernel spectral density estimators as in (\ref{['eq:KSD-sum']}) of the periodograms that are computed in the bottom panels of Figure \ref{['fig:motiv']}. Middle: $|\widehat{I}_{h,n}(\boldsymbol{\omega}) - f(\boldsymbol{\omega})|$ for each model. Bottom: $\widehat{f}_{n,b}^{(R)}(\boldsymbol{\omega}) - f(\boldsymbol{\omega})|$ for each model.
  • Figure H.2: Densities of the estimated parameters for the correctly specified Thomas clustering process model as in Section \ref{['sec:CS']}. Each row refers to different observation domains. Vertical dashed lines refer to the true parameter values.
  • ...and 3 more figures

Theorems & Definitions (48)

  • Theorem 3.1: Asymptotic uncorrelatedness of the DFT and periodogram
  • proof
  • Theorem 3.2: Asymptotic joint distribution of the DFTs and periodograms
  • proof
  • Remark 3.1
  • Theorem 3.3
  • proof
  • Theorem 4.1: Asymptotic distribution of the integrated periodogram
  • proof
  • Remark 4.1: Estimation of the asymptotic variance
  • ...and 38 more