Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimax estimation of the structure factor of spatial point processes

Gabriel Mastrilli

TL;DR

The paper studies the problem of estimating the structure factor $S$ for stationary spatial point processes, establishing minimax lower bounds that yield the classical nonparametric rate $|W|^{-\beta/(2\beta+d)}$ for Hölder regularity $\beta$. It then develops a multitaper estimator, based on Hermite tapers, that attains this minimax rate and proves non-asymptotic risk bounds and high-probability concentration under Brillinger-mixing. A data-driven, local cross-validation procedure selects the number of tapers with an oracle-type guarantee, combining theoretical rigor with practical adaptability. Numerical experiments on diverse models (clusters and repulsive patterns) demonstrate favorable performance and validate the proposed taper-selection scheme. Overall, the work provides both optimal spectral-inference guarantees and a practical methodology for structure-factor estimation in spatial point processes.

Abstract

We investigate the problem of estimating the structure factor, or spectra, of stationary spatial point processes. In the first part, we establish a minimax lower bound for this estimation problem, using an approach tailored to second-order properties of spatial point processes. Although not the main focus, this methodology also extends naturally to a minimax lower bound for the estimation of the pair correlation function of spatial point processes. In the second part, we construct a multitaper estimator that achieves the optimal rate of convergence in squared risk. Under a Brillinger-mixing condition, we further establish a chi-square-type concentration bound. Finally, we propose a data-driven procedure for selecting the number of tapers, supported by an oracle inequality, and we demonstrate the practical effectiveness of the method through numerical experiments.

Minimax estimation of the structure factor of spatial point processes

TL;DR

The paper studies the problem of estimating the structure factor for stationary spatial point processes, establishing minimax lower bounds that yield the classical nonparametric rate for Hölder regularity . It then develops a multitaper estimator, based on Hermite tapers, that attains this minimax rate and proves non-asymptotic risk bounds and high-probability concentration under Brillinger-mixing. A data-driven, local cross-validation procedure selects the number of tapers with an oracle-type guarantee, combining theoretical rigor with practical adaptability. Numerical experiments on diverse models (clusters and repulsive patterns) demonstrate favorable performance and validate the proposed taper-selection scheme. Overall, the work provides both optimal spectral-inference guarantees and a practical methodology for structure-factor estimation in spatial point processes.

Abstract

We investigate the problem of estimating the structure factor, or spectra, of stationary spatial point processes. In the first part, we establish a minimax lower bound for this estimation problem, using an approach tailored to second-order properties of spatial point processes. Although not the main focus, this methodology also extends naturally to a minimax lower bound for the estimation of the pair correlation function of spatial point processes. In the second part, we construct a multitaper estimator that achieves the optimal rate of convergence in squared risk. Under a Brillinger-mixing condition, we further establish a chi-square-type concentration bound. Finally, we propose a data-driven procedure for selecting the number of tapers, supported by an oracle inequality, and we demonstrate the practical effectiveness of the method through numerical experiments.

Paper Structure

This paper contains 29 sections, 33 theorems, 449 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. General notations
  3. Notations and definitions specific to point processes
  4. Second order properties and structure factor of stationary point processes
  5. Cumulants of stationary point processes
  6. Minimax lower bounds
  7. Minimax estimator of the structure factor
  8. Multitaper estimator: introduction
  9. Multitaper estimators: mean integrated squared risk
  10. Multitaper estimators: non asymptotic fluctuations
  11. Data-driven procedure for the choice of the number of tapers
  12. Description and theoretical guarantee
  13. Numerical study
  14. Description of the models
  15. Choice of the hyper-parameters of the cross-validation
  16. ...and 14 more sections

Key Result

Theorem 4.1

Let $h :[0, \infty) \to [0, \infty)$ be an non-identically null increasing function with $h(0) = 0$. Consider $\mathcal{W}$, a regular family of compact sets of $\mathbb{R}^d$. Let $(\beta, M, L) \in (0, \infty)^3$. For $S \in \Theta(\beta, L)$, we denote by $\mathcal{L}_{(S, M)}$ the set all laws o where $\sup_{S \in \Theta(\beta, L)}$ denotes the supremun over structure factors of regularity $\T

Figures (1)

  • Figure 1: Each model is simulated $500$ times in the observation window $[-20, 20]^2$. The Hölder regularity $\beta$ of the corresponding structure factors are indicated between parenthesis. The dashed black curve represents the theoretical structure factor. The red color corresponds to the multitaper method, with the number of tapers locally selected via the cross-validation procedure of Section \ref{['sec:dd']}. The blue color corresponds to the kernel estimator yang2024fourier, with the cross-validation method introduced in ding2025pseudo. For both methods, the solid line represents the average of the structure factor estimates, while the shaded region corresponds to their 0.05--0.95 empirical quantiles.

Theorems & Definitions (76)

  • Definition 1
  • Definition 2
  • Theorem 4.1
  • Theorem 4.2
  • Definition 3
  • Remark 1
  • Theorem 5.1
  • Definition 4
  • Corollary 5.1
  • Theorem 5.2
  • ...and 66 more