CS32: Advances in Statistical Inference for Spatial Point Processes

Organizer: Bartlomiej Blaszczyszyn (Inria/ENS)

Minimax Estimation of the Structure Factor of Spatial Point Processes

Gabriel Mastrilli

Spectral methods have recently gained attention in spatial statistics as an alternative to direct-space techniques for second-order inference of spatial point processes. Instead of analyzing second-order summary statistics like the Ripley K-function, one can estimate Bartlett’s spectral measure, also known in physics as the structure factor \(S\), which is the Fourier transform of the pair correlation function. Several estimators of \(S\) have been introduced \([1, 2, 3]\). However, without a reference minimax rate of convergence, it remains unclear whether better estimators could be constructed.

We address this gap by proving the following minimax result: for estimating a structure factor of Hölder regularity \(\beta\), the worst-case error of any estimator is at least the expected number of observed points raised to the power \(-\beta/(2\beta + d)\). We then construct multi-taper estimators that achieve this optimal rate and further show that they exhibit exponential concentration. These estimators are oracle in the sense that they depend on unknown parameters related to the smoothness of \(S\). To obtain a practical estimator, we propose a cross-validation method based on random thinning and decorrelation between distinct frequencies.

Bibliography

\([1]\) D. Hawat, G. Gautier, R. Bardenet and R. Lachièze-Rey, "On estimating the structure factor of a point process, with applications to hyperuniformity", Statistics and Computing, 2023.

\([2]\) T. Rajala, S. Olhede, J. P Grainger, and D. J Murrell, "What is the Fourier transform of a spatial point process ?", IEEE Transactions on Information Theory, 2023.

\([3]\) J. Yang and Y. Guan,"Fourier analysis of spatial point processes", arXiv:2401.06403, 2024.