Estimating the hyperuniformity exponent of point processes
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arXiv
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| Format: | Preprint |
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2024
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| author | Mastrilli, Gabriel Błaszczyszyn, Bartłomiej Lavancier, Frédéric |
| author_facet | Mastrilli, Gabriel Błaszczyszyn, Bartłomiej Lavancier, Frédéric |
| contents | We address the challenge of estimating the hyperuniformity exponent $α$ of a spatial point process, given only one realization of it. Assuming that the structure factor $S$ of the point process follows a vanishing power law at the origin (the typical case of a hyperuniform point process), this exponent is defined as the slope near the origin of $\log S$. Our estimator is built upon the (expanding window) asymptotic variance of some wavelet transforms of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator $\widehatα$. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for $α$ when $α< d$ and under Brillinger mixing. This construction is derived from a multivariate central limit theorem where the normalisations are non-standard and vary among the components. We also present a non-asymptotic deviation inequality providing insights into the influence of tapers on the bias-variance trade-off of $\widehatα$. Finally, we investigate the performance of $\widehatα$ through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16797 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimating the hyperuniformity exponent of point processes Mastrilli, Gabriel Błaszczyszyn, Bartłomiej Lavancier, Frédéric Statistics Theory Probability Methodology Primary: 62M15, 62M30, secondary: 62F12, 60F05, 60G55 We address the challenge of estimating the hyperuniformity exponent $α$ of a spatial point process, given only one realization of it. Assuming that the structure factor $S$ of the point process follows a vanishing power law at the origin (the typical case of a hyperuniform point process), this exponent is defined as the slope near the origin of $\log S$. Our estimator is built upon the (expanding window) asymptotic variance of some wavelet transforms of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator $\widehatα$. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for $α$ when $α< d$ and under Brillinger mixing. This construction is derived from a multivariate central limit theorem where the normalisations are non-standard and vary among the components. We also present a non-asymptotic deviation inequality providing insights into the influence of tapers on the bias-variance trade-off of $\widehatα$. Finally, we investigate the performance of $\widehatα$ through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae. |
| title | Estimating the hyperuniformity exponent of point processes |
| topic | Statistics Theory Probability Methodology Primary: 62M15, 62M30, secondary: 62F12, 60F05, 60G55 |
| url | https://arxiv.org/abs/2407.16797 |