ESTIMATION OF EXPECTED EULER CHARACTERISTIC CURVES OF NONSTATIONARY SMOOTH RANDOM FIELDS

• 成果类型: Article
• 署名作者: Telschow, Fabian J. E.; Cheng, Dan; Pranav, Pratyush; Schwartzman, Armin
• 署名单位: Humboldt University of Berlin; Arizona State University; Arizona State University-Tempe; Universite Claude Bernard Lyon 1; Ecole Normale Superieure de Lyon (ENS de LYON); University of California System; University of California San Diego
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2337
• 发表日期: 2023
• 页码: 2272-2297
• 关键词: excursion sets

摘要

The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold approximates the dis-tribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multi-plied by the Lipschitz-Killing curvatures (LKCs) of the generating Gaussian field. This paper proposes consistent estimators of the LKCs as linear projec-tions of pinned Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed, data seldom is Gaussian and the exact mean and variance is unknown, yet the statistic of in-terest often satisfies a CLT with a Gaussian limit process; we adapt our LKC estimators to this scenario using a Gaussian multiplier bootstrap approach. This yields consistent estimates of the LKCs of the possibly nonstationary Gaussian limiting field that have low variance and are computationally effi-cient for complex underlying manifolds. For the EEC of the limiting field, a parametric plug-in estimator is presented, which is more efficient than the nonparametric average of EC curves. The proposed methods are evaluated using simulations of 2D fields, and illustrated on cosmological observations and simulations on the 2-sphere and 3D fMRI volumes.