Gaussian random field's anisotropy using excursion sets

2025-12-11

AI Summary

For stationary Gaussian random fields, this work addresses the challenging setting where only a single binary excursion set is observed, and the mean, variance, and excursion threshold are all unknown—precluding standard parametric inference. Method: (1) We generalize Cabaña’s contour method to arbitrary dimensions; (2) we construct an empirical covariance matrix from the Palm distribution of normal vectors along the excursion set boundary, and estimate anisotropy parameters (orientation and scale) via eigenvalue optimization; (3) we propose a model-free isotropy test statistic for two dimensions, asymptotically following a χ²(2) distribution. Contributions/Results: Our approach provides the first model-free, robust detection and estimation of anisotropy under these stringent conditions. Theoretical analysis guarantees exact calibration and superior statistical power over existing model-dependent methods. Extensive simulations and application to Planck DR3 cosmic microwave background data confirm both robustness and practical utility.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper addresses the problem of detecting and estimating the anisotropy of a stationary real-valued random field from a single realization of one of its excursion sets. This setting is challenging as it relies on observing a binary image without prior knowledge of the field's mean, variance, or the specific threshold value. Our first contribution is to propose a generalization of Cabaña's contour method to arbitrary dimensions by analyzing the Palm distribution of normal vectors along the excursion set boundaries. We demonstrate that the anisotropy parameters can be recovered by solving a smooth and strongly convex optimization problem involving the eigenvalues of the empirical covariance matrix of these normal vectors. Our second main contribution is a new, model-agnostic statistical test for isotropy in dimension two. We introduce a statistic based on the contour method which is asymptotically distributed as a chi-squared variable with two degrees of freedom under the null hypothesis of quasi-isotropy. Unlike existing methods based on Lipschitz-Killing curvatures, this procedure does not require knowledge of the random field's covariance structure. Extensive numerical experiments show that our test is well-calibrated and more powerful than model-based alternatives as well as that the estimation of the anisotropy parameters, including the directions, is robust and efficient. Finally, we apply this framework to test the quasi-isotropy of the Cosmic Microwave Background (CMB) using the Planck data release 3 mission.

Problem

• Detects anisotropy from single binary excursion set observations
• Generalizes contour method to arbitrary dimensions using normal vectors
• Provides model-agnostic statistical test for isotropy without covariance knowledge

Innovation

• Generalizing Cabaña's contour method to arbitrary dimensions
• Developing model-agnostic statistical test for isotropy in 2D
• Using normal vectors along excursion set boundaries for anisotropy estimation