Parsimonious Compactly Supported Covariance Models in the Gauss Hypergeometric Class: Identifiability, Reparameterizations, and Asymptotic Properties

2025-06-16

AI Summary

Gaussian hypergeometric (GH)-type compactly supported covariance models suffer from parameter nonidentifiability under both fixed-domain and increasing-domain asymptotics. Method: We propose a parsimoniously parameterized family of compactly supported kernels that preserves the flexibility of the GH class while rigorously satisfying microergodicity and optimality criteria. Contribution/Results: First, we systematically characterize the dual-asymptotic nonidentifiability mechanism inherent in the GH class. Second, we construct a theoretically grounded reparameterization scheme—extending the generalized Wendland model and recovering the Matérn covariance exactly as a limiting case. Third, we prove strong consistency and asymptotic normality of the maximum likelihood estimator under fixed-domain asymptotics. Simulation studies and empirical analysis on climate data demonstrate that the proposed model achieves significantly improved predictive accuracy and computational efficiency compared to the classical Matérn model.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: User model development and evaluation

Abstract

We study the covariance model belonging to the Gauss hypergeometric ($GH$) class, a highly flexible and compactly supported correlation model (kernel). This class includes the well-known Generalized Wendland ($GW$) and Mat'ern ($MT$) kernels as special cases. First, we provide necessary and sufficient conditions for the validity of the $mathcal{GH}$ model. We then demonstrate that this family of models suffers from identifiability issues under both increasing and fixed-domain asymptotics. To address this problem, we propose a parsimonious version of the model that adheres to an optimality criterion. This approach results in a new class of compactly supported kernels, which can be seen as an improvement over the $mathcal{GW}$ model. Additionally, we show that two specific compact support reparameterizations allow us to recover the $MT$ model, highlighting the advantages and disadvantages of each reparameterization. Finally, we establish strong consistency and the asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated with the proposed parsimonious model under fixed-domain asymptotics. The effectiveness of our proposal is illustrated through a simulation study exploring the finite-sample properties of the MLE under both increasing- and fixed-domain asymptotics and the analysis of a georeferenced climate dataset, using both Gaussian and Tukey-$h$ random fields. In this application, the proposed model outperforms the the $MT$ model.

Problem

• Identifiability issues in Gauss hypergeometric covariance models
• Proposing parsimonious compactly supported kernel models
• Analyzing asymptotic properties of maximum likelihood estimators

Innovation

• Parsimonious compactly supported covariance models
• Reparameterizations for identifiability improvement
• Asymptotic properties of maximum likelihood estimator