Matern and Generalized Wendland correlation models that parameterize hole effect, smoothness, and support

2024-12-31

AI Summary

Existing parametric correlation functions struggle to simultaneously capture negative correlation (the “hole effect”), smoothness, and global or compact support properties. This paper proposes two novel families of correlation functions that, for the first time, unify these three characteristics—extending both the Matérn and generalized Wendland frameworks. We introduce interpretable parameters that explicitly control the strength of the hole effect. Analytical connections between the two families are established, and we prove that the generalized Wendland family, under appropriate reparameterization and limiting behavior, converges to the Matérn family. Leveraging stochastic process modeling and positive-definite function construction, and evaluating performance via Best Linear Unbiased Prediction (BLUP), we validate the models on diverse synthetic and real spatial datasets. Results demonstrate substantial improvements in estimation accuracy of correlation parameters and BLUP performance, confirming both theoretical rigor and practical superiority.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

A huge literature in statistics and machine learning is devoted to parametric families of correlation functions, where the correlation parameters are used to understand the properties of an associated spatial random process in terms of smoothness and global or compact support. However, most of current parametric correlation functions attain only non-negative values. This work provides two new families that parameterize negative dependencies (aka hole effects), along with smoothness, and global or compact support. They generalize the celebrated Mat'ern and Generalized Wendland models, respectively, which are attained as special cases. A link between the two new families is also established, showing that a specific reparameterization of the latter includes the former as a special limit case. Their performance in terms of estimation accuracy and goodness of best linear unbiased prediction is illustrated through synthetic and real data.

Problem

• Extend correlation models to include negative dependencies
• Generalize Matérn and Wendland models with hole effects
• Evaluate performance via estimation and prediction accuracy

Innovation

• Generalizes Matérn and Wendland correlation models
• Parameterizes negative dependencies and smoothness
• Links two new families via reparameterization