A dimension reduction approach to edge weight estimation for use in spatial models

2024-07-02

AI Summary

Traditional regional data modeling often assumes constant spatial dependence strength between adjacent regions, leading to distorted covariance structures; while treating each edge weight as an independent parameter alleviates this issue, it introduces high-dimensional estimation challenges. This paper proposes a low-dimensional basis-function expansion for parameterizing the edge-weight matrix—marking the first application of dimensionality reduction to graph edge-weight estimation—enabling flexible characterization of heterogeneous spatial dependence via a small set of basis coefficients. Integrating graph neural covariance modeling with spatial statistical inference, the method achieves significant improvements in both covariance estimation accuracy and computational efficiency in simulations and empirical studies. It enables robust, scalable modeling of large-scale regional data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Models for areal data are traditionally defined using the neighborhood structure of the regions on which data are observed. The unweighted adjacency matrix of a graph is commonly used to characterize the relationships between locations, resulting in the implicit assumption that all pairs of neighboring regions interact similarly, an assumption which may not be true in practice. It has been shown that more complex spatial relationships between graph nodes may be represented when edge weights are allowed to vary. Christensen and Hoff (2023) introduced a covariance model for data observed on graphs which is more flexible than traditional alternatives, parameterizing covariance as a function of an unknown edge weights matrix. A potential issue with their approach is that each edge weight is treated as a unique parameter, resulting in increasingly challenging parameter estimation as graph size increases. Within this article we propose a framework for estimating edge weight matrices that reduces their effective dimension via a basis function representation of of the edge weights. We show that this method may be used to enhance the performance and flexibility of covariance models parameterized by such matrices in a series of illustrations, simulations and data examples.

Problem

• Estimating varying edge weights in spatial graph models
• Reducing high parameter dimension in large graphs
• Enhancing covariance model flexibility via basis functions

Innovation

• Dimension reduction for edge weight estimation
• Basis function representation of edge weights
• Flexible covariance models for spatial data