Graphical Gaussian Process Models for Highly Multivariate Spatial Data.

Area of Science:

• Statistics
• Machine Learning
• Spatial Analysis

Background:

• Multivariate spatial Gaussian process (GP) models often lack conditional independence properties.
• Standard cross-covariance functions face computational challenges (curse of dimensionality) in high-dimensional settings.

Purpose of the Study:

• To propose a novel class of multivariate Gaussian Processes called Graphical Gaussian Processes (GGPs).
• To ensure process-level conditional independence among variables using relational inter-variable graphs.

Main Methods:

• Developed a general construction called "stitching" to create cross-covariance functions from graphs.
• Applied stitching to the Matérn family of functions to create a graphical Matérn GP.
• Ensured process-level conditional independence aligned with the specified graphical model.

Main Results:

• The proposed graphical Matérn GP ensures conditional independence and offers significant computational gains.
• Achieved massive parameter dimension reduction and computational efficiency, especially for decomposable graphical models.
• Demonstrated effectiveness in jointly modeling highly multivariate spatial data.

Conclusions:

• Graphical Gaussian Processes provide an efficient and scalable solution for high-dimensional spatial modeling.
• The "stitching" method effectively integrates graph structures into GP cross-covariance functions.
• The graphical Matérn GP is a powerful tool for complex spatial data analysis, as shown in air-pollution modeling.