Spatial Covariance Constraints for Gaussian Mixture Models

TL;DR

The paper tackles the challenge of clustering high-dimensional spatial data with Gaussian mixtures by introducing SpatGMM, a finite mixture model that imposes a spatially constrained covariance via a sigmoid-decay structure. This approach yields a parsimonious per-component covariance with a fixed, small number of parameters, enabling scalable clustering of tensor-valued spatial data while preserving spatial dependencies. Parameter estimation is achieved through a novel EM variant that incorporates generalized least squares (GLS), including E-step, three M-steps (updates for mixing proportions and means, GLS-based spatial parameters, and the sigmoid shape), and optimization of the sigmoid parameter. Through simulation studies and a Raman spectroscopy application, SpatGMM demonstrates superior clustering accuracy and effective model selection compared to competing methods, underscoring its practical value for spatially structured, multi-way data analysis.

Abstract

Although extensive research exists in spatial modeling, few studies have addressed finite mixture model-based clustering methods for spatial data. Finite mixture models, especially Gaussian mixture models, particularly suffer from high dimensionality due to the number of free covariance parameters. This study introduces a spatial covariance constraint for Gaussian mixture models that requires only four free parameters for each component, independent of dimensionality. Using a coordinate system, the spatially constrained Gaussian mixture model enables clustering of multi-way spatial data and inference of spatial patterns. The parameter estimation is conducted by combining the expectation-maximization (EM) algorithm with the generalized least squares (GLS) estimator. Simulation studies and applications to Raman spectroscopy data are provided to demonstrate the proposed model.

Figures (7)

• Figure 1: Decay curves for the quadratic (left) and sigmoid (right) models.
• Figure 2: Simulated spatial maps from Gaussian noise (left), quadratic decay (middle), and sigmoid decay (right) models.
• Figure 3: Visualization of the first observed tensor from each of the three groups in simulation design I.
• Figure 4: Raman spectra of dosimetric film at two different dosage levels.
• Figure 5: Schematic representation of the RS sampling protocol. (i) Macroscopic view of the irradiated EBT-3 film ($2.5 \times 2.5$ cm). (ii) The ROI comprises an array of 400 separate sub-ROIs (grids) arranged in a $20 \times 20$ pattern. (iii) Microscopic view of a sub-ROI, illustrating the $10 \times 10$ point-scan geometry.