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Abstract

This thesis presents a tensor-variate spatially constrained Gaussian mixture model that can be employed to perform model-based clustering while estimating spatial patterns in the data. As data becomes increasingly abundant and complex, the demand for effective clustering methodologies grows. Unsupervised learning techniques, particularly clustering, seek to uncover hidden patterns without labeled examples. Model-based clustering offers a mathematically rigorous approach, assigning probabilities for cluster membership, but it struggles with the curse of dimensionality. Our proposed model effectively captures positive spatial correlation in tensor-variate data that takes advantage of a spatial coordinate system through a linear covariance structure with sigmoid decay. By applying an appropriate decomposition, this highly constrained covariance structure offers an efficient way to model spatial information while maintaining a constant number of free parameters. Additionally, the factor analyzers model is utilized to model the dependence among different spatial systems for further dimensionality reduction. We present both simulation studies and applications using Raman spectroscopy data to demonstrate the model.