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.

翻译：尽管空间建模领域已有广泛研究，但针对空间数据的有限混合模型聚类方法却鲜有探讨。有限混合模型，特别是高斯混合模型，常因自由协方差参数数量过多而受高维问题困扰。本研究为高斯混合模型引入了一种空间协方差约束，该约束使每个组分仅需四个自由参数，且与数据维度无关。通过建立坐标系，空间约束高斯混合模型能够实现多维度空间数据的聚类与空间模式推断。参数估计通过将期望最大化算法与广义最小二乘估计器相结合来完成。本文通过模拟研究及拉曼光谱数据的应用实例，验证了所提出模型的有效性。