This work introduces a refinement of the Parsimonious Model for fitting a Gaussian Mixture. The improvement is based on the consideration of clusters of the involved covariance matrices according to a criterion, such as sharing Principal Directions. This and other similarity criteria that arise from the spectral decomposition of a matrix are the bases of the Parsimonious Model. We show that such groupings of covariance matrices can be achieved through simple modifications of the CEM (Classification Expectation Maximization) algorithm. Our approach leads to propose Gaussian Mixture Models for model-based clustering and discriminant analysis, in which covariance matrices are clustered according to a parsimonious criterion, creating intermediate steps between the fourteen widely known parsimonious models. The added versatility not only allows us to obtain models with fewer parameters for fitting the data, but also provides greater interpretability. We show its usefulness for model-based clustering and discriminant analysis, providing algorithms to find approximate solutions verifying suitable size, shape and orientation constraints, and applying them to both simulation and real data examples.

翻译：本文介绍了一种对拟合高斯混合模型的简约模型的改进方法。该改进基于根据某种准则（如共享主方向）对涉及的协方差矩阵进行聚类。此类准则以及其他源于矩阵谱分解的相似性准则构成了简约模型的基础。我们证明，通过对CEM（分类期望最大化）算法进行简单修改即可实现协方差矩阵的聚类。我们的方法基于简约准则对协方差矩阵进行聚类，从而提出用于基于模型的聚类和判别分析的高斯混合模型，并在十四个广为人知的简约模型之间创建了中间步骤。这种增加的灵活性不仅使我们能够获得参数更少的模型来拟合数据，还提供了更强的可解释性。我们展示了该方法在基于模型的聚类和判别分析中的实用性，提供了寻找满足适当尺寸、形状和方向约束的近似解的算法，并将其应用于模拟和实际数据示例。