This work introduces a refinement of the Parsimonious Model for fitting a Gaussian Mixture. The improvement is based on the consideration of groupings of the 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. The classification can be achieved with simple modifications of the CEM (Classification Expectation Maximization) algorithm, using in the M step suitable estimation methods known for parsimonious models. This 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。该改进基于根据一定准则（如共享主方向）对协方差矩阵进行分组的考虑。源自矩阵谱分解的这一准则及其他相似性准则构成了简约模型的基础。通过对CEM（分类期望最大化）算法进行简单修改，并在M步骤中使用已知适用于简约模型的适当估计方法，即可实现分类。该方法提出了基于模型聚类和判别分析的高斯混合模型，其中协方差矩阵根据简约准则进行聚类，在十四个广泛已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用更少的参数拟合数据，还提供了更强的可解释性。我们展示了其在基于模型聚类和判别分析中的实用性，提供了在满足适当大小、形状和方向约束条件下寻找近似解的算法，并将其应用于模拟数据和真实数据示例。