One of the challenges in cluster analysis is the evaluation of the obtained clustering results without using auxiliary information. To this end, a common approach is to use internal validity criteria. For mixtures of linear regressions whose parameters are estimated by maximum likelihood, we propose a three-term decomposition of the total sum of squares as a starting point to define some internal validity criteria. In particular, three types of mixtures of regressions are considered: with fixed covariates, with concomitant variables, and with random covariates. A ternary diagram is also suggested for easier joint interpretation of the three terms of the proposed decomposition. Furthermore, local and overall coefficients of determination are respectively defined to judge how well the model fits the data group-by-group but also taken as a whole. Artificial data are considered to find out more about the proposed decomposition, including violations of the model assumptions. Finally, an application to real data illustrates the use and the usefulness of these proposals.

中文翻译：

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通过总平方和分解对回归混合进行聚类验证

聚类分析的挑战之一是在不使用辅助信息的情况下评估获得的聚类结果。为此，一种常见的方法是使用内部有效性标准。对于参数通过最大似然估计的线性回归的混合，我们建议将总平方和的三项分解作为定义一些内部有效性标准的起点。特别地，考虑了三种类型的回归混合：具有固定协变量、具有伴随变量和具有随机协变量。还建议使用三元图，以便更容易地联合解释所提议分解的三个术语。此外，分别定义局部决定系数和整体决定系数，以判断模型对数据的逐组拟合程度以及整体的拟合程度。人工数据被认为是为了找出更多关于建议分解的信息，包括违反模型假设。最后，对真实数据的应用说明了这些建议的用途和有用性。