Mixtures of factor analyzers (MFA) provide a powerful tool for modelling high-dimensional datasets. In recent years, several generalizations of MFA have been developed where the normality assumption of the factors and/or of the errors were relaxed to allow for skewness in the data. However, due to the form of the adopted component densities, the distribution of the factors/errors in most of these models is typically limited to modelling skewness concentrated in a single direction. Here, we introduce a more flexible finite mixture of factor analyzers based on the class of scale mixtures of canonical fundamental skew normal (SMCFUSN) distributions. This very general class of skew distributions can capture various types of skewness and asymmetry in the data. In particular, the proposed mixtures of SMCFUSN factor analyzers (SMCFUSNFA) can simultaneously accommodate multiple directions of skewness. As such, it encapsulates many commonly used models as special and/or limiting cases, such as models of some versions of skew normal and skew t-factor analyzers, and skew hyperbolic factor analyzers. For illustration, we focus on the t-distribution member of the class of SMCFUSN distributions, leading to mixtures of canonical fundamental skew t-factor analyzers (CFUSTFA). Parameter estimation can be carried out by maximum likelihood via an EM-type algorithm. The usefulness and potential of the proposed model are demonstrated using four real datasets.

因子分析器（MFA）的混合物为建模高维数据集提供了强大的工具。近年来，已经对MFA进行了几种概括，其中放宽了因素和/或误差的正态性假设，以允许数据出现偏斜。但是，由于采用的组件密度形式，大多数这些模型中的因子/误差分布通常仅限于建模偏向于单个方向的模型。在这里，我们根据典型的基本偏正态正态分布（SMCFUSN）的比例混合类，介绍一种更加灵活的因子分析器有限混合体。这种非常普遍的偏斜分布类可以捕获数据中各种类型的偏斜和不对称。尤其是，建议的SMCFUSN因子分析仪（SMCFUSNFA）的混合物可以同时适应多个偏斜方向。因此，它封装了许多常用模型作为特殊情况和/或限制情况，例如某些版本的偏斜法线和偏斜模型t因子分析仪和斜双曲因子分析仪。为了说明，我们把重点放在上吨类SMCFUSN分布的-配送构件，导致典型基本歪斜的混合物吨-因子分析仪（CFUSTFA）。参数估计可以通过EM型算法以最大似然来执行。使用四个真实数据集证明了该模型的有效性和潜力。