Factor analysis is a well-known method for describing the covariance structure among a set of manifest variables through a limited number of unobserved factors. When the observed variables are collected at various occasions on the same statistical units, the data have a three-way structure and standard factor analysis may fail. To overcome these limitations, three-way models, such as the Parafac model, can be adopted. It is often seen as an extension of principal component analysis able to discover unique latent components. The structural version, i.e., as a reparameterization of the covariance matrix, has been also formulated but rarely investigated. In this article, such a formulation is studied by discussing under what conditions factor uniqueness is preserved. It is shown that, under mild conditions, such a property holds even if the specific factors are assumed to be within-variable, or within-occasion, correlated and the model is modified to become scale invariant.

中文翻译：

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结构 Parafac 模型的因子唯一性

因子分析是一种众所周知的方法，用于通过有限数量的未观察因子来描述一组明显变量之间的协方差结构。当观察变量在不同场合收集在同一统计单元上时，数据具有三向结构，标准因子分析可能会失败。为了克服这些限制，可以采用三向模型，例如 Parafac 模型。它通常被视为能够发现独特潜在成分的主成分分析的扩展。结构版本，即作为协方差矩阵的重新参数化，也已被制定，但很少被研究。在本文中，通过讨论在什么条件下保留因子唯一性来研究这样的公式。结果表明，在温和条件下，