Abstract

The correlational structure of a set of variables is often conveniently described by the pairwise partial correlations as they contain the same information as the Pearson correlations with the advantage of straightforward identifications of conditional linear independence. For mathematical convenience, multiple matrix representations of the pairwise partial correlations exist in the literature but their properties have not been investigated thoroughly. In this paper, we derive necessary and sufficient conditions for the eigenvalues of differently defined partial correlation matrices so that the correlation structure is a valid one. Equipped with these conditions, we will then emphasize the intricacies of algorithmic generations of correlation structures via partial correlation matrices. Furthermore, we examine the space of valid partial correlation structures and juxtapose it with the space of valid Pearson correlation structures. As these spaces turn out to be equal in volume for every dimension and equivalent with respect to rotation, a simple formula allows the creation of valid partial correlation matrices by the use of current algorithms for the generation and approximation of correlation matrices. Lastly, we derive simple conditions on the partial correlations for frequently assumed sparse structures.

摘要

一组变量的相关结构通常由成对偏相关方便地描述，因为它们包含与 Pearson 相关相同的信息，具有直接识别条件线性独立性的优点。为了数学上的方便，文献中存在成对偏相关的多个矩阵表示，但它们的性质尚未得到彻底研究。在本文中，我们推导了不同定义的偏相关矩阵的特征值的充要条件，使得相关结构是有效的。具备这些条件后，我们将通过偏相关矩阵强调相关结构的算法生成的复杂性。此外，我们检查有效偏相关结构的空间，并将其与有效 Pearson 相关结构的空间并列。由于这些空间在每个维度上的体积都是相等的，并且在旋转方面是相等的，所以一个简单的公式允许通过使用当前算法来生成和逼近相关矩阵来创建有效的偏相关矩阵。最后，我们推导出经常假设的稀疏结构的偏相关的简单条件。一个简单的公式允许通过使用当前算法生成和近似相关矩阵来创建有效的偏相关矩阵。最后，我们推导出经常假设的稀疏结构的偏相关的简单条件。一个简单的公式允许通过使用当前算法生成和近似相关矩阵来创建有效的偏相关矩阵。最后，我们推导出经常假设的稀疏结构的偏相关的简单条件。