The shape of partial correlation matrices

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.

Keywords:

• Partial correlations
• concentration matrix
• graphical models
• elliptical tetrahedron

Notes

1 Strictly speaking Lewis and Styan (1981) only proved that ${\tilde{\mathit{R}}}_{1,2}={\tilde{\rho }}_{12}.$ However, by pre- and post-multiplying $\mathbf{\Sigma }$ with permutation matrices U^T and U such that the i-th and j-th row/column respectively become the 1-st and 2-nd row/column we find that ${\tilde{\mathit{R}}}_{i,j}={\tilde{\rho }}_{ij}$ since ${(U^T\mathbf{\Sigma }U)}_{12}^{-1}=U^T{\mathbf{\Sigma }}^{-1}{U}_{12}={\mathbf{\Sigma }}_{ij}^{-1}.$

2 These results were generated by repeatedly generating random partial correlation matrices in d dimensions in R (R Core Team 2018) via cor2pcor(rcorrmatrix(d)).

Additional information

Funding

The research leading to the results reported in this paper was supported in part by the Research Fund of the KU Leuven (C14/19/054).