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Completion problem with partial correlation vines. (English) Zbl 1106.15011

An \(n \times n\) symmetric real matrix with off-diagonal elements in the interval \((-1,1)\) and with 1’s on the main diagonal is called a proto correlation matrix. The paper deals with a completion problem whose goal is to analyze when a partially specified correlation matrix \(A\) has a positive definite completion \(A_c\). The authors show how this completion problem can be solved using partial correlation vine.
A vine \(V\) on \(n\) variables is a nested set of connected trees \(V=\{T_1,\ldots,T_{n-1}\}\) where the edges of tree \(j\) are the nodes of tree \(j+1\), \(j=1,2,\ldots,n-2\). A regular vine on \(n\) variables is a vine in which two edges in tree \(j\) are joined by an edge in tree \(j+1\) only if these edges share a common node, \(j=1,2,\ldots,n-2\). The edges of a regular vine may be associated with partial correlations, with values chosen arbitrarily in the interval \((-1,1)\) in a particular way, obtaining a partial correlation vine.
The authors show that a partial correlation vine represents a factorization of the determinant of the correlation matrix. They prove that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an \(m\)-saturated incomplete regular vine, that is, an incomplete regular vine for which all edges corresponding to membership-descendents of a specified edge are specified.
By using this result the authors find the set of desired completions, and also the completion with maximal determinant for partially specified matrices corresponding to chordal graphs.

MSC:

15A29 Inverse problems in linear algebra
15B48 Positive matrices and their generalizations; cones of matrices
15A15 Determinants, permanents, traces, other special matrix functions
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
62H20 Measures of association (correlation, canonical correlation, etc.)
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References:

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