On the validity of parametric block correlation matrices with constant within and between group correlations

Abstract

We consider the set B p of parametric block correlation matrices with p blocks of various (and possibly different) sizes, whose diagonal blocks are compound symmetry (CS) correlation matrices and off-diagonal blocks are constant matrices. Such matrices appear in probabilistic models on categorical data, when the levels are partitioned in p groups, assuming a constant correlation within a group and a constant correlation for each pair of groups. We obtain two necessary and sufficient conditions for positive definiteness of elements of B p . Firstly we consider the block average map φ, consisting in replacing a block by its mean value. We prove that for any A ∈ B p , A 0 if and only if φ(A) 0. Hence it is equivalent to check the validity of the covariance matrix of group means, which only depends on the number of groups and not on their sizes. This theorem can be extended to a wider set of block matrices. Secondly, we consider the subset B bg-iso p of B p for which the between group correlation is the same for all pairs of groups. Positive definiteness then comes down to find the positive definite interval of a matrix pencil on S p . We obtain a simple characterization by localizing the roots of the determinant with within group correlation values.

Matrix compression along isogenic blocks

“…As shown in [24,25], positive definiteness of A is equivalent to the positivity of the much smaller compressed matrix Σ ↓ π (A), where π is the partition of {1, . .…”

“…A popular approach for constructing probabilistic models involving categorical data consists of grouping the input levels in such a way that correlation is constant within each group and across groups [2,8,17,24]. In particular, the rapidly developing area of group kernels exploits this idea to obtain useful Gaussian processes for categorical variables [22,25].…”

Matrix compression along isogenic blocks

“…As shown in [23,24], positive definiteness of A is equivalent to the positivity of the much smaller compressed matrix Σ ↓ π (A), where π is the partition of {1, . .…”

“…Block correlation matrices and group kernels. A popular approach for constructing probabilistic models involving categorical data consists of grouping the input levels in such a way that correlation is constant within each group and across groups [2,7,16,23]. In particular, the rapidly developing area of group kernels exploits this idea to obtain useful Gaussian processes for categorical variables [21,24].…”

“…[23, Theorem 1]). Let B = (B ij ) m i,j=1 be an arbitrary block matrix with B ij ∈ R n i ×n j for i, j = 1, .…”