tr(): trace of the matrix. Sum of diagonals is the same as the sum of the eigen-values.
averageCorr(): The average correlation is computed by summing the off-diagonal values in the correlation matrix. The sum of all elements in a matrix is \(g = \sum_{i,j} C_{i,j} = 1^T C 1 \), where \(1\) is a vector of \(p\) elements with all entries 1. This last term is a quadratic form of the correlation matrix that can be computed efficiently using the SVD and shrinkage parameter from eclairs(). Given the value of \(g\), the average is computed by subtracting the diagonal values and dividing by the number of off-diagonal values: \((g - p) / (p(p-1))\).
averageCorrSq(): The average squared correlation is computed using only the eigen-values. Surprisingly, this is a function of the variance of the eigen-values. The is reviewed by Watanabe (2022) and Durand and Le Roux (2017). Letting \(\lambda_i\) be the \(i^{th}\) sample or shrunk eigen-value, and \(\tilde{\lambda}\) be the mean eigen-value, then \(\sum_i (\lambda_i - \tilde{\lambda})^2 / p(p-1)\tilde{\lambda}^2\).
sumInverseCorr(): The 'effective number of independent features' is computed by summing the entires of the inverse covariance matrix. This has the form \(\sum_{i,j} C^{-1}_{i,j} = 1^T C^{-1} 1\). This last term is a quadratic form of the correlation matrix that can be computed efficiently using the SVD and shrinkage parameter from eclairs() as described above.
effVariance(): Compute a metric of the amount of variation represented by a correlation (or covariance) matrix that is comparable across matrices of difference sizes. Proposed by Peña and Rodriguez (2003), the 'effective variance' is \(|C|^\frac{1}{p}\) where \(C\) is a correlation (or covariance matrix) between \(p\) variables. The effective variance is the mean of the log eigen-values.