Compensating for the increase in the sum of eigenvalues and monitoring the bending performance for conditioning covariance matrices in multi-trait livestock evaluations

Abstract

Bending is a method for transforming symmetric non-positive-definite matrices to positive-definite (PD) to guarantee the invertibility of the matrix. Most of the bending approaches are based on eigendecomposition and eigenvalue modification of the matrix. Genetic and residual covariance matrices among traits used in multivariate analyses are among those matrices. Due to computational limitations, variance components for many traits are often estimated for multiple subsets of traits. Collating smaller matrices into a larger matrix may result in a non-PD matrix. Although the estimated covariance matrix from a single variance component estimation procedure is PD, the variance component estimation procedure requires a starting PD matrix. Aiming to modify the existing bending methods to improve bending performance, several tests were performed on a sample non-PD covariance matrix. Replacing negative eigenvalues with small positive values in decreasing order did not improve the bending performance (average absolute deviation between the upper triangle elements of the original matrix and the bent matrix) compared to replacing eigenvalues smaller than a small positive value with that small positive value (ε = 1e−4). Bending increases the sum of eigenvalues. Keeping the sum of eigenvalues constant (equal to the trace of the original matrix) did not improve the bending performance. Bending performance deteriorated when large eigenvalues were reduced to keep the sum of eigenvalues constant. In another attempt, besides increasing eigenvalues smaller than ε to ε, the smallest eigenvalue greater than ε was reduced. Reducing that eigenvalue to a certain level improved the bending performance. Therefore, a controlled reduction of the smallest eigenvalue greater than ε while simultaneously monitoring the improvement in bending performance is recommended.