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

M.A. Nilforooshan
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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.

补偿特征值和的增加和监测多性状家畜评价中调节协方差矩阵的弯曲性能
弯曲是将对称非正定矩阵转化为正定矩阵以保证矩阵可逆性的一种方法。大多数弯曲方法都是基于矩阵的特征分解和特征值修正。多变量分析中使用的性状间的遗传和残差协方差矩阵就是其中之一。由于计算的限制,许多性状的方差分量通常是对多个性状子集进行估计的。将较小的矩阵整理成较大的矩阵可能导致非pd矩阵。虽然从单方差分量估计过程估计的协方差矩阵是PD,但方差分量估计过程需要一个起始PD矩阵。为了改进现有的弯曲方法,提高弯曲性能,对一个样本非pd协方差矩阵进行了多次试验。与用小于一个小正值(ε = 1e−4)替换小于一个小正值的特征值相比,以递减的顺序用小正值替换负特征值并没有提高弯曲性能(原始矩阵的上三角形元素与弯曲矩阵之间的平均绝对偏差)。弯曲增加了特征值的和。保持特征值和不变(等于原矩阵的轨迹)并不能改善弯曲性能。减小较大的特征值以保持特征值和不变时,弯曲性能会下降。在另一种尝试中,除了将小于ε的特征值增大到ε外,还将大于ε的最小特征值减小。将该特征值减小到一定程度,可以提高弯曲性能。因此,建议在监测弯曲性能改善的同时,控制减小最小特征值大于ε。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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