mbend: an R package for bending non-positive-definite symmetric matrices to positive-definite

Area of Science:

• Quantitative genetics
• Statistical computing
• Bioinformatics

Background:

• Covariance matrices in multi-trait best linear unbiased prediction (BLUP) must be positive-definite (PD).
• Non-PD matrices can arise, necessitating transformation to PD.
• The R package mbend addresses this by implementing matrix bending techniques.

Purpose of the Study:

• To develop and evaluate methods for transforming symmetric non-positive-definite (non-PD) matrices into positive-definite (PD) matrices.
• To implement and compare different matrix bending algorithms within the R package mbend.
• To assess the effectiveness of weighted versus unweighted bending for various matrix types.

Main Methods:

• Implementation of two primary bending methods: LRS14 (unweighted) and HJ03 (weighted).
• Development of a weighted version of LRS14 and an unweighted version of HJ03.
• Testing of an additional unweighted method, DB88, designed for correlation matrices.
• Application of bending procedures to covariance (V), correlation (C), and genomic relationship (G) matrices of varying sizes.

Main Results:

• Weighted bending significantly improved the quality of matrix transformations compared to unweighted methods.
• The HJ03 method with a small positive eigenvalue replacement value (ϵ = 10⁻⁴) demonstrated superior performance for both covariance and correlation matrices.
• LRS14 showed the best performance for the large genomic relationship matrix (G), while HJ03-2 was better for the covariance matrix (V).
• The DB88 method ranked last in unweighted bending tests.

Conclusions:

• The R package mbend offers effective tools for converting non-PD matrices to PD matrices.
• Weighted bending approaches, especially HJ03, enhance the quality of matrix transformations.
• The optimal bending method depends on matrix characteristics such as type, size, and eigenvalue distribution.