mbend

mbend transforms symmetric non-positive-definite (non-PD) matrices into positive-definite (PD) matrices to enable analyses that require PD covariance or correlation matrices, such as multi-trait best linear unbiased prediction (BLUP).

Key Features:

• Unweighted bending (LRS14): Replaces negative eigenvalues with small positive values in descending order following the LRS14 approach.
• Precision-based weighted bending (HJ03): Replaces smaller eigenvalues with a custom small positive value (ε) while weighting adjustments by element precision to relax low-precision elements and minimize changes in high-precision elements.
• Weighted LRS14: Implements a weighted version of the LRS14 method combining weighting and the LRS14 eigenvalue replacement strategy.
• Unweighted HJ03: Provides an unweighted variant of HJ03 for cases where precision information is unavailable.
• DB88 (correlation-focused) method: Implements an unweighted DB88 approach that replaces eigenvalues less than ε with 100×ε, originally intended for correlation matrices and tested to perform best with lower ε values.
• Evaluation metrics and test matrices: Computes weighted distance statistics and tests methods on a 5×5 covariance matrix (V), its correlation matrix equivalent (C), and an ill-conditioned 1000×1000 genomic relationship matrix (G).

Scientific Applications:

• Covariance matrices: Ensures covariance matrices are positive-definite for use in multi-trait BLUP and other multivariate analyses.
• Correlation matrices: Restores positive-definiteness to correlation matrices used in statistical inference and modeling.
• Genomic relationship matrices: Corrects ill-conditioned genomic relationship matrices (e.g., 1000×1000 G) to enable genomic prediction and related analyses.

Methodology:

Methods include eigenvalue modification by replacing negative or small eigenvalues with a small positive value (ε) or 100×ε (DB88), application of precision-based weighting to scale eigenvalue adjustments (HJ03), a weighted variant of LRS14, and evaluation via weighted distance statistics on matrices V (5×5 covariance), C (correlation), and G (1000×1000 genomic relationship), with method variants tested at ε values such as 10^-4 and 10^-2.