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PDMATRIX

Programs to make matrices positive definite

• Quantitative genetic theory for continuous traits assumes a multivariate normal distribution with (semi-) positive definite covariance or correlation matrix, i.e. a matrix which has non-negative eigenvalues.
• However, estimates of these matrices frequently have negative eigenvalues.
• 'Bending' and related techniques which modify the eigenvalues of such matrices are widely used to 'correct' covariance matrices out of bounds of the parameter space.
• However, estimates of individual covariances or correlations often originate from different analyses and can be based on greatly differing amounts of information.
• 'Bending' etc. do not allow for differential weighing of individual estimates.

Purpose

PDMATRIX is a collection of - currently two - programs to help make invalid, i.e. non-positive definite covariance matrices positive (semi-) definite, allowing individual estimates to be given differential emphasis. The programs are :

1. FLBEND written by John Henshall ( John.Henshall@removeme.csiro.au)
   FLBEND modifies a non-positive definite matrix comprised of pair wise or block covariance estimates, where estimates may or may not be based on experimental results.
   FLBEND finds a positive definite covariance matrix at 'minimum distance' from a non-positive definite matrix, from of pair- or or block-wise covariance estimates. The distance function is the weighted Euclidean norm, i.e. the elementwise sum of squared differences, weighted by standard error estimates. Finding the positive definite matrix is a constrained minimisation problem with the boundary of the constraint not a simple function. A reparameterisation is used to simplify the minimisation problem.
   FLBEND has been used to obtain covariance matrices for use in Australian sheep genetic evaluations. Variance components for groups of 42 traits were combined into genetic and residual matrices which were close to, but not positive definite. Using FLBEND, positive definite matrices were obtained in which the largest deviations from the composite matrices were for those correlations with highest standard errors.
2. ITSUMCOV written by Karin Meyer (kmeyer.agbu@removeme.gmail.com)
   ITSUMCOV deals with specific case of multiple, multivariate analyses of different subsets of a number of traits. Say we have q traits in total, and S analyses involving k traits each.
   ITSUMCOV implements the 'iterative summing of expanded part matrices' approach of Mantysaari (1999; 50th Annual Meeting Europ. Ass. Anim. Prod.), modified to allow for differential weights for different analyses.
   ITSUMCOV has been used to combine estimates from 231 bivariate analyses for 22 traits recorded for Hereford cattle, resulting in an estimate of the complete genetic covariance matrix for traits in BREEDPLAN, the Australian genetic evaluation scheme for beef cattle.
   New: Pooling of estimates from part analyses using a penalized likelihood approach - provides an alternative with better properties; see: http://didgeridoo.une.edu.au/km/pool.php

Computing environment

Programs are written in standard Fortran 95 and self-contained, except for some public domain random number generation routines used in FLBEND. Compiled versions are available for Linux and Compaq Alpha stations.

Availability

Programs are available free of charge to the scientific community under the conditions that they remains our copyright, that they not modified other than to adapt it to the local computing environment or for personal research, that it is used for scientific purposes only, and that their use is credited in any publications.

Reference

Henshall, J.M. and Meyer, K. (2002). "PDMATRIX - Programs to make matrices positive definite". Seventh World Congress on Genetics Applied to Livestock Production, Montpellier, France, August 19-23, 2002, Communication No. 28-12.

View summary - download manuscript (2 pages, pdf file)

© 2007 K. Meyer · Last changed: July 03, 2018 · Design by Andreas Viklund