Restricted maximum likelihood estimation of covariances in sparse linear models

Abstract

This paper discusses the restricted maximum likelihood (REML) approach for the estimation of covariance matrices in linear stochastic models, as implemented in the current version of the VCE package for covariance component estimation in large animal breeding models. The main features are: 1) the representation of the equations in an augmented form that simplifies the implementation; 2) the parametrization of the covariance matrices by means of their Cholesky factors, thus automatically ensuring their positive definiteness; 3) explicit formulas for the gradients of the REML function for the case of large and sparse model equations with a large number of unknown covariance components and possibly incomplete data, using the sparse inverse to obtain the gradients cheaply; 4) use of model equations that make separate formation of the inverse of the numerator relationship matrix unnecessary. Many large scale breeding problems were solved with the new implementation, among them an example with more than 250 000 normal equations and 55 covariance components, taking 41 h CPU time on a Hewlett Packard 755. © Inra/Elsevier, Paris