## Genetic and PE matrices

Much time was spent looking up genetic parameters, either variances and covariances or
correlations. Estimates for many combinations of traits simply do not exist in the
literature, especially health traits. With 220 traits, there are 24,090 possible
covariances. Permanent environmental covariances and variances were made equal to
one half the value of the genetic values. Several people helped to obtain values
for the G matrix including Jarmila Bohmanova, Sven Konig, Jalal Fatehi, and LRS.
Values for lactose, MUN, and Omega 3 fatty acids were made equal or proportional to
those for other production traits.

With covariances and correlations coming from several dozen sources, the resulting
matrices were going to be non positive definite, meaning they would be unsuitable for
use in genetic evaluation or in a simulation program. The Schaeffer PD modification
procedure was used to make the matrices positive definite.

First, the eigenvalues (D), and eigenvectors (U), are calculated, where UU'=I.
There were 21 negative eigenvalues in D out of 220. Let X be the value of the smallest
positive eigenvalue, and Z_{big} be the largest negative eigenvalue. For each negative
eigenvalue compute a new value as X(1 - (Z_{i}*Z_{i})/(Z_{big}*Z_{big} + 1)).
Now all eigenvalues are
positive, they continue to get smaller than the last positive eigenvalue, and they
are all greater than zero. Lastly, reform G_{new} using U(d)U' where (d) is the new
set of eigenvalues. Then G_{new} is positive definite.

The variances in G_{new} are very similar to the old values,
and the covariances are only slightly modified in most cases. G can be
decomposed into the product of two triangular matrices for the simulation.
The same procedure was applied to the PE matrix. Not all traits have PE
effects associated with them, and were ignored for those traits.

## Residual Matrix

For production traits, the residual variances were matrices of
order 4, for the 4 traits of milk, fat, protein, and SCS. All other residual
covariances were assumed to be zero. Many of the estimates found in the literature
were not different from zero.

The availability of new values for these matrices should be
incorporated from time to time. A database should be created so that values can
be added easily, for the existing traits and new traits, and so that the
covariance matrix for a block of traits can be retrieved.