Page 292 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The solutions obtained are the same as those obtained from Jacobi iteration and by
direct inversion of the coefficient matrix in Example 3.1. In addition, the equations
converged faster than when using Jacobi iteration and no relaxation factor was
applied.
Iterating on the MME could be carried out as described above, once the equa-
tions have been set up, using only the stored non-zero elements of the coefficient
matrix. In practice, it may be necessary to store the non-zero elements and their rows
and columns on disk for large data sets because of the memory requirement, and
these are read in each round of iteration.

17.4 Iterating on the Data

This is the most commonly used methodology in national genetic evaluations, which
usually involve millions of records. Schaeffer and Kennedy first presented this method in
1986. It does not involve setting up the coefficient matrix directly, but it involves setting
up equations for each level of effects in the model as the data and pedigree files are read
and solved using either Gauss–Seidel or Jacobi iteration or a combination of both or a
variation of any of the iterative procedures such as second-order Jacobi. Presented below
are the basic equations for the solutions of various effects under several models and these
form the basis of the iterative process for each of the models.
The equation for the solution of level i for a fixed effect in the model in a
univariate animal situation is Eqn 3.5, which is derived from the MME and can be
generalized as:
m
ni
å y ki å w ˆ ij
-
=
=
ˆ i b = k1 j1 (17.5)
i n
where y is the kth record in level i, m is the total number of levels of other effects
ki
within subclass i of the fixed effect and w is the solution for the jth level, and n is
ˆ
ij i
the number of records in fixed effect subclass i. However, when there are many fixed
effects in the model, the above formula may be used to obtain solutions for the major
fixed effect with many levels such as HYS, while the vector of solutions (f) for other
minor fixed effects with few levels may be calculated as:
ˆ
−1
f = (X′X) X′(y − wˆ − b) (17.6)
−1
where y is the vector of observations, (X′X) is the inverse of the coefficient matrix
ˆ
ˆ
for the minor fixed effects, and w and b are vectors of solutions for effects as defined
in Eqn 17.4. The matrix X′X could be set up in the first round of iteration and stored
in the memory for use in subsequent rounds of iterations.
The solution (uˆ) for the level j (animal j) of the random animal effect in the uni-
variate animal model is calculated using Eqn 3.8, which can be rewritten (replacing
n by k) as:
3
u = [n a(uˆ + uˆ ) + n yd + S{k a(uˆ − 0.5(uˆ ))}]/diag (17.7)
ˆ
j 1 s d 2 o o o mo j
with:
diag = 2(n )a + n + S {(k /2)a}
j 1 2 o o
276 Chapter 17