Page 310 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Computing Cd then implies calculating:
N
N
å wR w d +′ i V d = å v + v d (17.13)
-1
-1
i
i
i
=
=
i 1 i 1
If solving the MME with iteration on the data for a univariate model without any
regression effects, this calculation can be achieved by accumulating for each individ-
−1
ual i, the product v = T d, where the coefficients in T = w R w ′ can be deduced
i i i i i i
without performing any of the products, as w contains zeros and ones only and R −1
i i
−1
is a scalar or R is factored out (Eqn 3.4). For a multivariate model, the principles
i
for computing T are essentially the same but with scalar contributions replaced by
i
matrix R . Strandén and Lidauer (1999) suggested the following three-step method
i
−1
for calculating the product w R d:
i i
*
−1
s ¬ w ′d; s ¬ R s ; v ¬ w s *
i i i i i i i i
*
where vectors s and s are of size equal to the number of traits observed on individual
i i
i (l ). They demonstrated that this three-step approach reduced substantially the num-
i
ber of floating point operations (multiplications) compared with a multivariate accu-
mulation technique as used by Groeneveld and Kovac (1990). For instance, given that
q is the number of effects over traits observed for individual i, the number of floating
i
point operations were 720 with l = 3 and q = 15 using the multivariate accumulation
i i
technique compared with 78 with the three-point approach. They also suggested that
−1
v = V d in Eqn 17.13 can be evaluated in a two-step approach:
d
−1
x ¬ (I Ä A )d; v ¬ (G −1 Ä I)x
d
17.5.2 Numerical application
Example 17.6
The application of PCG to solve MME is illustrated using data for Example 3.1 for
a univariate model and iterating on the data.
COMPUTING STARTING VALUES
−1
Initially, the pedigree is read and diagonal elements of A multiplied by a are
accumulated for animals, where the variance ratio a is 2, as in Example 3.1. This is
straightforward and has not been illustrated, but elements for animals 1 to 8 stored
in a vector h are:
h′ = [3.667 4.0 4.0 3.667 5.0 5.0 4.0 4.0]
Second, read through the data as shown in Table 3.1 and accumulate right-hand
side (r) for all effects, diagonals for the levels of sex of calf effect and add contribution
of information from data to diagonals from A a for animals. Assuming that diago-
−1
nals for all effects are stored as diagonal elements of M, such that the first two
elements are for the two levels of sex of calf effect and the remaining elements for
animals 1 to 8, then r and M are:
r′ = [13.0 6.8 0.0 0.0 0.0 4.5 2.9 3.9 3.5 5.0]

294 Chapter 17