Page 68 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

⎡ 0.0 0.0 0.0 1.0 0.0 0.0⎤
⎢ ⎥
⎢ 0.0 0.0 0.0 0.0 1.0 0.0 ⎥
W = 0.0 0.0 0.0 0.0 0.0 1.0⎥
⎢
⎢ ⎥
⎢ 0..0 0.0 0.0 0.5 0.5 0.0 ⎥
⎢ ⎣ 0.0 0.0 0.5 0.0 0.0 0.5 ⎥ ⎦

and:
⎡ 0.0 0.0 0.0 0.0 0.0 0.0 ⎤
⎢ ⎥
⎢ 0.0 0.0 0.0 0.0 0.0 0.0 ⎥
⎢ 0.0 0.0 0.005 0.00 0.0 0.005⎥
′
−1
WR W = ⎢ ⎥
⎢ 0.0 0.0 0.0 0.03 0.005 0.0 ⎥
⎢ 0.0 0.0 0.0 0.005 0.03 0.0 ⎥
⎢ ⎥
⎣ ⎢ 0.0 0.0 00.005 0.0 0.0 0.03 ⎥ ⎦
The transpose of the vector of observations, y, is as defined in Section 3.3.1.
−1
−1
−1
−1
The remaining matrices, X′R W, W′R X, X′R y and Z′R y can easily be calcu-
−1
lated through matrix multiplication since X, R , W and y have been set up.
Therefore:
⎡ 0.000 0.000 0.010 0.035 0.010 0.010⎤
′
−1
XR W = ⎢ ⎥
⎣ 0.000 0.000 0.000 0.000 00.025 0.025 ⎦
−1
−1
The matrix W′R X is the transpose of X′R W.
⎡ 0.000⎤
⎢ ⎥
⎢ 0.000 ⎥
⎡0.282 ⎤ −1 ⎢ 0.050⎥
′
−1
XR y = ⎢ ⎥ and W ′R y = ⎢ ⎥
⎣ 0.170 ⎦ ⎢ 0.148 ⎥
⎢ 0..107 ⎥
⎢ ⎥
⎣ ⎢ 0.148⎥ ⎦
The LSE are:
−1
ˆ ⎡ ⎤ ⎡ 0.0665 0.000 0.000 0.000 0.010 0.035 0.010 0.010⎤ ⎡ 0 0.282⎤
1 b
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
ˆ
b ⎢ ⎥ ⎢ 0.000 0.050 0.000 0.000 00.000 0.000 0.025 0.025 ⎥ ⎢ 0.170 ⎥
2
⎢ ⎥ ⎢ 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000⎥ ⎢ 0.000⎥ ⎥
1 ˆ a
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
2 ˆ a ⎢ ⎥ ⎢ 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 ⎥ ⎢ 0.000 ⎥
⎢ ⎥ = ⎢ 0.010 0.000 0.000 0.000 0.005 0.000 0.000 0.005 ⎥ ⎢ 0.050 ⎥
0
3 ˆ a
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ 0.035 0.000 0.000 0.000 0.000 0.030 0..005 0.000 ⎥ ⎢ 0.148 ⎥
4 ˆ a
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
5 ˆ a ⎢ ⎥ ⎢ 0.010 0.025 0.000 0.000 0.000 0.005 0.030 0.000 ⎥ ⎢ 0.107 ⎥
⎢ ⎥ ⎣ ⎢ 0.010 0.0225 0.000 0.000 0.005 0.000 0.000 0.03 ⎥ ⎣ ⎢ 0.148⎥ ⎦
⎦
6 ˆ a ⎣ ⎦
52 Chapter 3