Page 149 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The modelling of the fixed lactation curve by means of Legendre polynomials
implies the need to compute F, which is the matrix of Legendre polynomials evalu-
ated at the different DIM. The matrix F is of order t (the number of DIM) by k
(where k is the order of fit) with element f = f (a ), which is the jth Legendre poly-
ij j t
nomial evaluated at the standardised DIM t (a ). Therefore F = ML, where M is the
t
matrix containing the polynomials of the standardized DIM values and L is a matrix
of order k containing the coefficients of Legendre polynomials. The calculation of F
is outlined in Appendix G and matrix F for Example 9.1 is shown in Eqn g.1.

SETTING UP THE INCIDENCE MATRICES FOR THE MME
In Eqn 9.1, let Xb = X b + X b , then in Example 9.1, the matrix X , which relates
1 1 2 2 1
records to HTD effects, is of order n (number of TD records) and is too large to be
td
presented. However, X′X is diagonal and is:
1 1
X′ X = diagonal [3, 3, 3, 4, 4, 5, 5, 5, 5, 5]
1 1
The matrix X of order n by nf contains Legendre polynomials (covariables)
2 td
corresponding to the DIM of the ith TD yield. Thus the ith row of X contains
2
elements of the row of F corresponding to the DIM for the ith record. The matrix
X , with rows for the first three TD records of cow 4 and the last three TD records
2
of cow 8 is:
⎡ 0 7071 − 1 2247 1 5811 − 1 8704 2 1213⎤
.
.
.
.
.
⎢ 0 7071 − 0 6441 − − ⎥
.
.
.
⎢ . 0 9525 . 0 0176 −0 6205 ⎥
⎢ 0 7071 −0 6804 −0 0586 0 7573 −0 7757⎥
.
.
.
.
.
⎢ ⎥
⎢ ⎥
⎢ 0 7071 0 6804 − −0 0586 −0 7573 −0 7757 ⎥
.
.
.
.
.
⎢ ⎥
⎢ 0 7071 0 9525 0 6441 −0 0176 −0 6205 ⎥
.
.
.
.
.
⎢ ⎥
⎣ 070771 1 2247 1 5811 1 8704 2 1213 ⎦
.
.
.
.
.
and X′ X is:
2 2
é 20.9996 - 4.4261 4.0568 - 0.8441 8.7149ù
ê ú
ê - 4.4261 24.6271 - 4.77012 11.1628 - 3.0641 ú
X′ X = ê ê 4.0568 - 4.7012 31.0621 - 6.6603 19.0867ú ú
2
2
ê - 0.84441 11.1628 - 6.6603 38.6470 - 8.8550 ú
ê ë 8.7149 - 3.0641 19.0867 - 8.8550 448.2930 ú û
Considering only animals with records, Q = Z and is a matrix of order 5 (number
of animals) by n . The matrix Q′ could be represented as:
td
é q′ 4 0 0 0 0ù
ê ú
ê 0 q′ 5 0 0 0 ú
Q′ = ê 0 0 q ′ 6 0 0ú
ê ú
ê 0 0 0 q′ 7 0 ú
ê ú
ë 0 0 0 0 q ′ 8û
Analysis of Longitudinal Data 133