Definition of pedigree in package pedigreemm

The numerator relationship matrix \(A\)

6 x 6 sparse Matrix of class "dsCMatrix"
     1   2    3     4     5     6
1 1.00 .   .    0.500 .     0.250
2 .    1.0 .    0.500 0.500 0.500
3 .    .   1.00 .     0.500 0.250
4 0.50 0.5 .    1.000 0.250 0.625
5 .    0.5 0.50 0.250 1.000 0.625
6 0.25 0.5 0.25 0.625 0.625 1.125

The cholesky decomposition of \(A\)

     1   2    3         4         5         6
1 1.00 0.0 0.00 0.0000000 0.0000000 0.0000000
2 0.00 1.0 0.00 0.0000000 0.0000000 0.0000000
3 0.00 0.0 1.00 0.0000000 0.0000000 0.0000000
4 0.50 0.5 0.00 0.7071068 0.0000000 0.0000000
5 0.00 0.5 0.50 0.0000000 0.7071068 0.0000000
6 0.25 0.5 0.25 0.3535534 0.3535534 0.7071068

To get to the matrix \(L\), we have to use the definition of \(R = L * S\) and \(S\) is the diagnoal matrix with elements corresponding to the square root of matrix \(D\). Matrix \(D\) can be obtained as

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    0    0  0.0  0.0  0.0
[2,]    0    1    0  0.0  0.0  0.0
[3,]    0    0    1  0.0  0.0  0.0
[4,]    0    0    0  0.5  0.0  0.0
[5,]    0    0    0  0.0  0.5  0.0
[6,]    0    0    0  0.0  0.0  0.5

Matrix \(S\) is obtained from matrix \(D\)

     [,1] [,2] [,3]      [,4]      [,5]      [,6]
[1,]    1    0    0 0.0000000 0.0000000 0.0000000
[2,]    0    1    0 0.0000000 0.0000000 0.0000000
[3,]    0    0    1 0.0000000 0.0000000 0.0000000
[4,]    0    0    0 0.7071068 0.0000000 0.0000000
[5,]    0    0    0 0.0000000 0.7071068 0.0000000
[6,]    0    0    0 0.0000000 0.0000000 0.7071068

The matrix \(L\) is obtained from the defintion of matrix \(R = L * S\), therefore the matrix \(L = R * S^{-1}\)

  [,1] [,2] [,3] [,4] [,5] [,6]
1 1.00  0.0 0.00  0.0  0.0    0
2 0.00  1.0 0.00  0.0  0.0    0
3 0.00  0.0 1.00  0.0  0.0    0
4 0.50  0.5 0.00  1.0  0.0    0
5 0.00  0.5 0.50  0.0  1.0    0
6 0.25  0.5 0.25  0.5  0.5    1

The matrix \(P\) that we used in the simple decomposition, is computed from the relationship \(L^{-1} = I-P\) and from that we get \(P = I - L^{-1}\)

       1   2   3   4   5 6
[1,] 0.0 0.0 0.0 0.0 0.0 0
[2,] 0.0 0.0 0.0 0.0 0.0 0
[3,] 0.0 0.0 0.0 0.0 0.0 0
[4,] 0.5 0.5 0.0 0.0 0.0 0
[5,] 0.0 0.5 0.5 0.0 0.0 0
[6,] 0.0 0.0 0.0 0.5 0.5 0

The inbreeding coefficients \(F_i\) for all animals in the pedigree, can also be obtained with pedigreemm

[1] 0.000 0.000 0.000 0.000 0.000 0.125
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