Livestock Breeding and Genomics - Notebook 8
Peter von Rohr
2022-11-25
Problem 1: Inverse Numerator Relationship Matrix
During the lecture the method of computing the inverse numerator relationship matrix A−1 directly was introduced. The computation is based on the LDL-decomposition. As a result, we can write
A−1=(LT)−1⋅D−1⋅L−1 where L−1=I−P, and D−1 is a diagonal matrix with (D−1)ii∗σ−2u=var(mi)−1.
Tasks
- Use the example pedigree given below and compute the matrices L−1 and D−1 to compute A−1
- Verify your result using the function
getAinv()from package pedigreemm.
Pedigree
nr_animal <- 6
tbl_pedigree <- tibble::tibble(Calf = c(1:nr_animal),
Sire = c(NA, NA, NA, 1 ,3, 4),
Dam = c(NA, NA, NA, 2, 2, 5))
tbl_pedigreeCalf <int> | Sire <dbl> | Dam <dbl> | ||
|---|---|---|---|---|
| 1 | NA | NA | ||
| 2 | NA | NA | ||
| 3 | NA | NA | ||
| 4 | 1 | 2 | ||
| 5 | 3 | 2 | ||
| 6 | 4 | 5 |
Your Solution
- Construct matrix P from the pedigree
- Use matrix P to compute the matrix L−1
- Construct the matrix D−1
- Compute A−1 based on L−1 and D−1
Problem 2: Rules
The following diagram helps to illustrate the rules for constructing A−1
Tasks
- Go through the list of animals in the pedigree and write down the contributions that are made to the different elements of matrix A−1
- Based on the different contributions, try to come up with some general rules
Your Solution
Problem 3: Program using the Rules
Write a program in R that implements the rules found in the solution
of Problem 2. Test your program with the pedigree given in Problem 1.
Compare the results that you obtain with the result obtained from the
function pedigreemm::getAinv().
Your Solution
- Because the focus of this problem is the implementation of
Henderson’s rules in a function, we use then function
pedigreemm::Dmat()to obtain the values of the matrix D. - Write a function
get_A_inversewhich takes as input a pedigree and that returns the inverse numerator relationship matrix
Latest Changes: 2022-12-12 06:02:31 (pvr)
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