Page 108 - Applied Probability

5. Genetic Identity Coefficients
92
2
2
Next apply the inequality (a + b) ≥ 4ab to prove 4∆ 7 ≤ (4Φ ij ) ;
finally, rearrange.)
3. Calculate all nine condensed identity coefficients for the two inbred
siblings 5 and 6 of Figure 5.1.
4. The Cholesky decomposition of a positive definite matrix Ω is the
t
unique lower triangular matrix L =(l ij ) satisfying Ω = LL and
l ii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n
people numbered so that parents precede their children. The Cholesky
decomposition L of Φ can be defined inductively one row at a time
starting with row 1. Given that rows 1,... ,i − 1 have been defined
and that i has parents r and s, define [4, 10]
0 j> i


1 1
l
2
l ij = 2 rj + l sj j< i
i−1 1
 2
(Φ ii − l ) 2 j = i.
k=1 ik
Prove by induction that L is the Cholesky decomposition of Φ. Why
1
1
1
1
is l ii positive? (Hints: Φ ii > Φ ri + Φ si and Φ ij = Φ rj + Φ sj for
2 2 2 2
j< i.)
5. Explicit diagonalization of the kinship matrix Φ of a pedigree is an
unsolved problem in general. In this problem we consider the special
case of a nuclear family with n siblings. For convenience, number the
parents 1 and 2 and the siblings 3,... ,n+2. Let e i be the vector with
1 in position i and 0 elsewhere. Show that the kinship matrix Φ for the
1
nuclear family has one eigenvector e 1 − e 2 with eigenvalue ; exactly
2
n − 1 orthogonal eigenvectors 1 m−1 e j − e m, 4 ≤ m ≤ n +2,
m−3 j=3
1
with eigenvalue ; and one eigenvector
4
4λ − 2
e 1 + e 2 + (e 3 + ··· + e n+2 )
n
with eigenvalue λ for each of the two solutions of the quadratic equa-
tion
1 n +1 1
2
λ − ( + )λ + =0.
2 4 8
This accounts for n + 2 orthogonal eigenvectors and therefore diago-
nalizes Φ.
6. Continuing Problem 5, we can extract some of the eigenvectors and
eigenvalues of a kinship matrix of a general pedigree [16]. Consider a
set of individuals in the pedigree possessing the same inbreeding coef-
ficient and the same kinship coefficients with other pedigree members.
Typical cases are a set of siblings with no children and a married pair