Biology Reference
In-Depth Information
N
1
V
(
MMMM
3
3
)(
3
3
)
T
Z
i
i
N
i
1
N
1
V
(
MMMM
1
1
)(
3
3
)
T
XZ
,
i
i
N
i
1
N
1
V
(
MMMM
2
2
)(
3
3
)
T
YZ
,
i
i
N
i
1
Strictly speaking, the above formulae provide an estimate of the
variance covariance matrix. For the sake of simplicity of exposition, in
this chapter, we de-emphasize the difference between the true V and its
estimated value.
2.8.2 Further properties of the matrix valued normal distribution
Chapter 3 introduces statistical models to model variability between
different individuals within a population. We call such variability “the
biological variability.”The following concepts and notation are useful to
understand the discussion in Chapter 3 . In particular, we introduce a
special form for the variance-covariance matrix based on the
Kronecker product which is mathematically convenient and biological-
ly reasonable for modelling biological variability. We also introduce the
idea of transformation of a random variable and discuss the impact of
such transformation on the distribution of the random variable.
Kronecker product of two matrices
Let A be an m
n matrix and B be a p
q matrix. Then the Kronecker
product of A and B is defined as:
Then A
B
The resultant matrix is an mp
nq matrix.
For example, let and
.
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