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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