Biomedical Engineering Reference
In-Depth Information
Appendix C
Supplementary Mathematical Arguments
C.1
Multi-dimensional Gaussian Distribution
Let us define a column vector of
N
random variables as
x
. The multi-dimensional
Gaussian distribution for
x
is expressed as
2
exp
1
1
2
(
T
ʣ
−
1
p
(
x
)
=
−
x
−
μ
)
(
x
−
μ
)
.
(C.1)
(
2
ˀ)
N
/
2
|
ʣ
|
1
/
Here
μ
is the mean of
x
defined as
μ
=
E
(
x
)
and
ʣ
is the covariance matrix of
x
E
(
T
where
E
ʣ
=
−
μ
)(
−
μ
)
(
·
)
defined as
x
x
is the expectation operator. Also,
|
ʣ
|
ʣ
indicates the determinant of
. The Gaussian distribution in Eq. (
C.1
)isoften
written as
p
(
x
)
=
N(
x
|
μ
,
ʣ
).
When the random variable
x
follows a Gaussian distribution with mean
μ
and the
covariance matrix
ʣ
, the expression
x
∼
N(
x
|
μ
,
ʣ
)
(C.2)
is often used.
Two vector random variables
x
1
and
x
2
have the linear relationship,
x
2
=
Ax
1
+
c
,
where
A
is a matrix of deterministic variables, and
c
is a vector of deterministic
variables. Then, if we have
x
1
∼
N(
x
1
|
μ
,
ʣ
),
we also have
A
T
x
2
∼
N(
x
2
|
A
μ
+
c
,
A
ʣ
).
(C.3)
