Graphics Programs Reference
In-Depth Information
X
t
=
X
1
X
2
…
X
m
(13.81)
where the superscript indicates the transpose operation. The joint
pdf
for the
vector
X
is
f
x
()
f
x
1
x
2
…
x
m
=
(
x
1
,
x
2
…
x
m
,
,
)
(13.82)
,
,
,
The mean vector is defined as
EX
[]
EX
[]…
EX
[]
t
µ
x
=
(13.83)
and the covariance is an
mm
×
matrix given by
E
X X
t
µ
t
C
x
=
[
]
x
(13.84)
Note that if the elements of the vector
X
are independent, then the covariance
matrix is a diagonal matrix.
By definition a random vector
X
is multivariate Gaussian if its
pdf
has the
form
1
---
x
µ
x
1
2(
m
⁄
2
12
⁄
)
t
C
x
1
f
x
()
=
[
C
x
]
exp
(
(
x
µ
x
)
(13.85)
C
x
1
where
µ
x
is the mean vector,
C
x
is the covariance matrix,
is inverse of
the covariance matrix and
C
x
is its determinant, and
X
is of dimension
m
. If
A
is a
km
×
matrix of rank
k
, then the random vector
Y
=
X
is a k-variate
Gaussian vector with
µ
y
=
A
µ
x
(13.86)
A
C
x
A
t
C
y
=
(13.87)
The characteristic function for a multivariate Gaussian
pdf
is defined by
C
X
=
E
[
exp
{
j
ω
1
X
1
(
+
ω
2
X
2
…ω
m
X
m
+
+
)
}
]
=
(13.88)
1
j
µ
t
ω
---
ω
t
C
x
ω
exp
Then the moments for the joint distribution can be obtained by partial differen-
tiation. For example,
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