Image Processing Reference
In-Depth Information
4.4 Analysis of variance in experimental design
On the other hand, it is already shown that the analysis of variance can also be performed in
the model of experimental design on the basis of an orthonormal system (Ukita et al., 2010b).
We present three theorems with respect to the sum of squares needed in analysis of variance.
Theorem 5.
Let SS
Mean
be the sum of squares due to the mean in Sect.3.4, and let f
0...0
be the unbiased
estimator of the Fourier coefficient f
0...0
in the model of Sect.4.1. Then,
q
k
f
0...0
|
2
|
=
SS
Mean
,
(35)
where
1
q
k
f
0...0
∑
x
∈
C
⊥
(
x
)
X
0...0
(
x
)
=
y
.
(36)
f
0...0
a
l
0...0
be the
Theorem 6.
Let SS
F
l
be the sum of squares due to the effect of F
l
in Sect.3.4, and let
unbiased estimator of the Fourier coefficient f
0...0
a
l
0...0
in the model of Sect.4.1. Then,
∑
a
l
∈
q
k
f
0...0
a
l
0...0
2
|
|
=
=
···
SS
F
l
,
l
1, 2,
,
n
,
(37)
(
)
GF
q
a
l
=
0
where
1
q
k
f
0...0
a
l
0...0
∑
x
∈
C
⊥
(
x
)
X
0...0
a
l
0...0
(
x
)
=
y
.
(38)
Theorem 7.
Let SS
F
l
×
F
m
be the sum of squares due to the interaction effect of F
l
×
F
m
in Sect.3.4, and
let f
0...0
a
l
0...0
a
m
0...0
be the unbiased estimator of the Fourier coefficient f
0...0
a
l
0...0
a
m
0...0
in the model of
Sect.4.1. Then,
a
l
=
0
∑
f
0...0
a
l
0...0
a
m
0...0
|
q
k
2
|
=
SS
F
l
×
F
m
,
a
m
=
0
l
,
m
=
1, 2,
···
,
n
,
(
l
m
)
,
(39)
<
∈
(
)
where the sums are taken over a
l
,
a
m
GF
q
and
1
q
k
f
0...0
a
l
0...0
a
m
0...0
∑
x
∈
C
⊥
(
x
)
X
0...0
a
l
0...0
a
m
0...0
(
x
)
=
y
.
(40)
By these theorems,
SS
Mean
,
SS
F
l
and
SS
F
l
×
F
m
can be obtained in the proposed description of
experimental design. In addition, using the Parseval-Plancherel formula and these theorems,
SS
Error
can be computed as follows.
=
∑
x
∈
C
⊥
y
2
−
∑
l
∑
(
x
)
−
SS
F
l
−
SS
Error
SS
Mean
SS
F
l
×
F
m
.
(41)
∈
MF
{
l
,
m
}∈
IF
