Image Processing Reference
In-Depth Information
It is usually assumed that the set A satisfies the following monotonicity condition (Okuno &
Haga, 1969).
Definition 1. Monotonicity
a
A
b
A
b ( b a )
,
(7)
(
) (
)
indicates that if a i =
0 then b i =
=
where
b 1 , b 2 ,..., b n
a 1 , a 2 ,..., a n
0, i
1, 2, . . . , n.
Example 4. Consider A
= {
00000, 10000, 01000, 00100, 00010, 00001, 11000, 10100, 10010
}
.
Since the set A satisfies (7), A is monotonic.
Let y
( x )
denote the response of the experiment with level combination
x
. Assume the model
( x )= μ +
l
x l )+
{
MF α l (
I β l , m (
)+ x
y
x l , x m
,
(8)
l , m
}∈
where
is
the effect of the interaction of the x l -th level of Factor F l and the x m -th level of Factor F m and
x
μ
is the general mean,
α l
(
x l
)
is the effect of the x l -th level of Factor F l ,
β l , m
(
x l , x m
)
2 .
Since the model is expressed through the effect of each factor, it is easy to understand how
each factor affects the response variable. However, because the constraints
q
is a random error with a zero mean and a constant variance
σ
1
ϕ = 0 α l ( ϕ )= 0,
(9)
q
1
ϕ = 0 β l , m ( ϕ , ψ )= 0,
(10)
1
ψ = 0 β l , m ( ϕ , ψ )= 0,
q
(11)
are assumed, the model contains redundant parameters.
Example 5. Consider q
=
3, n
=
5 and A
= {
00000, 10000, 01000, 00100, 00010, 00001, 11000,
10100, 10010
}
. Then,
μ
,
α 1 (
0
)
,
α 1 (
1
)
,
α 1 (
2
)
,
α 2 (
0
)
,
α 2 (
1
)
,
α 2 (
2
)
,
α 3 (
0
)
,
α 3 (
1
)
,
α 3 (
2
)
,
α 4 (
0
)
,
α 4 (
1
)
,
α 4 (
2
)
,
α 5 (
0
)
,
α 5 (
1
)
,
α 5 (
2
)
,
β 1,2 (
0, 0
)
,
β 1,2 (
0, 1
)
,
β 1,2 (
0, 2
)
,
β 1,2 (
1, 0
)
,
β 1,2 (
1, 1
)
,
β 1,2 (
1, 2
)
,
β 1,2 (
2, 0
)
,
β 1,2 (
2, 1
)
,
β 1,2 (
2, 2
)
,,
β 1,3 (
0, 0
)
,
β 1,3 (
0, 1
)
,
β 1,3 (
0, 2
)
,
β 1,3 (
1, 0
)
,
β 1,3 (
1, 1
)
,
β 1,3 (
1, 2
)
,
β 1,3 (
2, 0
)
,
β 1,3 (
2, 1
)
,
β 1,3 (
2, 2
)
,
β 1,4 (
0, 0
)
,
β 1,4 (
0, 1
)
,
β 1,4 (
0, 2
)
,
β 1,4 (
1, 0
)
,
β 1,4 (
1, 1
)
,
β 1,4 (
1, 2
)
,
β 1,4 (
2, 0
)
,
β 1,4 (
2, 1
)
,
)
are parameters. The number of parameters is 43 , but the number of the independent parameters is 23
by the constraints.
β 1,4 (
2, 2
In experimental design, we are presented with a model of an experiment, which consists of a
set A
n . The set X
is referred to as a design. Then, we perform a set of experiments in accordance to the design
X and estimate the effects from the obtained results
n . First, we determine a set of level combinations
⊆{
0, 1
}
x
X , X
GF
(
q
)
.
An important standard for evaluating experimental design is the maximum of the variances of
the unbiased estimators of effects, as calculated from the results of the conducted experiments.
It is known that, for a given number of experiments, this criterion is minimized when using
orthogonal design (Takahashi, 1979). Hence, there has been extensive research focusing on
orthogonal design (Hedayat et al., 1999; Takahashi, 1979; Ukita et al., 2003; 2010a;b; Ukita &
Matsushima, 2011).
{ ( x
, y
( x )) | x
X
}
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