Image Processing Reference
In-Depth Information
3.3 Estimation of effects in experimental design
First, we adopt the following definitions.
=
x C
Y
y
( x )
,
(16)
C | =
q k .
|
where
x C l
( ϕ )=
( x )
Y l
y
,
(17)
( ϕ )
where C l
C }
C l
q k 1 .
( ϕ )= { x |
x l = ϕ
x
|
( ϕ ) | =
,
and
x C l , m ( ϕ , ψ )
Y l , m ( ϕ
,
ψ )=
y
( x )
,
(18)
where C l , m ( ϕ
C }
C l , m ( ϕ
q k 2 .
,
ψ )= { x |
x l
= ϕ
, x m
= ψ
,
x
and
|
,
ψ ) | =
1
q k Y , y l ( ϕ )=
1
q k 1 Y l ( ϕ )
1
q k 2 Y l , m ( ϕ
Let y
=
, y l , m ( ϕ
,
ψ )=
,
ψ )
. Then, the unbiased estimators of
the parameters in (8) are given as
ˆ
μ =
y ,
(19)
α l
ˆ
( ϕ )=
y l
( ϕ )
μ
ˆ
,
(20)
ˆ
β l , m
( ϕ
,
ψ )=
y l , m
( ϕ
,
ψ )
ˆ
α l
( ϕ )
α m
ˆ
( ψ )
μ
ˆ
.
(21)
Example 7. Consider the case that a set A is given by (14) and the result of experiments is given by
Table 1.
x
y
( x )
x
y
( x )
x
y
( x )
00000
93
10000
99
20000
87
00112
97
10112 109
20111
86
00221
98
10221 112
20221
90
01011
90
11011 102
21011
85
01120
96
11120 111
21120
82
01202 102
11202 111
21202
94
02022
97
12022 105
22022
84
02101
95
12101 104
22101
88
02210
95
12210 101
22210
83
Table 1. Result of experiments
First, using (16)-(18),
=
Y 1 (
)=
Y 1 (
)=
Y 1 (
)=
Y
2596,
0
863,
1
954,
2
779,
(
)=
(
)=
(
)=
(
)=
Y 2
0
871,
Y 2
1
873,
Y 2
2
852,
Y 3
0
842,
(
)=
(
)=
(
)=
(
)=
Y 3
1
868,
Y 3
2
886,
Y 4
0
873,
Y 4
1
848,
(
)=
(
)=
(
)=
(
)=
Y 4
2
875,
Y 5
0
847,
Y 5
1
864,
Y 5
2
885,
(
)=
(
)=
(
)=
(
)=
Y 1,2
0, 0
288, Y 1,2
0, 1
288, Y 1,2
0, 2
287, Y 1,2
1, 0
320,
(
)=
(
)=
(
)=
(
)=
Y 1,2
1, 1
324, Y 1,2
1, 2
310, Y 1,2
2, 0
263, Y 1,2
2, 1
261,
(
)=
(
)=
(
)=
(
)=
Y 1,2
2, 2
255, Y 1,3
0, 0
280, Y 1,3
0, 1
288, Y 1,3
0, 2
295,
(
)=
(
)=
(
)=
(
)=
Y 1,3
1, 0
306, Y 1,3
1, 1
324, Y 1,3
1, 2
324, Y 1,3
2, 0
256,
(
)=
(
)=
(
)=
(
)=
Y 1,3
2, 1
256, Y 1,3
2, 2
267, Y 1,4
0, 0
290, Y 1,4
0, 1
282,
(
)=
(
)=
(
)=
(
)=
Y 1,4
0, 2
291, Y 1,4
1, 0
314, Y 1,4
1, 1
312, Y 1,4
1, 2
328,
Y 1,4
(
)=
(
)=
(
)=
2, 0
269, Y 1,4
2, 1
254, Y 1,4
2, 2
256.
 
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