Biomedical Engineering Reference
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(i.e.
g
(p(C| T1))), where it is broken down to the
fuzzy labels L and H, for L;
C
L
) =
1.000
is above the threshold truth value
0.75 employed, so a leaf node is created. For the
path T1 = H the largest subsethood value is less
then 0.75 so it is not a leaf node, instead further
augmentation of this path is considered.
With three condition attributes included in
the example data set, the possible augmentation
to T1
H
is with either T2 or T3. Concentrating on
T2, where with
G
(T1
H
) = 0.347, the ambiguity
with partition evaluated for T2 (
G
(T1
H
and T2|
C)) has to be less than this value, where:
p(C
|
T1
L
) =
,
S
(
T
,
C
)
/
max
S
(
T
,
C
)
L
i
L
j
j
considering C
L
and C
H
with the information in
Table 1
S
(T1
L
, C
L
)
min(
(
u
),
(
u
))
(
u
)
∑
∑
=
T
C
T
L
L
L
u
∈
U
u
∈
U
(min(
0
1
+
min(
0
0
75
)
+
min(
0
75
,
1
+
min(
0
0
25
))
G
(T1
H
and T2| C) =
∑
k
=
.
w
(
T2
|
T1
)
G
(
T1
∩
T2
)
(
0
+
0
+
0
75
+
0
i
H
H
i
i
=
1
(
0
+
0
+
0
75
+
0
=
1
25
=
= 1.000,
Starting with the weight values, in the case of
T1
H
and T2
L
, it follows:
1
25
1
25
whereas,
w
(T2
L
| T1
H
) =
S
(T1
L
, C
H
)
k
∑
min(
(
u
),
(
u
))
min(
(
u
),
(
u
))
∑
∑
T2
T1
T2
T1
L
H
j
H
u
∈
U
j
=
1
u
∈
U
(min(
0
1
+
min(
0
0
25
)
+
min(
0
75
,
0
+
min(
0
0
75
))
=
(
0
+
0
+
0
75
+
0
where
=
(
0
+
0
25
+
0
+
0
=
0
25
= 0.200.
1
25
1
25
∈
U
min(
(
u
),
(
u
))
= 1.792
T2
T1
H
L
u
Hence p = {1.000, 0.200}, giving the ordered
normalized form of p
*
= {1.000, 0.200}, with
and
k
∑
∑
min(
(
u
),
(
u
))
3
=
0
, then;
= 2.917,
∗
T2
T1
j
H
j
=
1
u
∈
U
2
(
∗
−
∗
+
)
ln[
i
]
G
(T1
L
) =
g
(p(C| T1
L
)) =
∑
so
w
(T2
L
| T1
H
) = 1.792/2.917 = 0.614. Similarly
w
(T2
H
| T1
H
) = 0.386, hence
i
i
1
i
=
1
=
= 0.139,
(
1.000
−
0.200
)
ln[
1
+
(
0.200
−
0
000
)
ln[
2
G
(T1
H
and T2| C)
along with
G
(T1
H
) = 0.347, then
G
(T1) = (0.139 +
0.347)/2 = 0.243. Compared with
G
(T2) = 0.294
and
G
(T2) = 0.338, the condition attribute T1, with
the least classification ambiguity, forms the root
node for the desired fuzzy decision tree.
The subsethood values in this case are; for T1:
S
(T1
L
, C
L
) =
1.000
and
S
(T1
L
, C
H
) = 0.200, and
S
(T2
H
, C
L
) =
0.727
and
S
(T2
H
, C
H
) = 0.363. For
T2
L
and T2
H
, the larger subsethood value (in bold),
defines the possible classification for that path.
In the case T1 = L, the subsethood value
S
(T1
L
,
= 0.614 ×
G
(T1
H
∩ T2
L
) + 0.386 ×
G
(T1
H
∩ T2
H
)
= 0.614 × 0.499 + 0. 86 × 0.154
= 0.366,
A concomitant value for
G
(T1
H
and T3| C) =
0.250, the lower of these (
G
(T1
H
and T3| C)) is
lower than the concomitant
G
(T1
H
) = 0.347, so less
ambiguity would be found if the T3 attribute was
augmented to the path T1 = H. The subsequent
subsethood values in this case for each new path
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