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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