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our approach, we assign a fuzzy subset of
V
m
as the class label of a node,
whereas in [2] this subset is further defuzzified into a crisp subset of
V
m
.
The other difference is the stopping condition. In our approach, the stopping
condition is based on
ldc
s
, whereas in [2] a criterion based purely on
µ
s
is
given. In [2], with a user-specified parameter
σ
, the set
V
m
is partitioned into
large
s
and
small
s
in a node
s
,where
large
s
=
{
v
∈
V
m
|
µ
s
(
v
)
≥
σ
}
and
small
s
=
V
m
−
large
s
. A node
s
is called clear if min
{
µ
s
(
v
)
|
v
∈
large
s
}−
max
>δ
,where
δ
is again a user-specified parameter.
Roughly speaking, if a node
s
is clear, then no further expansion is needed
and the class label assigned to the node is
large
s
.
{
µ
s
(
v
)
|
v
∈
small
s
}
5.3 Fuzzy Decision Trees
For an instance of fuzzy decision trees, we use the following parameter setting:
1. data format in the FDT:
f
i
(
x
) is a singleton subset of
V
i
for all
f
i
∈
A
U
.
2. rule form in the FDL: we only restrict that
and
x
∈
A
.
3. interpretation of wffs: we use disjunctive interpretation, and still choose
the t-norm
L
i
is finite for all
f
i
∈
= min. However,
E
(
x,
(
a
i
,l
)) =
µ
ct
(
l
)
(
f
i
(
x
)) is now a real
number in [0
,
1]. Therefore,
U
s
is a fuzzy subset of
U
for each node
s
of
the decision tree.
4. assignment of class labels to decision tree nodes: we use average support,
and still assume
V
m
=
⊗
{
v
1
,...,v
k
}
.Let
SC
denote the sigma count of
U
s
and
r
i
denote
x
:
f
m
(
x
)=
v
i
µ
U
s
(
x
)for1
k
. Then, the class label of
s
is a fuzzy subset of
V
m
with the membership function
µ
s
(
v
i
)=
r
i
≤
i
≤
SC
=
p
i
,
1
≤
i
≤
k.
Note that
i
=1
p
i
=1holdsinthiscase.
5. computation of degrees of concentration: we use the
sim
defined in (10)
and compute the
gdc
according to (5). Then, analogous to the case of
classical decision trees,
k
p
i
gdc
s
=
p
i
.
2
−
i
=1
6 Conclusion
Decision tree approach is important since decision trees provide solutions to
classification problems and extract rules effectively. To deal with different
kinds of data, decision trees have been generalized along different directions in
the past. In this paper, we propose a quite general framework for fuzzy decision
trees. Some particular instances of this framework prove to be interesting
