Databases Reference
In-Depth Information
or by quantitative means:
ldc
s
=
1
≤i<j≤n
µ
U
s
(
x
i
)
sim
(
x
i
,x
j
)
1
≤i<j≤n
µ
U
s
(
x
i
)
·
µ
U
s
(
x
i
)
·
(7)
·
µ
U
s
(
x
i
)
recalling that
U
=
. The qualitative
ldc
s
measures the degree of
truth of the statement “decision classes of all objects left at
s
are pairwisely
similar” in a fuzzy logic sense, whereas the quantitative
ldc
s
measures the
average similarity of the decision values. Unlike in the global case, we do
not have to assign decision labels to the internal nodes of the decision tree.
However, in the local case, the similarity of decision classes of each pair of
objects has to be calculated.
If we let
dc
s
denote either
gdc
s
or
ldc
s
, then the stopping condition is
satisfied by a node
s
if
dc
s
≥
{
x
1
,
···
,x
n
}
θ
for some preset threshold value
θ
.Thechoice
of an appropriate definition of
dc
s
depends on the problem context and the
complexity consideration.
Choice of the Best Attribute
To choose the best attribute for split each time, we simply split a node
s
with
each attribute in
A
s
to find which attribute results in the maximum average
concentration degree. Let
f
i
be an attribute in
A
s
and for each
l
∈L
i
,
s
l
be the child node of
s
corresponding to (
a
i
,l
), provided that
s
is split with
attribute
f
i
. We denote the sigma count of
U
s
l
∈L
i
.The
resultant average degree of concentration after splitting
s
with attribute
f
i
is
as
SC
s
l
for any
l
dc
i,s
=
l∈L
i
SC
s
l
l∈L
i
SC
s
l
·
dc
s
l
.
(8)
Therefore, the a
ttr
ibute chosen for next split should be the attribute
f
i
∈
A
s
that maximizes
dc
i,s
, i.e.,
arg max
f
i
∈A
s
dc
i,s
.
4.2 The Decision Phase
Once a general fuzzy decision tree has been constructed, we can use it to
classify new data. Let
x
be an object with attributes
f
1
,...,f
m−
1
such that
f
i
(
x
)
˜
1. Since labels on the edges of the decision tree
are atomic formulas based on attributes
f
1
,...,f
m−
1
, the evaluation function
E
can be applied to the object
x
and the atomic formulas for assignment of
a decision value
f
m
(
x
)to
x
.
We associate an FDL wff,
ϕ
s
, with each leaf node
s
of the decision tree.
The wff
ϕ
s
is the conjunction of all atomic formulas appearing on the edges
∈
P
(
V
i
)for1
≤
i
≤
m
−
