Biomedical Engineering Reference
In-Depth Information
Equation 1.
Description of a Fuzzy Decision Tree
Technique
0
if
x
≤
j
,
x
−
j
,
The idea of FDTs has been around since the late
1970s (Chang and Pavlidis, 1977), since then a
number of different techniques have been devel-
oped to undertake a decision tree analysis in a
fuzzy environment. For information on the types
of FDTs introduced studies like, Yuan and Shaw
(1995), Janikow (1998) Chiang and Hsu. (2002),
Podgorelec
et al
. (2002) and Olaru and Wehenkel
(2003), describe the pertinent issues.
This section of the chapter outlines the FDT
approach first presented in Yuan and Shaw (1995)
and later references (Beynon
et al
., 2004a, 2004b).
For simplicity the notation used in the example
data set, and its fuzzification, are continued here.
The underlying knowledge within a FDT, related
to a linguistic term
C
j
for a decision attribute
C
,
can be represented as a set of fuzzy '
if
..
then
..'
decision rules, each of the form;
0
if
<
x
≤
j
,
j
,
2
−
j
,
2
j
,
x
−
j
,
2
0
+
0
if
<
x
≤
j
,
2
j
,
3
−
j
,
3
j
,
2
(
x
)
1
if
x
=
j
,
3
T
j
x
−
j
,
3
1
−
0
if
<
x
≤
j
,
3
j
,
4
−
j
,
4
j
,
3
x
−
j
,
4
0
−
0
if
<
x
≤
j
,
4
j
,
5
−
j
,
5
j
,
4
0
if
<
x
j
,
5
indicating this value of C is more associated with
being low rather than high.
Two MFs are also used to fuzzify each con-
dition attribute, here only T1 and T2 are shown,
see Figure 2.
The MFs described in Figure 2 are each found
from a series of defining values, in this case for;
T1: [[−∞,−∞, 6, 10, 11], [6, 10, 11, ∞, ∞]], T2:
[[−∞,−∞, 12, 16, 19], [12, 16, 19, ∞, ∞]] and T3:
[[−∞,−∞, 16, 20, 22], [16, 20, 22, ∞, ∞]]. Apply-
ing these MFs on the example data set achieves
a fuzzy data set, see Table 2.
In Table 2, each condition attribute, T1, T2
and T3, is described by two values associated
with two linguistic terms, with the value in bold
the larger fuzzy value describing each condition
attribute.
If (
A
1
is
T
) and (
A
2
is
T
) … and (
A
k
is
k
if0
T
) then
1
2
C
is
C
j
,
where
A
1
,
A
2
, ..,
A
k
and
C
are linguistic variables,
and
T
(
A
k
) = {
T
2
, ..
S
T
} and {
C
1
,
C
2
,
…,
C
L
} are
their respective linguistic terms. Each linguistic
term
k
T
1
,
k
ì
j
T
, which trans-
forms a value in its associated domain to a grade
of membership value to between 0 and 1. The
MFs,
k
j
T
is defined by the MF
(
x
)
(
x
)
and
j
C
, represent the grade of
membership of an object's attribute value for
A
j
being
(
y
)
k
T
j
j
T
and
C
being
C
j
, respectively.
k
Table 1. Example small data set
Object
T1
T2
T3
C
Object
T1
T2
T3
C
u
1
13
15
26
7
u
3
8
18
19
5
u
2
10
20
28
8
u
4
15
12
11
10
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