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^
`
"
are represented in input and output form. The objective is to
generate, from the above data set, the IF-THEN rules that will construct the rule
base of a fuzzy predictor system. This is carried out in the following steps.
X
XX
,
,
,
X
1
2
q
Step 1:
formation of fuzzy input and output regions
Suppose that the domain interval of
X
ki
is [
X
i_
lo
,
X
i
_hi
] and that of (
Y
k
) is [
Y
lo
,
Y
hi
],
where
k
= 1, 2, 3, ... ,
m
; and
i
= 1, 2, 3, 4; corresponding to the four inputs
respectively.
S2 S1 CE B1 B2
1.0
0.8
0.6
0.4
0.2
0.0
X
lo
X
ki
X
11
X
12
X
13
X
14
Y
1
X
hi
Figure 4.3(a).
Division of input and output range in fuzzy regions
Taking into account that all input values and the output value belong to the same
time series
X
= {
X
1
,
X
2
,
X
3
, ...,
X
q
}, for
t
= 1, 2, 3, ... ,
q
, the domain intervals of all
inputs and the output can be taken to be the same, say [
X
lo
,
X
hi
] or [
Y
lo
,
Y
hi
], for
i
=
1, 2, 3, 4. Each domain interval can be divided into (2
N
+ 1) fuzzy regions (see
Figure 4.3(a) and 4.3(b)) like
SN(Small N), ..., S2(Small 2), S1(Small 3), CE(Center), B1(Big 1), B2(Big
2), ..., BN(Big N).
Step 2:
data fuzzification and rules generation
This includes the determination of the degrees of membership of
X
k
1
,
X
k
2
,
X
k
3
,
X
k
4
,
Y
k
in different fuzzy regions and assignment of a given
X
k
1
,
X
k
2
,
X
k
3
,
X
k
4
,
Y
k
for
k
=
1, 2, ... ,
m
; to the region with the maximum degree. For example, for
k
= 1,
X
k
1
in
Figure 4.3(a) has degree of membership 0.8 in S1, 0.2 in S2 and 0 degrees in all
other regions. Similarly, for
k
= 1,
X
k
2
in Figure 4.3(a) has degree of membership
0.6 in CE, 0.4 in S1 and 0 degrees elsewhere. Again, for
k
= 1,
X
k
1
in Figure 4.3(a)
is considered to be S1 and
X
k
2
in Figure 4.3(a) is considered to be CE.
Similarly, the fuzzy regions with maximum degree should be assigned to the
X
k
3
,
X
k
4
,
Y
k
for
k =
1. Now, the corresponding rules can be obtained from the input-
output data sets. According to Figure 4.3(a), for
k
= 1,
X
k
1
,
X
k
2
,
X
k
3
,
X
k
4
, and
Y
k
give
[
X
11
(0.8 in S1, max),
X
12
(0.6 in CE, max),
X
13
(0.8 in CE, max),
X
14
(0.8 in B1,
max);
Y
1
(0.6 in B2, max)], and rule R1 is
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