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Once the fuzzy rule base with well-consistent and non-redundant rules is
determined, the final step is to check the quality of the rule base generated. For this
purpose, the first 50% data from the remaining data sets (
XIO
matrix) are used as
validation data and, thereafter, by applying the Mamdani rules inferencing
mechanism described above, the corresponding crisp output values for the given
input data sets are determined. The crisp values generated are then compared with
the desired output data and, consequently, the SSE or RMSE values are computed
for these validation data sets. If the computed values of SSE or RMSE are less than
the acceptable limit, then the rule base generated is considered as final and is
stored for the forecasting test. Otherwise, with finer or coarser partitioning of
universes of discourse of inputs and outputs and adopting a similar procedure, a
new rule base is built.
Alternatively, after the rule generation, in the final step the defuzzification
strategy recommended by (Wang and Mendel, 1992), usually the center of area
strategy, can be selected and, consequently, the output control
Y
for given inputs
(
X
k
1
,
X
k
2,
X
k
3 ,
X
k
4
) is determined by computing the degree of fulfilment of rule or,
degree P
O
of the output control corresponding to (
X
k
1
,
X
k
2,
X
k
3,
X
k
4
) as:
µ
l
O
l
=
µI
l
1
(
X
k1
).
µI
l
2
(
X
k2
).
µI
l
3
(
X
k3
).
µI
l
4
(
X
k4
).
(4.4)
where
O
l
denotes the output region of rule
l
, and
I
l
i
represents the input region for
i
th component of the rule
l
. For example,
µ
B2
=
µ
S1
(
X
11
).
µ
CE
(
X
12
).
µ
CE
(
X
13
).
µ
B1
(
X
14
).
(4.5)
represents the degree of fulfilment of the Rule-1. The crisp output value
y
is then
determined using the center average defuzzification formula
M
M
l
l
l
y
y
P
P
(4.6)
¦
O
¦
O
l
l
l
1
l
1
where
y
l
is the center value of region
O
l
and
M
represents the number of fuzzy
rules in the combined fuzzy rule base.
4.5.3 Estimation of Takagi-Sugeno Rule's Consequent Parameters
Using Wang and Mendel's approach, or it's proposed modifications, the
antecedent's fuzzy sets of the Takagi-Sugeno rules similar to Mamdani rules can
be determined easily. Once the IF parts (antecedents) of the Takagi-Sugeno type of
fuzzy rules are determined, the linear rule's consequent parameters of the Takagi-
Sugeno rule can be estimated by applying the least squares error (LSE) technique.
In order to describe the LSE method for rule's consequent parameter
estimation, Takagi-Sugeno type of fuzzy rules of a multi-input single-output
system are once again considered:
1
1
R
1
: IF x
1
is G
1
and ..... and x
n
is G
n
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