Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 160

(because r = 5), 0.98 in G 2 (because r = 2), 0.9 in G 6 (because r = 6), and 0.97 in
G 2 (because r = 2) respectively. Hence, the corresponding fuzzy rule R3 will be:
R3: IF X 11 is G 3 AND X 12 is G 5 AND X 13 is G 2 AND X 14 is G 6 THEN Y 1 is G 2
The same fuzzy rule can also be written as [3 5 2 6 2 Drule FOP ], where the
numbers correspond to G 3 , G 5 , G 2 , G 6 , G 2 of the rule respectively whereas, ( Drule )
and FOP stand for the degree of the rule and fuzzy operator (AND) respectively.
Drule = 1 if no degree of rule is specified or all rules have the same degree. If only
the AND operator is used, then FOP has the value 1. Otherwise, for the OR
operator the value 0 is used. Thus, for no degree of rule and for the AND operator
the same rule is rewritten as [3 5 2 6 2 1 1]. The rules built in this way are used
to build the rule list of the fuzzy system. If any two rules in the rule list create a
conflict situation, the rule with the higher Drule value is taken and the other one is
rejected from the rule list. For example, for the conflicting rules
[3 5 2 6 2] and [3 5 2 6 4]
Drule3 = P G 3 ( X 11 ).P G 5 ( X 12 ).P G 2 ( X 13 ).P G 6 ( X 14 ).P G 2 ( Y 1 )
= (0.95).(0.80).(0.98).(0.90).(0.97) = 0.65 (say),
Drule4 = P G 3 ( X 21 ).P G 5 ( X 22 ).P G 2 ( X 23 ).P G 6 ( X 24 ).P G 4 ( Y 2 )
= (0.90).(0.50).(0.80).(0.60).(0.70) = 0.15 (say),
so that because Drule3 > Drule4 the rule3 is selected and the second one rejected.
However, for redundant (duplicate rules) rules from a list of several such rules any
one is selected. Rules generated in this way are actually Mamdani-type fuzzy rules
and the complete procedure of such automated rule generation is summarized in
Algorithm 4.1.
However, a small modification in the final stage will also generate fuzzy
relational rules (fuzzy relational model/Pedrycz model), i.e. to generate the fuzzy
relational rule, the r values from the Mu-matrix are recorded for all four inputs as
mentioned earlier and these generate as usual the antecedent part of the fuzzy
relational rule. Now, the consequent part of the fuzzy relational rule is generated
from the complete last column (fifth column) of the Mu-matrix that contains the
degree of membership of output Y k in the G 1 to G n fuzzy sets. Note that for the
last column of the Mu-matrix we do not record the r value for the output Y ,
whereas the entire column is recorded for rule generation. Therefore, the
corresponding fuzzy relational rule can be written as
R5: IF X k1 is G 3 AND X k2 is G 5 AND X k3 is G 2 AND X k4 is G 6
THEN Y k is G 1 ( µ G 1 ( Y k )) , Y k is G 2 ( µ G 2 ( Y k )), ..., Y k is G n ( µ G n ( Y k )) .
Similarly, the antecedent part of the Takagi-Sugeno fuzzy rule is also generated in
the same way, whereas the linear consequent parameters of the TS rules are
generated by least squares error (LSE) estimation as described in Section 4.5.3.