Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 283

singleton fuzzy sets may represent exceptions. Interaction from the user is typically
needed in such cases to handle such situations. Since our interest is to develop an
automated simplification method, these types of rule reduction are not considered
here.

7.5.3 Removing Redundant Inputs

Figure 7.7 shows a non-similar partitioning of two input-domains. However, in
systems identification and time series modelling, highly similar partitioning of two
or more inputs can sometimes occur. An assessment of the similarity S pq between
the partitions of a pair of inputs ( x p , x q ) can be obtained by measuring the similarity
S ( A lp , A lq ) between all corresponding pairs of fuzzy sets l = 1, 2, ..., M , and taking
the minimum occurring similarity for each pair of inputs as the partition similarity:

S S A , A , l 1, 2,..., M . (7.6)

min

pq lp lq

l

If the partition similarity S pq is above an acceptable threshold value predefined by
the user, then one of the two inputs, x p or x q , can be removed from the model's
premise part. Depending upon the model type and it's performance, e.g . in a
Takagi-Sugeno fuzzy model, it might still be necessary to keep all variables in the
consequent part of the rule base.

7.5.4 Merging Rules

Given a Mamdani-type fuzzy model with k identical rules, if k t , then the rule

base simplification will result in the removal of k -1 rules, and thereby reducing rule
base. However, if only the premises of the rules (antecedent fuzzy sets) are equal,
but not the consequents, then this may indicate a rule conflict situation in the rule
base and that has to be solved by assigning a degree to each conflict rule (Wang
and Mendel, 1992). In the following, only the fuzzy models of Takagi-Sugeno type
are considered.
As in the case of Takagi-Sugeno models, the rule-consequents are not fuzzy;
therefore, the similarity concept is applied here only in the premise (antecedents)
part of the rules. When the premise parts of

2

k t Takagi-Sugeno rules are equal,
these rules are removed and replaced by one general rule R g . This general rule has
the same premise part as the rules that it replaces. However, the consequent
parameters of the general rule are re-estimated taking into account the total
influence of all the k -rules in fuzzy inferencing that it replaces. This can be done by
weighting R G with k and letting it's consequent be an average of the consequents of
all the k -rules with equal premise parts.
Let Q be a set of indices

2

" of the k rules R l with equal premise
parts. These rules are replaced by a single rule R g with weight k and consequent
parameters

l 1, 2, , M