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result in identical rules that can be subsequently removed from the rule base,
leading to a reduction of the number of rules too. Also, the number of dimensions
(features) in the model's premise can be reduced in the case of partition similarity.
In the rule base simplification method presented here, initially the set-theoretic
similarity between two fuzzy sets is defined, based on which the similarity between
the same sets can be numerically calculated. If the calculated similarity measure is
larger than a threshold value (say 0.7) predefined by the fuzzy model designer,
then the similar fuzzy sets are merged together, resulting in a unique fuzzy set
representative of both fuzzy sets. By selecting different values of similarity
threshold from the same initial (non-transparent/non-interpretable) rule base,
several final (transparent/interpretable) fuzzy models can be generated in which the
degree of acceptability of the final model is a trade-off between the three model
competitive issues: modelling accuracy, transparency, and compactness.
Setnes
et al
.(1998a, 1998b) have pointed out that several methods have been
proposed for optimizing the size of the rule base. However, the fuzzy set-theoretic
similarity-based rule base simplification method differs from other fuzzy rule base
reduction methods mainly in the way that its main objective is
to reduce the
number of fuzzy sets
used in the model and
not the number of rules
. Furthermore,
the method can favourably be combined with any data-driven modelling tools, such
as fuzzy clustering, or even the neuro-fuzzy approach of Palit and Babuška (2001)
and genetic algorithms in order to obtain a tool for transparent, yet reasonably
accurate and compact fuzzy modelling (Setnes and Roubos, 2000; Roubos and
Setnes, 2001).
In what follows, we will briefly discuss the transparent modelling procedure
followed by a general data-driven modelling scheme in which fuzzy set-theoretic
similarity-driven simplification is included. The concepts of similarity and
redundancy to be described here are illustrated through a similarity-driven rule
base simplification method, applied to the example of forecasting a nonlinear time
series using a fuzzy model.
7.3 Fuzzy Modelling with Enhanced Transparency
In the fuzzy modelling scheme presented below, our objective is to achieve a good
approximation accuracy and model transparency in a data-driven fuzzy modelling
approach. In order to make the model transparent and computationally more
efficient, an initial fuzzy model is extracted from observation data. In order to
remove the unnecessary redundancy in the knowledge learnt from the data, the
principle of
set-theoretic similarity-driven fuzzy rule base simplification
will be
used.
7.3.1 Redundancy in Numerical Data-driven Modelling
In the recent past a variety of numerical data-driven fuzzy modelling tools have
been developed for automated building of data-driven models (Roubos
et al.
, 2001;
Setnes, 2001). Usually, when building a fuzzy model, the model premise space is
partitioned by means of fuzzy sets. However, rule-based models obtained from
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