Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 269

7.2 Model Transparency and Compactness
Fuzzy models are often referred to as white-box models , in contrast to the neural-
networks-based models which are considered as black-box models . This is because
fuzzy models are, to some extent, transparent to interpretation and analysis,
implying that the model's output can be justified through developed IF-THEN
linguistic rules. However, the transparency of a fuzzy model cannot be achieved
automatically, unless some special measure is taken a priori . This is especially,
true for the automated data-driven fuzzy modelling technique, where the fuzzy
models generated are not at all or to a restricted degree transparent to
interpretation.
A system can be described by a few fuzzy rules using distinct, i.e. non-
overlapping, interpretable fuzzy sets. It can, of course, also be described by a few
fuzzy rules, but with a large number of highly overlapping fuzzy sets that hardly
allow for any interpretation. Alternatively, if a system is described by a large
number of rules but with a few (or many) distinct and non-overlapping fuzzy sets,
then the fuzzy models generated in such a case could also be unclear or close to
non-interpretable because of the large number of rules. This situation can occur
practically when the fuzzy rules are generated using the Wang and Mendel (1992a)
approach, or by its modification as proposed by Palit and Popovic (1999a),
presented in Chapter 4. In both rule-generation approaches a large number of
input-output data pairs (or training samples) generate a large number of rules, even
though fuzzy domains are partitioned by large (or small) numbers of distinct and
non-overlapping /partially overlapping fuzzy sets such as Small( N ), Small( N -1), ...,
Small(1), Centre (CE), Big(1), ..., Big( N ), etc . The reason for loss of
interpretability in the above case is mainly because the large number of rules fire
simultaneously for an unknown input condition (within the fuzzy domain) to infer
the corresponding output decision. Therefore, the corresponding output decision
cannot be easily justified by human reasoning.
Yet, in contrast to the above case when a fuzzy model is developed using expert
knowledge, the model designer usually takes care that neither the number of rules
nor the fuzzy sets, which are used to partition the domains, are large at all, besides
maintaining the proper distinguishability of applied fuzzy sets for domain partition.
On the other hand, when automated data-driven techniques are applied to build
fuzzy models from data, a certain degree of redundancy, and thus unnecessary
complexity, cannot be avoided.
In the following, we present a rule base simplification and reduction method
proposed by Setnes et al . (1998a and 1998b) and Setnes (2001) that seeks to
simplify an already available rule base by reducing redundant information present
in the form of similar fuzzy sets. Similar fuzzy sets are overlapping fuzzy sets that
describe almost the same region in the domain of some model variable. In such
cases, the model uses more fuzzy sets than necessary, since these fuzzy sets
represent more or less the same concept. We intend to use the concept of set
theoretic similarity measure , as extensively used by Setnes et al . (1998a, 1998b),
that helps to identify the similar fuzzy sets, and to replace these similar fuzzy sets
by a common fuzzy set representative of those original fuzzy sets. If the
redundancy in the model is very high, then merging the similar fuzzy sets might