Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 285

threshold O which represents the degree to which the user allows for equality
between the two fuzzy sets used in the model, depends on the application. The
lower the value of O the more fuzzy sets are combined, thereby decreasing the
term set of the model. In general, one can expect the numerical accuracy of the
model to decrease as the O value decreases.
However, this need not always be the case. If the model is highly redundant or
overdetermined, then the numerical accuracy may improve as a result of merging
the fuzzy sets and thereby possible reduction in rule base. As a general practice,
one may carry out the trial with several values of O for a particular application
with the training samples, and the O value that gives the best result with the
validation data set for a particular application should be finally selected. For
instance, in order to explain the operation of a particular system, e.g . operator's
training or expert's validation, a comprehensible linguistic description is important.
In such cases, it is reasonable to trade some accuracy for extra transparency and
readability. Consequently, this implies the use of a lower value of O so that more
fuzzy sets can be found to meet this similarity threshold, and which can, in turn, be
merged. In contrast to this, an application that aims at prediction or simulation
(function approximation) means that one can probably select much higher values
of O as in this case accuracy is more important. To obtain rules sufficiently
distinguishable to describe the system qualitatively, a O value around 2/3 has been
found to give good results in the various experiments of Setnes (2000). Since this
part of the simulation requires no additional data acquisition or computationally
expensive optimization, the effect of different thresholds can be easily investigated.
The simplification part of the algorithm can be performed in two ways:
x by iterative merging
x using similarity relations.
The main difference lies in the computational effort, and the sensitivity to changes
in the threshold O Iterative merging requires more computations than similarity
relations, but it is more transparent to user interaction. Both approaches are
presented below.
7.6.1 Iterative Merging
The algorithm is illustrated in Figure 7.8 and summarized in Algorithm 7.1. The
algorithm starts by iteratively merging similar fuzzy sets. In each iteration, the
similarities between all pairs of fuzzy sets for each variable are considered, and the
pair of fuzzy sets having the highest similarity S > O is merged to create a new
fuzzy set. Then, the rule base is updated by substituting this new fuzzy set for the
fuzzy sets merged to create it. The algorithm then again evaluates the similarities in
the updated rule base. This continues until there are no more fuzzy sets for which S
> O. Then the fuzzy sets that have similarity S > J to the universal fuzzy set are
removed. Thereafter, the rule base premise is checked for redundant inputs. If
present, such inputs are removed. The rule base is then checked for rules with
equal premise parts. Such rules are merged as discussed in Section 7.5.4. Finally,
the rule consequents are re-estimated.