Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 292

advantageous for another one. In what follows, a few suggestions are given for
selecting the fuzzy model for some applications.
An extremely complex but very accurate model (high level of similarity
acceptable) can be useful for off-line simulation (function approximation) or
prediction application, because in this case accuracy is more important than model
transparency and compactness. On the other hand, in order to explain the operation
of a particular system, i.e . for operator training, operator interaction, expert
validation, and to understand the basic concepts of the system a transparent model
with a comprehensible linguistic description (where a little similarity is accepted)
is needed. In such cases, it is reasonable to trade some accuracy for extra
transparency and better readability of the fuzzy model. Consequently, this actually
implies the use of a lower value of similarity threshold so that more fuzzy sets can
be found to meet this similarity threshold, which in turn can be merged to result in
fewer fuzzy sets. A model with fewer fuzzy sets and fewer rules is also
computationally less-expensive. Thereby, computationally less-expensive models
are more suitable for applications like model predictive control, memory-expensive
implementations, and fast, on-line model adaptation.

7.8 Application Examples

In order to illustrate the similarity-based rule simplification algorithm presented in
this chapter, the second-order nonlinear plant model (Wang and Yen, 1999) that
was modelled using the neuro-fuzzy approach in Chapter 6, is once again
considered here.

Table 7.1. Performance comparison of fuzzy model after neuro-fuzzy network training and
similar fuzzy sets merging

Training data Evaluation data No. of rules and no.

of fuzzy sets

SSE = 0.0090
MSE = 8.972e -05

SSE = 0.0069
MSE = 6.856e -05

Rules = 5
GMFs/input = 5

MSE (after merging) MSE (after merging) Rules = 2

= 0.0093 = 0.0147 GMFs/input = 2

The neuro-fuzzy trained model generated has five Takagi-Sugeno-type fuzzy
rules and the antecedent fuzzy sets generated for first input ( u ) and second input ( y )
respectively are shown in Figure 7.11(c) and Figure 7.11(d). From Figure 7.11(c)
and Figure 7.11(d) it is seen that the antecedent fuzzy sets are not interpretable, as
they largely overlap each other. However, the accuracy of this fuzzy model is very
high, as the MSE value with the training and validation data are respectively
8.9720e -05 and 6.8560e -05 (see Table 7.1).
In order to improve the model transparency, similar fuzzy sets are merged
together and the corresponding final interpretable fuzzy sets are shown in Figure