Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 291

antecedent fuzzy sets, such as (low, small), (low, moderate), (low, fast), ..., and
(high, fast) of the two input variables. However, for a multivariable system with a
large number of input and output variables and with a reasonably large number of
antecedent fuzzy sets this is not feasible, as it will explode the fuzzy rule base,
making the model non-transparent, computationally very expensive, and non-
compact.

Mu(x) Mu(x)

1.0 1.0

x x

Not moderate number of sets Not easily distinguishable sets

Mu(x)

Mu(x)

1.0

1.0

x x

Transparent partitioning of UD

Bad coverage, subnormal set

Figure 7.10. Transparent partitioning of domain by distinguishable fuzzy sets

Accuracy of the model, of course, depends on the type of fuzzification
(singleton or non-singleton) and of defuzzification (mean of maxima or centre of
gravity) method, as well as of the inference mechanism used. For inferencing a
Mamdani-type fuzzy model one can select the product/min operator for degree of
firing of rule computation with Mamdani's inferencing mechanism. Similarly, for
relational matrix computation (which is used in min-max compositional rule of
inference), Mamdani implication (min operator), or the alternative Larsen
implication (product operator), can be used (see Chapter 4). The different choices
of all those possibilities result in different accuracy of the model even though the
model type (Mamdani), number of inputs and outputs, and their partitioning fuzzy
sets numbers and types of membership function (Gaussian/triangular) may be the
same.
However, assuming that for the identical type of model and using identical
fuzzification, defuzzification, and inferencing mechanisms we obtain fuzzy model
1, which is the most accurate, model 2, which is the most transparent and model 3,
which is the most compact, the question that arises now is which model is to be
selected for a particular situation. There is no unique answer to this question,
because each model has it's own advantage for a particular application, but is less