Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 290

models are less prone to overfitting, and provide better control over the
interpolation and extrapolation properties of the mapping obtained.
In order to deal effectively with multivariable complex systems, hybrid
approaches should be applied which can use the available prior knowledge about
the system, and allow for decomposition of a large problem into a number of
simpler subproblems. Furthermore, if different fuzzy models of the same type (say,
of Mamdani and Takagi-Sugeno type) are only considered, then the accuracy,
transparency, or complexity and compactness of the generated model may also,
based on the various factors, vary. In addition, for a set of fuzzy models of the
same type (Mamdani) representative of an identical process and even with identical
model inputs and output(s) besides their identical domain representation for all
input and output variables, the accuracy, transparency and compactness of these
models, generated by the same or a different data-driven automated approach, may
be totally different. This is particularly because the model accuracy, transparency,
and compactness are influenced by many factors like

x Number of antecedent (or consequent) fuzzy sets assigned to each variable.
x Coverage of the antecedent (consequent) fuzzy sets.
x Number of fuzzy rules.
x Fuzzification/defuzzification or inference mechanism

The first factor suggests that the accuracy of the model may generally increase if
the input universes (and also output universes for a Mamdani model) are fine
partitioned using a large number of membership functions or antecedent (also
consequents) fuzzy sets. In fact, it was observed in Chapter 4 that when the input
and output universes of discourse are partitioned by 27 Gaussian membership
functions instead of an initially chosen 17 Gaussian membership functions, the
accuracy of the generated fuzzy chaotic time series forecaster model has
significantly increased.
Coverage means that each domain element is assigned at least one fuzzy set
with H(non-zero) membership degree, i.e .

xX i , , P x ! H .

G

So, coverage actually insists on there being a certain amount of overlapping
between the adjacent fuzzy set, so that entire universe of discourse is well covered
by the input/antecedent (output/consequent) fuzzy sets (see Figure 7.10). Optimum
selection of this coverage (small) value can result in both an accurate and a
transparent model. However, large coverage may result in indistinguishable fuzzy
sets, creating a model that is completely non-transparent. It is also observed that
the accuracy of the model may generally increase if the number of rules are such
that all possible combinations of inputs (antecedents) and output fuzzy sets are
covered by at least one rule (for a Mamdani model).
Suppose that for a two-input and one-output system the first input and second
input universes are partitioned respectively by antecedent fuzzy sets such as ( low ,
medium and high ) and ( slow , moderate and fast ). In this case at least (3 2 = 9) nine
fuzzy rules are required to take into account all possible combinations of