Environmental Engineering Reference
In-Depth Information
Fuzzy models take into account all possible combinations of membership func-
tions (characteristics) of all variables, thus the number of rules increases exponen-
tially with number of variables and membership functions. Simple models display a
few rules, but there can be fuzzy models with hundreds or thousands of rules. Single
rules are generally assessed by expert judgement, but this may become complicated
when a large number of rules is involved. Methods for automatic rule assessment
have been proposed, in order to overcome the problems of subjectivity (Tscherko
et al. 2007; Marchini et al. 2009).
10.2.3 Defuzzification
The fuzzy output is determined by the degrees of fulfilment of several rules. Fuzzy
rules in fact allow their partial and simultaneous fulfilments. Depending on the
values of the input variables, some rules will be fulfilled more than others, and their
own output will have more importance in the aggregation process for the computa-
tion of the final output.
Whilst in Takagi-Sugeno models the output is already in the form of scalar, the
output of Mamdani models is in fuzzy form, i.e. membership grades to linguistic
characteristics, and has to be defuzzified. Defuzzification is the process of combin-
ing several partial memberships to produce a crisp set, or a crisp single-valued
quantity, compatible to non-fuzzy approaches. It can be performed through differ-
ent techniques (Table 10.2 ). The selection of the most opportune technique depends
on the type of information desired in the output. In ecological models the output sets
are usually defuzzified by calculating their center of gravity, weighted average, and
maximum; less frequently by other methods. When qualitative classes need to be
evaluated, defuzzification is avoided, as it entails a loss of information.
10.2.4 How to Create and Use a Fuzzy Model
The previous sections have shown that fuzzy models can be developed using a
variety of strategies. It is almost impossible to specify a detailed guideline to create
a fuzzy modal that can have general validity. However, all models have essential
steps to be followed, which can be implemented choosing amongst the possible
options presented in Table 10.2 :
1. Select input variables and their characteristics
2. Design membership functions for each variable characteristic
3. Select an output variable and its classes
4. Generate if ... then rules, combining all variable characteristics (antecedent of
implication) and output classes (consequent of implication)
5. Define logical connectives used to manipulate rules ( and , if ... then , or )
6. Define defuzzification strategy (optional)
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