Environmental Engineering Reference
In-Depth Information
a
b
1
1
0
0
variable
v
variable
v
Fig. 10.1 Examples of linear (a) and non-linear (b) membership functions for the characteristics
low , medium and high of a variable v
10.2.2 Model Rules
A fuzzy model is a collection of linguistic rules, thus it is transparent and easily
understandable. A fuzzy rule is a logical implication, “ if antecedent, then conse-
quent”. The action of the implication is to reduce the truth of the consequent
according to the degree of fulfilment of the antecedent. The antecedent is a combina-
tion of several conditions connected with the logical operator of conjunction, and .To
produce a fuzzy output, all fuzzy rules are aggregated, generally with the logical
operator of disjunction, or . The logical operators and , or , if ... then can be implemen-
ted with different mathematical operations (Table 10.2 ). Typically, minimum is used
for and , maximum for or and product for if ... then .
There are two main types of fuzzy models: Mamdani and Takagi-Sugeno. In the
Mamdani type, both rule antecedent and consequent are in the form of membership
functions: “ if (variable v is characteristic C, and
... , and variable z is characteristic
Z), then output is characteristic O”. The model output is expressed in linguistic
terms: it is a membership grade to a linguistic characteristic. In some cases, a
linguistic output (for example a quality class) is desirable, thus, defuzzification can
be avoided. In Mamdani models, defuzzification becomes necessary only when a
numerical output is required. In Takagi-Sugeno models, the rule conclusion is
already a crisp value, linear function of the inputs: “ if (variable v is characteristic
C, and
, z)”. Therefore,
this model type does not require an explicit defuzzification procedure.
In ecological applications, the linguistic character of the Mamdani-type output is
suitable to model qualitative ecosystem features, for example classes of soil or
water quality (Ocampo-Duque et al. 2006; Tscherko et al. 2007; Icaga 2007),
whereas the behaviour of environmental variables can be easily forecast or
modelled by means of the Takagi-Sugeno approach (Jorquera et al. 1998; Ryoke
et al. 2000; Altunkaynak et al. 2005; Lin and Cobourn 2007).
...
, and variable z is characteristic Z), then output is f(v
...
 
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