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Statistical Methods for Fuzzy Data by Reinhard Viertl

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3.6 Fuzzy valued functions

Fuzzy valued functions f*(·) are functions whose range of values are fuzzy intervals, i.e.

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Definition 3.2:

Let f*(·) be a fuzzy valued function with domain M, and xM.

Denoting the δ-cuts of f *(x) by

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for variable x the real valued functions and are called δ-level functions.

In the case M = the δ-level functions are called δ-level curves.

Fuzzy valued functions can be graphically displayed by depicting some δ-level curves. An example is given in Figure 3.4.

Figure 3.4 δ-Level curves of a fuzzy valued function.

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Remark 3.7:

Fuzzy valued functions are important in environmetrics when modeling emissions to the environment. For calculations concerning total amounts of emissions integration of emission rates are necessary. Therefore a calculus of generalized integration is needed. This is possible based on δ-level functions with existing integrals.

Definition ...

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