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Statistical Methods for Fuzzy Data by Reinhard Viertl

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2.1 Fuzzy numbers and characterizing functions

In order to model one-dimensional fuzzy data the best up-to-date mathematical model is so-called fuzzy numbers.

Definition 2.1: A fuzzy number x* is determined by its so-called characterizing function ξ(·) which is a real function of one real variable x obeying the following:

1. ξ : → [0; 1].

2. ∀δ ∈ (0; 1] the so-called δ-cut Cδ(x*) :={x ∈ : ξ(x) ≥ δ} is a finite union of compact intervals, .

3. The support of ξ(·), defined by supp[ξ(·)] :={x ∈ : ξ(x) > 0} is bounded.

The set of all fuzzy numbers is denoted by .

For the following and for applications it is important that characterizing functions can be reconstructed from the family (Cδ(x*); δ ∈ (0; 1]), in the way described in Lemma 2.1.

Lemma 2.1:

For the characterizing function ξ(·) of a fuzzy number x* the following holds true:

Unnumbered Display Equation

Proof:

For fixed x0 ∈ we have

Therefore we have for every ...

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