5.1 Fuzzy frequency of a fixed class

Due to the possible inability to decide in which class a fuzzy data point xi* lies it seems natural to consider fuzzy values of relative frequencies. For n fuzzy observations x1*,…,xn* and classes K1,…,Kk the fuzzy relative frequencies of the classes are denoted by hn*(Kj), j = 1(1)k. The characterizing function of hn*(Kj) is constructed via its δ-cuts

where the limit points (Kj) and (Kj) are defined in the following way:

and

for all δ ∈ [0; 1].

The characterizing function ηj(·) of hn*(Kj) is obtained by Lemma 2.1.

Remark 5.1:

The characterizing function of a fuzzy relative frequency is a step-function. In Figure 5.2 the characterizing functions of a sample of size 8 are given. In Figure 5.3 the characterizing function of the fuzzy relative frequency of the class [1; 2] is depicted.

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