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:
for all δ ∈ [0; 1].
The characterizing function ηj(·) of hn*(Kj) is obtained by Lemma 2.1.
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.