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Statistical Methods for Fuzzy Data by Reinhard Viertl

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5.2 Fuzzy frequency distributions

Considering all classes K1,…,Kk of a decomposition of M, the fuzzy relative frequencies hn*(Kj), j = 1(1)k are defining a special fuzzy valued function f* : M from Section 3.6. For every xM the fuzzy value f*(x) is defined in the following way:

Determine the class Kj with xKj, and define the fuzzy value

Unnumbered Display Equation

Remark 5.2:

The abstract mathematical model for idealized fuzzy histograms are so-called fuzzy probability distributions, especially fuzzy probability densities which are described in Part III, and especially in Chapter .Graphically fuzzy histograms can be depicted using δ-level functions, and δ-level curves for one-dimensional distributions.An alternative way of representing fuzzy histograms is given in the next section.

Remark 5.3:

Different from histograms for precise data, where relative frequencies hn(·) are additive, i.e. for two disjoint sets A and B we have

Unnumbered Display Equation

for fuzzy relative frequencies hn*(Kj) less restrictive relations for the boundaries (Kj) and (Kj) of the δ-cuts are valid, i.e.

(5.1)

and

(5.2)

for all δ ∈ (0; 1].

Moreover the fuzzy ...

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