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 x ∈ M the fuzzy value f*(x) is defined in the following way:
Determine the class Kj with x ∈ Kj, and define the fuzzy value
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.
Different from histograms for precise data, where relative frequencies hn(·) are additive, i.e. for two disjoint sets A and B we have
for fuzzy relative frequencies hn*(Kj) less restrictive relations for the boundaries (Kj) and (Kj) of the δ-cuts are valid, i.e.
for all δ ∈ (0; 1].
Moreover the fuzzy ...