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Statistical Methods for Fuzzy Data by Reinhard Viertl

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5

Histograms for fuzzy data

Classical histograms are based on precise data x1,…,xn in order to explain the distribution of the observations xi. In order to construct histograms the set M of possible values xi is decomposed into so-called classes K1,…,Kk with KiKj = ij, and

Unnumbered Display Equation

Then for each class Kj the so-called relative frequency hn(Kj) is calculated, i.e.

Unnumbered Display Equation

The display of the relative frequencies is called a histogram.

For fuzzy data the following problem arises: By the fuzziness of observations it is not always possible to decide in which class Kj a fuzzy observation x* with characterizing function ξ(·) lies. This is depicted in Figure 5.1.

Figure 5.1 Fuzzy observation and classes of a histogram.

c05f001.eps

Therefore a generalization of histograms is necessary.

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