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Statistical Methods for Fuzzy Data by Reinhard Viertl

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10.2 Combination of fuzzy samples

Let x1*, … , xn* be n fuzzy elements of the observations space Mx, and ξ1(·), … , ξn(·) the corresponding characterizing functions. In order to obtain a fuzzy element * of the sample space Mxn, the vector-characterizing function ξ(., … , .) of * has to be constructed. The vector-characterizing function ξ(., … , .) is defined via its values ξ(x1, … , xn) for xiMx in the following way:

Let T be a t-norm,

Unnumbered Display Equation

For statistical inference the most suitable t-norm is the minimum t-norm, because using this t-norm the fuzziness of data is propagated in generalized inference procedures. Therefore the vector-characterizing function ξ(., … , .) is given by its values

Unnumbered Display Equation

and the corresponding fuzzy vector * is called the combined fuzzy sample. This combined fuzzy sample is basic for the generalization of statistical procedures to fuzzy data. The imprecision of fuzzy data is propagated by functions of data, i.e. s(x1*, … , xn*) = s(*).

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