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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 7

a. For data in the form of two-dimensional intervals (x, y)i* which has vector-characterizing function ξi(., .) = I[a1, i; b2,i] × [a1, i; b2, i](., .), i = 1(1)n the vector-characterizing function of the fuzzy combined sample is given by ξ(x1, y1,…,xn, yn) = min{ξi(xi, yi) : i = 1(1)n} and the δ-cuts of ξ(.,…,.) are the Cartesian products

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Therefore ξ(x1, y1,…,xn,yn) is an indicator function. From that it follows that also the characterizing function ψr* (·) of the generalized correlation coefficient r* is an indicator function.

b. In this situation the vector-characterizing function of (x, y)i*, i = 1(1)n is a function ξi(x, y) of the following form:

The δ-cuts of (x, y)i* are given by

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where ξi(·) is the characterizing function of the fuzzy y-component yi* of (x, y)i*.

The δ-cuts of the fuzzy combined sample (x1, y2, …, xn, yn)* are given by

Unnumbered Display Equation

The vector-characterizing function ξ(x1, y1, …, xn, yn) can be obtained by the construction lemma for membership functions.

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