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Statistical Methods for Fuzzy Data by Reinhard Viertl

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2.2 Vectors of fuzzy numbers and fuzzy vectors

For multivariate continuous data measurement results are fuzzy too. In the idealized case the result is a k-dimensional real vector (x1,…, xk). Depending on the problem two kinds of situations are possible.

The first is when the individual values of the variables xi are fuzzy numbers xi*. Then a vector (x1*,…, xk*) of fuzzy numbers is obtained. This vector is determined by k characterizing functions ξ1(·),…,ξk(·). Methods to obtain these characterizing functions are described in Section 2.1.

The second situation yields a fuzzy version of a vector, for example the position of a ship on a radar screen. In the idealized case the position is a two-dimensional vector (x, y) ∈ 2. In real situation the position is characterized by a light point on the screen which is not a precise vector. The result is a so-called fuzzy vector, denoted as (x, y)*.

The mathematical model of a fuzzy vector is given in the following definition, using the notation = (x1,…, xk).

Definition 2.3:

A k-dimensional fuzzy vector * is determined by its so-called vector-characterizing function ξ(,…,) which is a real function of k real variables x1,…, xk obeying the following: ...

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