3.1 Functions of fuzzy variables

In order to extend statistical functions f (x1,…xn) of samples x1,…,xn the following definition is useful.

Definition 3.1:

(Extension principle): Let f : M → N be an arbitrary function from M to N. For a fuzzy element x* in M, characterized by an arbitrary membership function μ : M → [0; 1] the generalized (fuzzy) value y* = f (x*) for the fuzzy argument value x* is the fuzzy element y* in N whose membership function ψ(·) is defined by

Remark 3.1:

The extension principle is a natural way to model the propagation of imprecision. It corresponds to the engineering principle ‘to be on the safe side’.

The following statements are important for the generalization of statistical functions.

Proposition 3.1

Let f : M → N be a classical function, and x* a fuzzy element of M with membership function μ(·). Then the fuzzy element y* = f (x*) defined by the extension principle obeys the following:

Proof:

Cδ(y*) = {y ∈ N : ψ(y) ≥ δ} where ψ(·) is defined as in Definition 3.1. By f (Cδ(x*)) = {f (x) : x ∈ Cδ(x*)}, for y ∈ (Cδ(x*)) there exists x ∈ (Cδ(x*)) with f (x) = y. Therefore μ(x) ≥ δ and sup {μ(x) : f (x) = y} ≥ δ and y ∈ (Cδ (y*)).

Classical statistical functions are frequently functions f : n → . If such statistics are continuous functions, the following ...

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