In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two part. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X⊖HC exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C′ chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize all fuzzy random translation; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C′ (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.

A decomposition theorem for fuzzy set–valued random variables / G. Aletti, E.G. Bongiorno. - In: FUZZY SETS AND SYSTEMS. - ISSN 0165-0114. - 219(2013 May 16), pp. 98-112. [10.1016/j.fss.2012.11.005]

A decomposition theorem for fuzzy set–valued random variables

G. Aletti
Primo
;
E.G. Bongiorno
Ultimo
2013-05-16

Abstract

In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two part. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X⊖HC exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C′ chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize all fuzzy random translation; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C′ (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.
Decomposition theorem; Fuzzy random variable; Hukuhara difference; Randomness defuzzification
Settore MAT/06 - Probabilita' e Statistica Matematica
Settore SECS-S/01 - Statistica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/213788
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