Random fuzzy set

A random fuzzy set (Engl. Random fuzzy set ) is a fuzzy set whose characteristics (eg. As size, shape, position) dependent on chance. If z. B. randomly selected test persons characterize the acceptance of a new product by linguistic expressions such as "high", "moderate" or "low" and these fuzzy linguistic expressions are sensibly modeled by fuzzy sets, then we have a random fuzzy set with the possible values ​​"high" , "Moderate" and "low". A random fuzzy set is a generalization of the term “ random set ”. The first investigations into random fuzzy sets were carried out in 1976 by R. Féron and in 1986 by ML Puri and DA Ralescu, although at that time they were still referred to as "fuzzy random variables".

Definitions

Be the dimensional Euclidean space and the set of all fuzzy sets in with the properties ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle d}$ ${\displaystyle {\mathfrak {F}}(\mathbb {R} ^{d})}$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{d}}$

• The membership function is semi-continuous from above . ${\displaystyle m_{A}}$
• The carrier of , namely the closure of , is compact . ${\displaystyle A}$ ${\displaystyle \operatorname {supp} A=\{x\in \mathbb {R} ^{d}:m_{A}(x)>0\}}$
• ${\displaystyle A}$is normal, d. H. . ${\displaystyle \exists a\in \mathbb {R} ^{d}:m_{A}(a)=1}$

Now be a probability space . The mapping is called a random fuzzy set if for each of the - intersection is a compact random set (see). Note that the concept of the random fuzzy set is reduced to the concept of the random compact set. On the one hand , this avoids the need to construct a suitable sigma algebra with respect to which the random variable can be measured , but on the other hand it also loses generality because one is limited to fuzzy sets with compact -cuts . ${\displaystyle (\Omega ,{\mathcal {B}},P)}$ ${\displaystyle Y|\Omega \to {\mathfrak {F}}(\mathbb {R} ^{d})}$ ${\displaystyle \alpha \in (0,1]}$ ${\displaystyle \alpha }$ ${\displaystyle Y_{\alpha }}$ ${\displaystyle Y}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \alpha }$

Expected value of a random fuzzy set

Let be a random fuzzy set. The expected value is the fuzzy set whose -cuts are equal to the Aumann expected values ​​of the compact -cuts , i.e. H. ${\displaystyle Y}$ ${\displaystyle EY}$ ${\displaystyle \alpha }$ ${\displaystyle (EY)_{\alpha }}$ ${\displaystyle E_{A}Y_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle Y_{\alpha }}$

${\displaystyle (EY)_{\alpha }=E_{A}Y_{\alpha }}$.

For a random triangular fuzzy number , for example, the result is quite simply (see e.g.) ${\displaystyle Y=(a,\Delta )}$

${\displaystyle E(a,\Delta )=(Ea,E\Delta )}$.

additional

Using a suitable metric between fuzzy sets and taking into account the Fréchet principle , a variance according to ${\displaystyle d}$

${\displaystyle VarY=Ed^{2}(Y,EY)}$

To be defined. This variance is real-valued, in contrast to the fuzzy set- valued variance of a fuzzy random variable . It is currently worth reading, especially when it comes to the methodological differences between random fuzzy sets and fuzzy random variables.

Individual evidence

1. ^ Féron, R. (1976). Ensembles aléatoires flous . CRAcad.Sci.Paris.Ser.A 282, 903-906
2. ↑ ^{a b c} Puri, ML and DA Ralescu (1986). Fuzzy random variables . Journ.Math.Anal.Appl. 114, 409-422
3. ^ Näther, W. (2000). On Random Fuzzy Variables of Second Order and Their Application to Linear Statistical Inference with Fuzzy Data . Metrika 51, 201-221.
4. ↑ Koerner, R. (1997). On the variance of fuzzy random variables . Fuzzy Sets and Systems 92, 83-93
5. ↑ I. Couso, D. Dubous and Sanchez, L. (2014). Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables . Jumper