Fuzzy random variable

A fuzzy random variable . Fuzzy Random Variable is a (sharp) random variable that can only be perceived in a vague manner. Fuzzy random variables were first treated in 1978 by H. Kwakernaak and in 1987 by R. Kruse and KD Meyer. Consider e.g. B. a flock of mayflies (the example is from). The current age of a randomly picked fly is a (positive real-valued) random variable , but this cannot be precisely observed because the exact date of birth of the mayfly is not known. So the age is only fuzzy z. B. perceive as "young", "middle age" or "old".

Definitions

Often be a probability space and a set of membership functions on a basic set . A fuzzy random variable is now a mapping , with the resulting membership functions on or on ${\displaystyle (\Omega ,{\mathfrak {B}},P)}$ ${\displaystyle S}$ ${\displaystyle U}$ ${\displaystyle U=\mathbb {R} }$ ${\displaystyle Y}$ ${\displaystyle Y|\Omega \to S}$ ${\displaystyle S}$

${\displaystyle m_{Y(\omega )}(x),\quad \omega \in \Omega ,\quad x\in U}$

(here suppressed) demands are still made that secure the measurability of , but see. Let the set of all sharp random variables continue , here called the set of originals . The fuzzy set of all possible originals of the fuzzy random variables now plays an essential role, especially for the basics of statistics with fuzzy data . This fuzzy set has the membership function ${\displaystyle Y}$ ${\displaystyle O}$ ${\displaystyle U}$ ${\displaystyle O_{Y}}$ ${\displaystyle Y}$

${\displaystyle m_{O_{Y}}(Z)=\inf _{\omega \in \Omega }m_{Y(\omega )}(Z(\omega )),\quad Z\in O}$.

${\displaystyle m_{O_{Y}}(Z)}$describes the degree of possibility that an (unobservable) original is from . To put it more casually: Which (sharp) random variables are possible if one can only observe the fuzzy . Referring to the example above with the swarm of mayflies: Which (sharp) age constellations are possible if one can only observe random variables with the values ​​"young", "middle age" and "old". ${\displaystyle Z}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle Y}$

Expected value of a fuzzy random variable

The expectation value of a fuzzy random variables is based on the fuzzy set , which according to the so-called extension principle by ${\displaystyle EY}$ ${\displaystyle Y}$ ${\displaystyle U}$

${\displaystyle m_{EY}(x)=\sup _{Z\in O:EZ=x}m_{O_{Y}}(Z),\quad x\in U}$

is obtained. This expected value agrees with the expected value of a random fuzzy set . In special cases, the alpha cuts can be used to construct through the intervals ${\displaystyle U=\mathbb {R} }$ ${\displaystyle EY}$ ${\displaystyle (EY)_{\alpha }}$

${\displaystyle (EY)_{\alpha }=[E\inf(Y(\omega ))_{\alpha },E\sup(Y(\omega ))_{\alpha }],\quad \alpha \in (0,1]}$.

additional

A variance can also be defined for fuzzy random variables . It is a fuzzy set. In contrast to this, the variance of a random fuzzy set is a real number.

Individual evidence

1. ↑ Kwakernaak, H. (1978). Fuzzy random variables. Part I: Definitions and theorems . Information Sciences 15, 1-15
2. ↑ ^{a b c d e} Kruse, R. and Meyer, KD (1987). Statistics with vague data . Reidel, Dordrecht
3. ↑ D. Dubois and H. Prade (1980) Fuzzy Sets and Systems . Academic Press, New York
4. ↑ Koerner, R. (1997). On the variance of fuzzy random variables . Fuzzy Sets and Systems 92, 83-93

Bibliography

• I. Couso, D. Dubois and Sanchez, L. (2014). Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables . Jumper
• I. Couso and Dubois, D. (2009). On the variability of the concept of variance for fuzzy random variables . IEEE Trans. Fuzzy Syst. 17, 1070-1080
• V. Krätschmer (2001). A unified approach to fuzzy random variables . Fuzzy Sets and Systems 123, 1-9