Extension principle in the theory of fuzzy sets

The extension principle (Engl. Extension principle ) in the theory of fuzzy sets going to Lotfi Zadeh back 1965th

It is the attempt to "expand" classical mathematical concepts in order to be able to work with fuzzy sets. At its core, the principle of enlargement is nothing more than a principle of propagation of uncertainty. It answers the question what fuzzy value a classical function has when the fuzzy argument is present, i.e. H. what is meant by ? ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle f(A)=B}$

Definitions

First, let us be a single-digit real-valued function and a fuzzy set with the membership function . If is one-of-a-kind , then the membership function for simply is given by ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle m_{A}}$ ${\displaystyle f}$ ${\displaystyle m_{B}}$ ${\displaystyle B=f(A)}$

${\displaystyle m_{B}(y)=m_{A}(f^{-1}(y))=m_{A}(x),\quad y=f(x)}$,

d. H. through the membership value is transferred directly to . The more interesting case is when is not unambiguous, i.e. H. when several can lead to the same thing . Then it's to Zadeh ${\displaystyle y=f(x)}$ ${\displaystyle m_{A}(x)}$ ${\displaystyle m_{B}(y)}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle y}$

Extension principle for a function that is not one-to-one: three x-values ​​lead to the same y

${\displaystyle m_{B}(y)=\sup _{x\in \mathbb {R} :f(x)=y}m_{A}(x)}$

to form, d. H. is equal to the largest possible membership value with . In general, let us now be a multi-digit real-valued function; H. and be the fuzzy arguments. Then the fuzzy function value is defined by ${\displaystyle m_{B}(y)}$ ${\displaystyle m_{A}(x)}$ ${\displaystyle y=f(x)}$ ${\displaystyle f\colon \mathbb {R} ^{d}\to \mathbb {R} }$ ${\displaystyle f(x_{1},...,x_{d})=y}$ ${\displaystyle A_{1},...,A_{d}}$ ${\displaystyle B}$

${\displaystyle m_{B}(y)=\sup _{(x_{1},...,x_{d})\in \mathbb {R} ^{d}:f(x_{1},...,x_{d})=y}\min[m_{A_{1}}(x_{1}),...,m_{A_{d}}(x_{d})]}$,

see e.g. B. Another T-norm can be used for in the last formula . ${\displaystyle \min }$

Applications

• Arithmetic with fuzzy numbers : The extension principle, applied to the functions defines addition, subtraction, multiplication and division of fuzzy numbers, see e.g. B. ${\displaystyle f_{1}(x_{1},x_{2})=x_{1}+x_{2},\quad f_{2}(x_{1},x_{2})=x_{1}-x_{2},\quad f_{3}(x_{1},x_{2})=x_{1}x_{2},\quad f_{4}(x_{1},x_{2})={\frac {x_{1}}{x_{2}}}}$
• Compatibility of fuzzy sets : The compatibility of a fuzzy set with the fuzzy set indicates the degree to which the fuzzy element to hear. For example, to what degree does a woman around 30 belong to the fuzzy set of young women? is obtained by applying the principle of expansion to the function . ${\displaystyle C_{B}(A)}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C_{B}(A)}$ ${\displaystyle f(x)=m_{B}(x)}$
• Statistics with fuzzy data : Let be a sampling function , e.g. B. an estimator or a test statistic . The extension principle applied to this sampling function leads to a sampling function for fuzzy data , see e.g. B. ${\displaystyle f(x_{1},...,x_{n})}$ ${\displaystyle A_{1},...,A_{n}}$

Individual evidence

1. ↑ ^{a b} L.A. Zadeh (1965): Fuzzy sets . Information and Control 8: 338-353. doi : 10.1016 / S0019-9958 (65) 90241-X
2. ↑ ^{a b c} D. Dubois and H. Prade (1980) Fuzzy Sets and Systems . Academic Press, New York
3. ↑ Bandemer, H. and Näther, W. (1992): Fuzzy Data Analysis , Kluwer Dordrecht