Fuzzy crowd

A fuzzy set (also fuzzy set , English fuzzy set ) is a set whose elements are not necessary with certainty, but only gradually to the crowd.

So z. For example, the “amount of higher earners in Germany”, the “amount of young people in Berlin” or the “amount of ripe apples on a tree” are better described by a fuzzy set than by a (sharp) crowd with a classic yes-no - Affiliation of the elements.

The term fuzzy set was coined in 1965 by Lotfi Zadeh (1921–2017), but has conceptual forerunners well into antiquity (e.g. the Sorites problem ), but also in multi-valued logic . Fuzzy sets are basic elements of fuzzy logic and the fuzzy controller and have been introduced there in partly special terminology.

Definitions

Be a lot. It is the basic set (also: universe ) on which the investigations are carried out. A classic example of a universe is the set of real numbers . During a classical (sharp) amount by their indicator function will be described, a fuzzy set is to so-called due to their membership function (engl. Membership function ) characterized: ${\displaystyle U}$ ${\displaystyle A\subset U}$ ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle m_{A}}$

${\displaystyle m_{A}\quad |\quad U\to [0,1]}$.

${\displaystyle m_{A}(x),x\in U}$Thus, lies between 0 and 1 and is interpreted as the degree of acceptance ( possibility or truth that) to hear ${\displaystyle x}$ ${\displaystyle A}$ . An important role is played by the so-called cuts , (Engl. -Cuts) ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

${\displaystyle A_{\alpha }=\{x\in U:m_{A}(x)\geq \alpha \}\quad \alpha \in (0,1]}$,

d. H. the sharp sets of all elements that have a minimum membership of to of. ${\displaystyle \alpha }$ ${\displaystyle A}$

Operations with fuzzy sets

The intersection and the union of two fuzzy sets is usually defined by ${\displaystyle A\cap B}$ ${\displaystyle A\cup B}$ ${\displaystyle A,B\subset U}$

${\displaystyle m_{A\cap B}(x)=\min[m_{A}(x),m_{B}(x)],\quad x\in U}$,

${\displaystyle m_{A\cup B}(x)=\max[m_{A}(x),m_{B}(x)],\quad x\in U}$.

Instead of and , however, other T-standards or T-Conormen can be used, see e.g. B. The complement formation to happens mostly according to ${\displaystyle \min }$ ${\displaystyle \max }$ ${\displaystyle {\bar {A}}}$ ${\displaystyle A}$

${\displaystyle m_{\bar {A}}(x)=1-m_{A}(x),\quad x\in U}$,

but can also be designed differently, e.g. B. by the so-called complement, which goes back to Sugeno. In contrast to sharp sets, here and are not necessarily disjoint and, when combined, do not necessarily give the universe, i.e. H. ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle {\bar {A}}}$

${\displaystyle A\cap {\bar {A}}\neq \emptyset \quad ,\quad A\cup {\bar {A}}\neq U}$,

In the case of fuzzy sets, the rule of the excluded third party does not apply .

Fuzzy numbers

Fuzzy numbers are special fuzzy sets. The universe is the set of real numbers. is called a fuzzy number if there is exactly one where the membership function takes the value 1, i.e. H. ${\displaystyle U}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle a\in \mathbb {R} }$

${\displaystyle \exists a\in \mathbb {R} :m_{A}(a)=1\quad \forall x\neq a:m_{A}(x)<1}$.

Then it can be interpreted as the fuzzy set that describes the expression "approximately " . If z. For example, if the room temperature is "around 20 degrees Celsius", the amount of possible room temperatures could be modeled by a fuzzy number whose membership function is one at 20 degrees Celsius and which drops to zero on the left or right. The simplest form of a fuzzy number is the triangular fuzzy number , the membership function of which looks like an equilateral triangle with the tip at , i.e. H. ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle (a,\Delta )}$ ${\displaystyle x=a}$

${\displaystyle m_{A}(x)=\max \left\{(1-{\frac {|x-a|}{\Delta }}),0\right\}}$.

It is the so-called spreading parameters, d. H. is only greater than zero within the interval . In the case of a fuzzy controller , the necessary fuzzy quantities are mostly modeled using triangular fuzzy numbers. ${\displaystyle \Delta }$ ${\displaystyle m_{A}(x)}$ ${\displaystyle (a-\Delta ,a+\Delta )}$

Addition of fuzzy numbers

Illustratively, “about 3” plus “about 4” should equal “about 7”, but the question arises as to what exactly is meant by this. With the help of the very general expansion principle (see e.g.) the sum of two fuzzy numbers is obtained by ${\displaystyle A\oplus B}$

${\displaystyle m_{A\oplus B}(x)=\sup _{y_{1},y_{2}:y_{1}+y_{2}=x}\min[m_{A}(y_{1}),m_{A}(y_{2})]}$.

For sharp and this formula is reduced to the Minkowski addition . For two triangular fuzzy numbers, z. B. very simple ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle (a,\Delta _{1})\oplus (b,\Delta _{2})=(a+b,\Delta _{1}+\Delta _{2})}$.

additional

• In addition to addition, other algebraic operations for fuzzy numbers such as subtraction, multiplication, division, etc. a. are introduced, see e.g. B.
• An important generalization of fuzzy sets are intuitionistic fuzzy sets .
• So-called probabilistic fuzzy sets are fuzzy sets where the membership values are random variables , see
• With so-called type 2 fuzzy sets , the membership values ​​are not real numbers between zero and one, but rather fuzzy values ​​such as B. "high" or "low", see e.g. B.
• see also random fuzzy set , fuzzy random variable , extension principle

literature

• D. Dubois, H. Prade: Fuzzy Sets and Systems . Academic Press, New York 1980.
• GJ Klir, Bo Yuan: Fuzzy sets and fuzzy logic: theory and applications . Prentice Hall, 1995.
• H.-J. Zimmermann : Fuzzy set theory - and its applications . 4th ed. Kluwer, 2001.
• H. Bandemer , S. Gottwald : Fuzzy sets, fuzzy logic, fuzzy methods: With applications. Wiley, Chichester 1995.

Individual evidence

1. ↑ LAZadeh : Fuzzy sets . In: Information and Control , 8, 1965, pp. 338-353, doi: 10.1016 / S0019-9958 (65) 90241-X
2. ↑ EP Klement, R. Mesiar, E. Pap: Triangular Norms . Kluwer Dordrecht 2000.
3. ↑ M. Sugeno: Theory of fuzzy integrals and its applications . Ph.D. thesis. Tokyo Institute of Technology, Tokyo 1974.
4. ↑ H. Bandemer , S. Gottwald : Introduction to Fuzzy Methods . 4th revised and expanded edition. Akademieverlag, Berlin 1993
5. ↑ D. Dubois, H. Prade: Fuzzy Sets and Systems . Academic Press, New York 1980
6. ↑ K. Hirota: Concepts of probabilistic sets . In: Fuzzy Sets and Systems , 5, 1981, pp. 31-46
7. ↑ M. Mizumoto, K. Tanaka: Some properties of fuzzy sets of type-2 . In: Information and Control , 30, 1976, pp. 312-340