Jan Jantzen www.inference.dk 2013 A set is a collection of objects A special kind of set Fuzzy Sets Jan Jantzen www.inference.dk 2013

Summary: A set ... This object is not a member of the set This object is a member of the set A classical set has a sharp boundary

... and a fuzzy set This object is not a member of the set This object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1. A fuzzy set has a graded boundary

Example: High and low pressures

Example: "Find books from around 1980" This could include 1978, 1979, 1980, 1981, and 1982

Maybe even 41 or 42 could be all right?

Example: A fuzzy washing machine If you fill it with only a few clothes, it will use shorter time and thus save electricity and water. There is a computer inside that makes decisions depending on how full the machine is and other information from sensors. Samsung J1045AV capacity 7 kg

A rule (implication) IF the machine is full THEN wash long time Condition Action. The internal computer is able to execute an if—then rule even when the condition is only partially fulfilled.

IF the machine is full … Example: Load = 3.5 kg clothes Classical set true false Example: Load = 3.5 kg clothes Classical set Linear fuzzy set Nonlinear fuzzy set Three examples of functions that define ‘full’. The horizontal axis is the weight of the clothes, and the vertical axis is the degree of truth of the statement ‘the machine is full’.

… THEN wash long time long time could be t = 120 minutes The duration depends on the washing program that the user selects.

Decision Making (inference) Rule. IF the machine is full THEN wash long time Measurement. Load = 3.5 kg Conclusion. Full(Load) × t = 0.5 × 120 = 60 mins The machine is only half full, so it washes half the time. Analogy The dots mean 'therefore'

A rule base with four rules IF machine is full AND clothes are dirty THEN wash long time IF machine is full AND clothes are not dirty THEN wash medium time IF machine is not full AND clothes are dirty THEN wash medium time IF machine is not full AND clothes are not dirty THEN wash short time There are two inputs that are combined with a logical 'and'.

Theory of Fuzzy Sets

Lotfi Zadeh’s Challenge Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh 1965).

{Live dinosaurs in British Museum} =  Sets The set of positive integers The set of belonging to for which {Live dinosaurs in British Museum} =  The empty set

Fuzzy Sets {nice days} {adults} Membership function Much greater than

Tall Persons Degree of membership 150 160 170 180 190 200 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1 Height [cm] Membership fuzzy crisp Membership function Universe

Fuzzy (http://www.m-w.com) adjective Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague Unfortunately, they all carry a negative connotation.

'Around noon' Triangular Trapezoidal Smooth versions of the same sets.

The 4 Seasons 1 Membership 0.5 Time of the year Spring Summer Autumn Seasons have overlap; the transition is fuzzy. 0.5 1 Time of the year Membership Spring Summer Autumn Winter we are here

Summary A set A fuzzy set

Operations on Fuzzy Sets

Set Operations Union Intersection Negation Classical Fuzzy

Fuzzy Set Operations A B Union Intersection Negation

Example: Age The negation of 'very young' Primary term Primary term The square root of Old The square of Young

Operations Here is a whole vocabulary of seven words. Each operates on a membership function and returns a membership function. They can be combined serially, one after the other, and the result will be a membership function.

Cartesian Product The AND composition of all possible combinations of memberships from A and B The curves correspond to a cut by a horizontal plane at different levels

Example: Donald Duck's family Suppose, nephew Huey resembles nephew Dewey nephew Huey resembles nephew Louie nephew Dewey resembles uncle Donald nephew Louie resembles uncle Donald Question: How much does Huey resemble Donald?

Solution: Fuzzy Composition of Relations Dewey Louie Huey = Donald 0.8 0.9 0.5 0.6 Relation ? ? 0.8 0.5 Huey Dewey Donald Huey Louie Donald 0.9 0.6 ?

If—Then Rules If x is Neg then y is Neg If x is Pos then y is Pos approximately equal

Key Concepts Universe Membership function Fuzzy variables Set operations Fuzzy relations All of the above are parallels to classical set theory

Application examples Database and WWW searches Matching of buyers and sellers Rule bases in expert systems