Introduction to fuzzy sets Andrea Bonarini Artificial Intelligence and Robotics Lab Department of Electronics and Information Politecnico di Milano E-mail: bonarini@elet.polimi.it URL:http://www.dei.polimi.it/people/bonarini
A bit of history Fuzzy sets have been defined by Lotfi Zadeh in 965, as a tool to model approimate concepts In 972 the first linguistic fuzzy controller is implemented In the Eighties boom of fuzzy controllers first in Japan, then USA and Europe In the Nineties applications in many fields: fuzzy data bases, fuzzy decision making, fuzzy clustering, fuzzy learning classifier systems, neuro-fuzzy systems Massive diffusion of fuzzy controllers in end-user goods Now, fuzzy systems are the kernel of many intelligent devices Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 2 of 33
Main characteristics Fuzzy sets: precise model in a finite number of points, smooth transition approimation among them. E.g.: control of a power plant. We can define what to do in standard operating conditions e.g., steam temperature =20, steam pressure 2 atm, and when in critical situations e.g., steam temperature= 00, and design a model that smoothly goes from one point to the other. Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 3 of 33
What is a fuzzy set? A fuzzy set is a set whose membership function may range on the interval [0,]. Males Adults Engineers Children Crisp sets Fuzzy sets Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 4 of 33
Fuzzy membership functions A membership function defines a set Defines the degree of membership of an element to the set : U [0, ] A 35 years old person is: Young with membership 0,3 Young not very Young Old Old with membership 0,2 not very Young with membership 0,6 0,5 0 35 Age Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 5 of 33
How to define MFs. Select a variable 2. Define the range of the variable 3. Identify labels 4. For each label identify characteristic points 5. Identify function shapes 6. Check Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 6 of 33
Let s try to define some MFs First of all, the variable Range of the variable Labels Characteristic points Function shape Distance [0..0] Close, Medium, Far 0, ma, where MF=, Linear Close Medium Far 0 2 4 6 0 Distance [m] Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 7 of 33
MFs and concepts MFs define fuzzy sets Labels denote fuzzy sets Fuzzy sets can be considered as conceptual representations Symbol grounding: reason in terms of concepts and ground them on objective reality Hot! T=00 C Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 8 of 33
Some conceptual differences A fuzzy set with only one member with the maimum membership a- α a a + β A fuzzy set with a set of members with the maimum membership a- α a b b -β Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 9 of 33
Some conceptual differences A fuzzy set with only one member a A fuzzy set with all the members having the maimum membership a b Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 0 of 33
Some variations Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - of 33
Fuzzy sets on ordinal scales 0 - no education - elementary school 2 - high school 3 - two year college 4 - bachelor's degree 5 - masters's degree 6 - doctoral degree poorly educated highly educated very highly educated 0 2 3 4 5 6 Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 2 of 33
Fuzzy sets and intervals very low low medium high very high Smoother transition in labeling a value T very low low medium high very high T Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 3 of 33
Frame of cognition Fuzzy sets covering the universe of discourse very low low medium high very high T Each fuzzy set is a granule Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 4 of 33
Properties of a frame of cognition Coverage Each element of the universe of discourse is assigned to at least a granule with membership > 0 Unimodality of fuzzy sets There is a unique set of values for each granule with maimum membership Fuzzy partition: for each value of the universe of discourse the sum of membership degrees to the corresponding granules is Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 5 of 33
Robustness Let s consider a punctual error as the sum of the errors in interpretation of a point by fuzzy sets due to imprecise measurements, noise, e â = â - a' +... + n â - n a' and the integral error, as the integral of ea over the range of a e i = ea da It can be demonstrated that the integral error of a fuzzy partition is smaller than that of a boolean partition, and that it is minimum w.r.t. any other frame of cognition. Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 6 of 33
α-cuts The α-cut of a fuzzy set is the crisp set of the values of such that X α α X= { α} α α X X Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 7 of 33
Support of a fuzzy set The crisp set of values of X such that f > 0 is the support of the fuzzy set f on the universe X support X Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 8 of 33
Height of a fuzzy set The height ha of a fuzzy set A on the universe X is the highest membership degree of an element of X to the fuzzy set Heigth X A fuzzy set f is normal iff h f = Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 9 of 33
Conve fuzzy sets A fuzzy set is conve iff λ + -λ 2 min [, 2 ] for any, 2 in R and any λ belonging to [0,] X X 2 X X 2 Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 20 of 33
Standard operators on fuzzy sets Complement f =- f Union f f 2 = ma f, f2 Intersection f f 2 = min f, f2 Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 2 of 33
Eamples of operator application Complement Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 22 of 33
Union Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 23 of 33
Union Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 24 of 33
Fundamental property of standard operators Using the standard operators the maimum error is the one we have on the operand s MFs X Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 25 of 33
Complement Aioms: c : [0,] -> [0,] c A = A. c0=; c=0 boundary conditions 2. For all a and b in [0,], if a < b then ca cb monotonicity 3. c is a continuous function 4. c is involutive, i.e., cca=a for all a in [0,] Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 26 of 33
Intersection and T-norms A B = i[ A, B ] Aioms:. i[a, ]=a boundary conditions 2. d b implies ia,d ia,b monotonicity 3. ib,a = ia,b commutativity 4. iia,b,d = ia,ib,d associativity 5. i is continuous 6. a ia,a sub-idempotency 7. a < a 2 and b < b 2 implies that i a,b <ia 2,b 2 strict monotonicity Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 27 of 33
Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 28 of 33 T-norms: eamples ],, ma[ α b a ab for α= we have ab for α=0 we have mina, b, ma0,, 2.5 t t B A B A B A B A B A B A + = + =
Union and T-conorms S-norms A B = u[ A, B ] Aioms:. u[a, 0]=a boundary conditions 2. b d implies ua,b ua,d monotonicity 3. ua,b = ub,a commutativity 4. ua,ub,d = uua,b,d associativity 5. u is continuous 6. ua,a a super-idempotency 7. a < a 2 e b < b 2 implies that ua,b <ua 2,b 2 strict monotonicity Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 29 of 33
Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 30 of 33 T-conorms: eamples,, ma, min,, min{,, 3 / s s s p s B A B A B A B A B A B A B A p p B p A B A + = = + = + = +
Aggregation Aioms: A = h[ A,..., An ]. h[0,..., 0]=0, h[,..., ]= boundary conditions 2. monotonicity 3. h is continuous 4. ha,..,a = a idempotency 5. simmetricity Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 3 of 33
Properties of aggregation min a,..., a n ha,..., a n ma a,..., a n Eample of aggregation operator: generalized average ha,..., a n = a α + + a nα /α / n Introduction to Fuzzy Sets A. Bonarini bonarini@elet.polimi.it - 32 of 33