Fuzzy Sets IntroductionOverview Material for these slides obtained

Fuzzy Sets Introduction/Overview Material for these slides obtained from: n. Modern Information Retrieval by Ricardo Baeza-Yates and Berthier Ribeiro-Neto http: //www. sims. berkeley. edu/~hearst/irbook/ n. Data Mining Introductory and Advanced Topics by Margaret H. Dunham http: //www. engr. smu. edu/~mhd/book n. Introduction to “Type-2 Fuzzy Logic by Jenny Carter

Fuzzy Sets and Logic n n Fuzzy Set: Set membership function is a real valued function with output in the range [0, 1]. f(x): Probability x is in F. 1 -f(x): Probability x is not in F. EX: n T = {x | x is a person and x is tall} n Let f(x) be the probability that x is tall n Here f is the membership function 2

Fuzzy Sets 3

Fuzzy Set Theory n n A fuzzy subset A of U is characterized by a membership function (A, u) : U [0, 1] which associates with each element u of U a number (u) in the interval [0, 1] Definition n Let A and B be two fuzzy subsets of U. Also, let ¬A be the complement of A. Then, n n n (¬A, u) = 1 - (A, u) (A B, u) = max( (A, u), (B, u)) (A B, u) = min( (A, u), (B, u)) 4

The world is imprecise. n Mathematical and Statistical techniques often unsatisfactory. n n n Experts make decisions with imprecise data in an uncertain world. They work with knowledge that is rarely defined mathematically or algorithmically but uses vague terminology with words. Fuzzy logic is able to use vagueness to achieve a precise answer. By considering shades of grey and all factors simultaneously, you get a better answer, one that is more suited to the situation. © Jenny Carter 5

Fuzzy Logic then. . . n n n is particularly good at handling uncertainty, vagueness and imprecision. especially useful where a problem can be described linguistically (using words). Applications include: n n n robotics washing machine control nuclear reactors focusing a camcorder information retrieval train scheduling © Jenny Carter 6

Crisp Sets n if you are tall and can run fast you should consider basketball Figure 1: A crisp way of modeling tallness © Jenny Carter 7

Crisp Sets Figure 2: The crisp version of short © Jenny Carter 8

Crisp Sets n Different heights have same ‘tallness’ © Jenny Carter 9

Fuzzy Sets n The shape you see is known as the membership function © Jenny Carter 10

Fuzzy Sets n Now we have added some possible values on the height - axis © Jenny Carter 11

Fuzzy Sets Shows two membership functions: ‘tall’ and ‘short’ © Jenny Carter 12

Notation n For any fuzzy set, A, the function µA represents the membership function for which µA(x) indicates the degree of membership of x (of the universal set X) in set A. It is usually expressed as a number between 0 and 1: © Jenny Carter 13

Notation For the member, x, of a discrete set with membership µ we use the notation µ/x. In other words, x is a member of the set to degree µ. Discrete sets are written as: A = µ 1/x 1 + µ 2/x 2 +. . + µn/xn Or where x 1, x 2. . xn are members of the set A and µ 1, µ 2, . . , µn are their degrees of membership. A continuous fuzzy set A is written as: © Jenny Carter 14

Fuzzy Sets n n n The members of a fuzzy set are members to some degree, known as a membership grade or degree of membership. The membership grade is the degree of belonging to the fuzzy set. The larger the number (in [0, 1]) the more the degree of belonging. (N. B. This is not a probability) The translation from x to µA(x) is known as fuzzification. A fuzzy set is either continuous or discrete. Graphical representation of membership functions is very useful. © Jenny Carter 15

Fuzzy Sets - Example “numbers close to 1” © Jenny Carter 16

Fuzzy Sets - Example Again, notice the overlapping of the sets reflecting the real world more accurately than if we were using a traditional approach. © Jenny Carter 17

Imprecision Words are used to capture imprecise notions, loose concepts or perceptions. © Jenny

Rules n Rules often of the form: IF x is A THEN y is B where A and B are fuzzy sets defined on the universes of discourse X and Y respectively. n n if pressure is high then volume is small; if a tomato is red then a tomato is ripe. where high, small, red and ripe are fuzzy sets. © Jenny Carter 19

Example - Dinner for two (p 2 -21 of FL toolbox user guide) Dinner for two: this is a 2 input, 1 output, 3 rule system Rule 1 Input 1 Service (0 -10) Rule 2 If service is poor or food is rancid, then tip is cheap Output If service is good, then tip is average Tip (5 -25%) Input 2 Food (0 -10) Rule 3 The inputs are crisp (nonfuzzy) numbers limited to a specific range If service is excellent or food is delicious, then tip is generous All rules are evaluated in parallel using fuzzy reasoning © Jenny Carter The results of the rules are combined and distilled (de-fuzzyfied) The result is a crisp (nonfuzzy) number 20