Fuzzy Sets Introduction/Overview Material for these slides obtained from: Modern Information Retrieval by Ricardo Baeza-Yates and Berthier Ribeiro-Neto Data Mining Introductory and Advanced Topics by Margaret H. Dunham Introduction to “Type-2 Fuzzy Logic by Jenny Carter

CSE 5331/7331 F07 2 Fuzzy Sets and Logic 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: T = {x | x is a person and x is tall} Let f(x) be the probability that x is tall Here f is the membership function

CSE 5331/7331 F07 3 Fuzzy Sets

CSE 5331/7331 F07 4 Fuzzy Set Theory 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 Let A and B be two fuzzy subsets of U. Also, let ¬A be the complement of A. Then,  (¬A,u) = 1 -  (A,u)  (A  B,u) = max(  (A,u),  (B,u))  (A  B,u) = min(  (A,u),  (B,u))

CSE 5331/7331 F07 5 The world is imprecise. Mathematical and Statistical techniques often unsatisfactory. 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

CSE 5331/7331 F07 6 Fuzzy Logic then... is particularly good at handling uncertainty, vagueness and imprecision. especially useful where a problem can be described linguistically (using words). Applications include: robotics washing machine control nuclear reactors focusing a camcorder information retrieval train scheduling © Jenny Carter

CSE 5331/7331 F07 7 Crisp Sets if you are tall and can run fast you should consider basketball Figure 1: A crisp way of modeling tallness © Jenny Carter

CSE 5331/7331 F07 8 Crisp Sets Figure 2: The crisp version of short © Jenny Carter

CSE 5331/7331 F07 9 Crisp Sets Different heights have same ‘tallness’ © Jenny Carter

CSE 5331/7331 F07 10 Fuzzy Sets The shape you see is known as the membership function © Jenny Carter

CSE 5331/7331 F07 11 Fuzzy Sets Now we have added some possible values on the height - axis © Jenny Carter

CSE 5331/7331 F07 12 Fuzzy Sets Shows two membership functions: ‘tall’ and ‘short’ © Jenny Carter

CSE 5331/7331 F07 13 Notation For any fuzzy set, A, the function µ A represents the membership function for which µ A (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

CSE 5331/7331 F07 14 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 µ n /x n Or where x 1, x 2....x n are members of the set A and µ 1, µ 2,...., µ n are their degrees of membership. A continuous fuzzy set A is written as: © Jenny Carter

CSE 5331/7331 F07 15 Fuzzy Sets 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 µ A (x) is known as fuzzification. A fuzzy set is either continuous or discrete. Graphical representation of membership functions is very useful. © Jenny Carter

CSE 5331/7331 F07 16 Fuzzy Sets - Example “numbers close to 1” © Jenny Carter

CSE 5331/7331 F07 17 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

CSE 5331/7331 F07 18 Imprecision Words are used to capture imprecise notions, loose concepts or perceptions. © Jenny Carter

CSE 5331/7331 F07 19 Rules 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. 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

CSE 5331/7331 F07 20 Example - Dinner for two ( p2-21 of FL toolbox user guide) Rule 2If service is good, then tip is average Rule 3If service is excellent or food is delicious, then tip is generous The inputs are crisp (non- fuzzy) numbers limited to a specific range All rules are evaluated in parallel using fuzzy reasoning The results of the rules are combined and distilled (de-fuzzyfied) The result is a crisp (non- fuzzy) number Output Tip (5-25%) Dinner for two: this is a 2 input, 1 output, 3 rule system Input 1 Service (0-10) Input 2 Food (0-10) Rule 1If service is poor or food is rancid, then tip is cheap © Jenny Carter

CSE 5331/7331 F07 21 Dinner for two 1. Fuzzify the input: 2. Apply Fuzzy operator © Jenny Carter

CSE 5331/7331 F07 22 Dinner for two 3. Apply implication method © Jenny Carter

CSE 5331/7331 F07 23 Dinner for two 4. Aggregat e all outputs © Jenny Carter

CSE 5331/7331 F07 24 Dinner for two 5. defuzzify Various approaches e.g. centre of area mean of max © Jenny Carter