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IMAT 3406 Fuzzy Logic Weeks 5 and 6

IMAT 3406 Fuzzy Logic Weeks 5 and 6. Determining membership functions and rules Designing a fuzzy system. If you attend lectures and labs and if you work hard you get good mark If you don’t attend lectures and labs and if you don’t work hard you can’t get good mark.

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IMAT 3406 Fuzzy Logic Weeks 5 and 6

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  1. IMAT 3406 Fuzzy Logic Weeks 5 and 6 Determining membership functions and rules Designing a fuzzy system If you attend lectures and labs and if you work hard you get good markIf you don’t attend lectures and labs and if you don’t work hard you can’t get good mark

  2. Determining Membership Functions • Discussion with an expert or group of experts • Exemplification builds a membership function from a number of samples • Direct rating presents randomly selected members of the fuzzy set where some measure is available for that member • Polling a number of experts is involved. For the question “Do you agree that a 5-hour study a day is enough to get a good mark?”, membership value = ratio of yes responses to all responses

  3. Machine learning methods (e.g., clustering) y Crisp/Hard Clustering A If dA(p)<dB(p) p A If dB(p)<dA(p) p B B dA(p) dB(p) p “d” is any metric (e.g. Euclidean distance) x

  4. clustering y y 1 B A yB yA yC C x 0 1 xC xB xA x

  5. Gaussian membership functions µ A B C X Y

  6. IF x AND y THEN p A O = [1 0 0] IF x AND y THEN p B O = [0 1 0] IF x AND y THEN p C O = [0 0 1]

  7. Other methods: • 100s of methods for clustering • Genetic algorithms • …. • Experts + machine learning methods • Number of membership functions and rules • # of membership functions for each rule # of parameters (variables) • # of rules: no established method • Optimum number of rules by means of SVD (studied by H. Seker et al., “An Intelligent Hybrid Neuro-Fuzzy Rule-Based System for Prognostic Decision Making in Prostate Cancer Patients”, Proc. of the 4th Annual IEEE Engineering in Medicine and Biology Society Special Topic Conference on Information Technology Applications in Biomedicine, April 2003)

  8. Rule 1 Rule r Rule 2 Fuzzy System Design (Fuzzy) x is A1 w1 y is B1 X Fuzzy Composition (Fuzzy) x is A2 w2 y is B2 Defuzzifier (Fuzzy) y x is Ar wr y is Br (Fuzzy) (Crisp) A fuzzy system is a computer system that uses fuzzy sets in either the antecedent and/or the consequent of fuzzy if-then rules. Main components of the Fuzzy System: (a) The ‘base’ fuzzy sets that describe the problem (b) A set of sensible “if-then rules” (c) Rule composition (d) Defuzzification

  9. The Mamdani Model for two rules IF x is A1and y is B1THEN z is C1 IF x is A2 and y is B2THEN z is C2

  10. The Takagi-Sugeno Model for two rules IF x is A1and y is B1THEN z1= p1x + q1y + r1 IF x is A2and y is B2THEN z2= p2x + q2y + r2

  11. Defuzzification • Without the defuzzification phase, the final output of the FIS is a fuzzy set • Defuzzification is used to obtain a crisp output from the FIS • Methods for Defuzzification • The Centre of Area (COA) • The Mean of Maximum (MOM) • Bisector of Area (BOA) • Smallest of Maximum (SOM) • Largest of Maximum (LOM) zBOA zLOM zSOM zMOM zCOA

  12. Rules to model the relationship between salary, period of employment and mortgage: Rule 1 IF salary is low AND period of employment is low THEN mortgage is low Rule 2 IF salary is high AND period of employment is high THEN mortgage is high Example: “salary, period of employment and mortgage”

  13. With the Mamdani Model Salary Period of Employment Mortgage (M) THEN IF AND min 1 0.95 0.95 50K Example (cont.) : “salary, period of employment and mortgage” 0.12 0.1 0.1 150K 30K 15 years

  14. Adaptive Fuzzy System (AFS)introduction FS output input AFS

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