1. Fuzzy Rule-based Models *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

2. A Classification of Fuzzy Rule-based models for function approximation *Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999 Fuzzy Rule-based Models NonAdditive Rule Models Additive Rule Models Mamdani Model (Mamdani) TSK Model (Takagi-Sugeno-Kang) Standard Additive Model (Kosko) Tsukamoto Model (Tsukamoto)

3. Mamdani model *Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999 -Named after E.H. Mamdani who developed first fuzzy controller. -The inputs may be crisp or fuzzy numbers - Uses rules whose consequent is a fuzzy set, i.e. If x 1 is A i1 and … and x n is A in then y is C i, where i=1,2 ….M, M is the number of the fuzzy rules - Uses clipping inference - Uses max aggregation

4. Why TSK? *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Main motivation –to reduce the number of rules required by the Mamdani model For complex and high-dimensional problems develop a systematic approach to generate fuzzy rules from a given input- output data set TSK model replaces the fuzzy consequent, (then part), of Mamdani rule with function (equation) of the input variables

5. TSK Fuzzy Rule *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 If x is A and y is B then z = f(x,y) –Where A and B are fuzzy sets in the antecedent, and –Z = f(x,y) is a crisp function in the consequence, e.g f(x,y)=ax+by+c. Usually f(x,y) is a polynomial in the input variables x and y, but it can be any function describe the output of the model within the fuzzy region specified by the antecedence of the rule.

6. First order TSK Fuzzy Model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 f(x,y) is a first order polynomial Example: a two-input one-output TSK IF x is A j and y is B k then z i = px+qy+r The degree the input matches i th rule is typically computed using min operator: w i = min(  A j (x),  B k (y))

7. First-Order TSK Fuzzy Model (Cont) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Each rule has a crisp output Overall output is obtained via weighted average (reduce computation time of defuzzification required in a Mamdani model) z =  i w i z i /  i w i Where W i is matching degree of rule R i (result of the if … part evaluation) To further reduce computation, weighted sum may be used, I.e. z =  i w i z i

8. First-Order: TSK Fuzzy Model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

9. Example #1: Single-input *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 A single-input TSK fuzzy model can be expresses as –If X is small then Y = 0.1 X +6.4. –If X is medium then Y = -0.5X +4. –If X is large then Y = X-2.

10. Example #1: Non fuzzy rule set *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 If “small”, “medium.” and “large” are non fuzzy sets, then the overall input-output curve is piecewise linear.

11. Example #1: Fuzzy rule set *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 If “small”, “medium,” and “large” are fuzzy sets (smooth membership functions), then the overall input-output curve is a smooth one.

12. Example #2 : Two-input *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 A two-input TSK fuzzy model with 4 rules can be expresses as –If X is small and Y is small then Z = -X +Y +1. –If X is small and Y is large then Z = -Y +3. –If X is large and Y is small then Z = -X+3. –If X is large and Y is large then Z = X+Y+2.

13. Example #2 : Two-input *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

14. Zero-order TSK Fuzzy Model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 When f is constant, we have a zero-order TSK fuzzy model ( a special case of the Mamdani fuzzy inference system which each rule’s consequent is specified by a fuzzy singleton or a pre defuzzified consequent) Minimum computation time

15. Summary: TSK Fuzzy Model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Overall output via either weighted average or weithted sum is always crisp Without the time-consuming defuzzification operation, the TSK (Sugeno) fuzzy model is by far the most popular candidate for sample-data-based fuzzy modeling. Can describe a highly non-linear system using a small number of rules Very well suited for adaptive learning.

16. Tsukamoto Fuzzy Models *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 The consequent of each fuzzy if-then rule is represented by a fuzzy set with monotonical MF –As a result, the inferred output of each rule is defined as a crisp value induced by the rules’ firing strength. The overall output is taken as the weighted average of each rule’s output.

17. Tsukamoto Fuzzy Models *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

18. Example: Single-input Tsukamoto fuzzy model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 A single-input Tsukamoto fuzzy model can be expresses as –If X is small then Y is C1 –If X is medium then Y is C2 –If X is large then Y is C3

19. Example: Single-input Tsukamoto fuzzy model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

20. Standard Additive Model (SAM) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Introduced by Bart Kosko in 1996 Efficient to compute Similar to Mamdani model, but –Assumes the inputs are crisp –Uses the scaling inference method (prod.] –Uses addition to combine the conclusions of rules –Uses the centroid defuzzification technique

21. Standard Additive Model (SAM) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 IF x is A i and y is B i then z is C i then for crisp inputs x=x 0 and y=y 0 Z* = Centroid(  i  A i (x 0 )   B i (y 0 )   C i (z) )

22. Standard Additive Model (SAM) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Z* = Centroid(  i  A i (x 0 )   B i (y 0 )   C i (z) ) then Z* can be represented Z* =  i (  A i (x 0 )   B i (y 0 ) )  Area i  g i /  i (  A i (x)   B i (y) )  Area i Where Area i =   C i (z) dz, {Area of C i } g i =  z x  C i (z) dz /   C i (z) dz{Centroid of C i }

23. Standard Additive Model (SAM) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997 Main Advantage is the efficiency of its computation, i.e. –Both Area i and g i can be pre computed!