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Fuzzy Rule-based Models

Fuzzy Rule-based Models. *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997. A Classification of Fuzzy Rule-based models for function approximation. Fuzzy Rule-based Models. Additive Rule Models. NonAdditive Rule Models. TSK Model (Takagi-Sugeno-Kang).

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Fuzzy Rule-based Models

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  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 Rule-based Models Additive Rule Models NonAdditive Rule Models TSK Model (Takagi-Sugeno-Kang) Tsukamoto Model (Tsukamoto) Mamdani Model (Mamdani) Standard Additive Model(Kosko) *Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999

  3. Mamdani model • 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 x1 is Ai1 and … and xn is Ain then y is Ci, where i=1,2 ….M, M is the number of the fuzzy rules • Uses clipping inference • Uses max aggregation *Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999

  4. Why TSK? • 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 *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  5. TSK Fuzzy Rule • 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. *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  6. First order TSK Fuzzy Model • f(x,y) is a first order polynomial Example: a two-input one-output TSK IF x is Aj and y is Bk then zi= px+qy+r The degree the input matches ith rule is typically computed using min operator: wi = min(mAj(x), mBk(y)) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

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

  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 • 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. *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

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

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

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

  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 • 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 *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  15. Summary: TSK Fuzzy Model • 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. *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

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

  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 • 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 *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  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) • 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 *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  21. Standard Additive Model (SAM) IF x is Ai and y is Bi then z is Ci then for crisp inputs x=x0 and y=y0 Z* = Centroid(SimAi(x0) mBi(y0) mCi(z) ) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

  22. Standard Additive Model (SAM) Z* = Centroid(SimAi(x0) mBi(y0) mCi(z) ) then Z* can be represented Z* = Si (mAi(x0) mBi(y0) )  Areai gi/ Si (mAi(x) mBi(y) )  Areai Where Areai =  mCi(z) dz, {Area of Ci} gi =  z x mCi(z) dz /  mCi(z) dz {Centroid of Ci} *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

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

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