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 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)

Mamdani model *Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 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

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

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.

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))

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

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

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 –If X is medium then Y = -0.5X +4. –If X is large then Y = X-2.

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.

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.

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.

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

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

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.

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.

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

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

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

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

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) )

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 }

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!