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1 (Lecture Note #14) (Lecture Note #14)

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2 Outline Fuzzy Logic and Intelligent Systems Progress of Fuzzy Logic Generalized Modus Ponens Fuzzy Reasoning Triangular Norms Triangular Conorms Theoretical Foundations of Fuzzy Inference

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3 Saturn Fuzzy Logic Transmission Saturn automobiles smart transmission When the car is moving uphill or downhill Employs shift stabilization Fuzzy logic control is used to avoid hunting, that is frequent shifting of gears Shifting decisions are made by weighing many input variables at once and using fuzzy if-then rules to generate an output control signal

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4 Fuzzy Logic and Intelligent Systems Objective: Develop a cost-effective approximate solution to a problem Approach: Exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low cost Preciseness Utility (Usefulness of system) Cost of system Low cost, high usefulness

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5 Fuzzy Logic Applications Diagnosis Financial Analysis and Prediction Robotics Data Compression and Pattern Recognition Consumer electronics –Typical KB size: rules –Typical KB number: 1 Automobile/Transportation system –Typical KB size: rules –Typical KB number: 1-3 Locomotive, Subways, Aircraft engines, Helicopters –Typical KB size: rules –Typical KB number: supervisory (hierarchical)

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6 Progress of Fuzzy Logic 1965, Prof. Loft A. Zadeh at UCB developed fuzzy set theory and fuzzy logic 1974, Fuzzy logic controller for a steam engine (Prof. Mamdani, London University) 1980, Control of Cement-Kiln with monitor capability (Smidth, Denmark) 1987, Automatic train operation for Sendai subway (Hitachi) 1988, Stock trading expert system (Yamichi security) 1989, Laboratory for International Fuzzy Engineering , 389 patents in U.S. regarding fuzzy systems

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7 Fuzzy Logic is Not … Is not a clever disguise of probability theory The behavior of a fuzzy system is not fuzzy (it is deterministic) The founder of fuzzy logic (Prof. Zadeh) is not a mathematician. Fuzzy logic is not to replace conventional techniques.

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8 Generalized Modus Ponens In fuzzy logic –the truth value of a statement becomes a matter of degrees –Reasoning in fuzzy logic is based on generalized modus ponens Modus Ponens Generalized Modus Ponens – Given A B Given A B – A A – – Deduce B Deduce B – where A is a fuzzy set that partially matches A Example –If a person is self-confident then he/she has a happy life –Jack is somewhat self-confident –What can we conclude using conventional logic? –What can we conclude using generalized modus ponens? –=> Jack is somewhat happy

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9 Fuzzy Reasoning Fuzzy Rules: If x is A 1 and y is B 1 then z is C 1 If x is A 2 and y is B 2 then z is C 2 –x and y are inputs, z is an output –A i, B i, and C i are fuzzy sets Input data: x is A, y is B –A and B are also fuzzy sets Question: z is ?

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10 Step 1: Compatibility Calculate the degree that input data (A, B ) matches each rule premise (A 1, A 2, B 1, B 2 ) –(A 1, A ), (A 2, A ), (B 1, B ), (B 2, B ) Compatibility between A and A : –(A, A ) = sup x min{ A (x), A (x)} x A 0 A

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11 Step 2: Combine Compatibilities Combine the degree of matching for the inputs –for and, usually take min –for or, usually take max –min{(A 1, A ), (B 1, B )} –min{(A 2, A ), (B 2, B )} x A 0 y B 0 A B 0.3

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12 Step 3: Derive Output Fuzzy Sets The (combined) degree of matching i is propagated to the consequent to form an inferred fuzzy subset C i –Type I: C (z) = C (z) [ usually take min ] –Type II: C (z) = x C (z) –C 1 and C 2 are derived z 1 C C z 1 C C Type IType II

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13 Step 4: Combine Output Fuzzy Sets Combine the inferred fuzzy values (C 1 and C 2 ) of z –max { C1 (z), C2 (z)} z 1 0 C 2 C 1

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14 Step 5: Defuzzification Perform defuzzification to obtain z s final value –Mean of Maximum method (MOM) ( j=1,k w j )/k where w j is peak and k is the number of peaks –Center of Area method (COA) ( j=1,n z (w j ) x w j )/ j=1,n z (w j ) w1w1 1 0 wkwk MOM 1 0 COA w1w1 wnwn

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15 What about Crisp Input? Inputs are x 0 and y 0 rather than A and B. Compatibility between A and x 0 (Step 1): –(A, x 0 ) = A (x 0 ) x0x A

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16 Triangular Norms (Conjunction) Four methods to calculate and Min – A and B (x) = min{ A (x), B (x)} Algebraic product – A and B (x) = A (x) x B (x) Bounded difference – A and B (x) = max {0, A (x) + B (x) – 1} Drastic product – A and B (x) = A (x) if B (x) = 1 B (x) if A (x) = 1 0 otherwise min algebraic product bounded difference drastic product

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17 Triangular Conorms (Disjunction) Four methods to calculate or Max – A OR B (x) = max{ A (x), B (x)} Bounded sum – A OR B (x) = min{1, A (x) + B (x)} Algebraic sum – A OR B (x) = A (x) + B (x) - A (x) x B (x) Drastic sum – A OR B (x) = A (x) if B (x) = 0 B (x) if A (x) = 0 1 otherwise max bounded sum algebraic sum drastic sum

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18 Theorectical Foundation of Fuzzy Inference A fuzzy rule = A fuzzy relation Inference is a composition of relations Relation R: U X V = {0, 1} –R(x, y) = 1: the relation holds between x and y –R(x, y) = 0: the relation does not hold between x and y –e.g., Take(Kim, ICE607) = 1 Take(Kim, ICE608) = 0 Fuzzy relation R: U X V = [0, 1] –E.g., Friendly(Jack, Joe) = 1 Friendly(Clinton, Hussain) = 0 Friendly(Clinton, Gingrich) = 0.2

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19 Fuzzy Rule as Fuzzy Relation Fuzzy rules are fuzzy relations over the Cartesian product of the domains of antecedent and consequent variables Semantically the fuzzy relation captures the degree of association between a pair of antecedent variables and consequent variables Example –If height is tall then IQ is high H\I R(6.5,135) High Tall R(6.5,135): The possibility that a person 6.5 tall has a 135 IQ.

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20 Methods to Calculate the Degree of Association Two families of calculating the degree of association –Fuzzy implication operators –Fuzzy conjunction operators

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21 Families of Fuzzy Implication Operators Material implication –A B = (not A) + B –e.g., Tall (6.25)=0.25, High (125)=0.5 TallHigh (6.25,125)=max{1-0.25,0.5}=0.75 Propositional calculus –A B = (not A) + (A * B) ( A (AB) = (AA) (AB) = AB ) Extended propositional calculus –A B = (A x B) + B

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22 Families of Fuzzy Implication Operators (2) Generalization of modus ponens –A B = sup { c[0,1] | A x c B } –e.g., Tall (6.25)=0.25, High (125)=0.5 TallHigh (6.25,125) = sup { c[0,1] | min(0.25,c) 0.5 } = 1 –If x is min, generalization of mp is –A B = 1 if A B B otherwise Generalization of modus tollens –A B = inf { t[0,1] | B+t A } All fuzzy implication operations share the fundamental implication property –00=1, 01=1, 10=0, 11=1

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23 Fuzzy Relation Using Conjunction Operator Use conjunction to represent relation (implication) –A B = min {A, B} –i.e., AB (x, y) = min { A (x), B (y)} Does this hold the fundamental implication property? –00=1, 01=1, 10=0, 11=1

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24 Summary Fuzzy Logic and Intelligent Systems Progress of Fuzzy Logic Generalized Modus Ponens Fuzzy Reasoning Triangular Norms Triangular Conorms Theoretical Foundations of Fuzzy Inference

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