October 14, 2004 1 Demola Popoola Department of Computing University of Surrey Fuzzy Expert Systems Fuzzy Expert Systems CS364 Artificial Intelligence October 14, 2004 2  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy Expert Systems October 14, 2004 3  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy Expert Systems October 14, 2004 4 The operation of a fuzzy expert system depends on the execution of FOUR major tasks: Introduction Introduction • Fuzzification of input variables • Inference/rule evaluation • Composition/Aggregation • Defuzzification October 14, 2004 5 Introduction Introduction • Fuzzification: definition of fuzzy sets, and determination of the degree of membership of crisp inputs in appropriate fuzzy sets. • Inference: evaluation of fuzzy rules to produce an output for each rule. • Composition: aggregation or combination of the outputs of all rules. • Defuzzification: computation of crisp output October 14, 2004 6  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy Expert Systems October 14, 2004 7 Mamdani fuzzy inference Mamdani fuzzy inference Example: a simple two-input one-output problem with three rules. Rule: 1 Rule: 1 IF x is A3 IF project_funding is adequate OR y is B1 OR project_staffing is small THEN z is C1 THEN risk is low Rule: 2 Rule: 2 IF x is A2 IF project_funding is marginal AND y is B2 AND project_staffing is large THEN z is C2 THEN risk is normal Rule: 3 Rule: 3 IF x is A1 IF project_funding is inadequate THEN z is C3 THEN risk is high October 14, 2004 8 Mamdani fuzzy inference Mamdani fuzzy inference Crisp Input y1 0.1 0.7 1 0 y1 B1 B2 Y Crisp Input 0.2 0.5 1 0 A1 A2 A3 x1 x1 X µ (x = A1) = 0.5 µ (x = A2) = 0.2 µ (y = B1) = 0.1 µ (y = B2) = 0.7 Fuzzification: determine degree of membership of crisp inputs x1 and y1 in appropriate fuzzy sets October 14, 2004 9 Mamdani fuzzy inference Mamdani fuzzy inference Inference: apply fuzzified inputs, µ(x=A1) = 0.5, µ(x=A2) = 0.2, µ(y=B1) = 0.1 and µ(y=B2) = 0.7, to the antecedents of the fuzzy rules. For fuzzy rules with multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. This number (the truth value) is then applied to the consequent membership function. October 14, 2004 10 Mamdani fuzzy inference Mamdani fuzzy inference Inference: to evaluate i) the disjunction of rule antecedents, we use the OR fuzzy operation, typically defined by the classical fuzzy operation union: µA∪B(x) = max [µA(x), µB(x)] ii) the conjunction of rule antecedents, we apply the AND fuzzy operation intersection: µA∩B(x) = min [µA(x), µB(x)] [...]... nonlinear systems October 14, 2004 25 Summary • The operation of a fuzzy expert system is in four major stages: fuzzification, inference, composition and defuzzification • Mamdani- and Sugeno-style fuzzy inference systems are two commonly employed methods • Mamdani fuzzy inference systems use fuzzy sets in the rule consequent while Sugeno systems use mathematical functions, most often a constant • Mamdani systems. .. 0.2 + 0.2 + 0.5 + 0.5 + 0.5 + 0.5 Degree of Membership 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70 80 90 67.4 October 14, 2004 17 100 Z Fuzzy Expert Systems  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary October 14, 2004 18 Sugeno fuzzy inference Mamdani-style inference is, in general, not computationally efficient This is because it involves finding the centroid of a... which generally loses less information, can be very useful in fuzzy expert systems October 14, 2004 13 Mamdani fuzzy inference Degree of Membership 1.0 Degree of Membership 1.0 C2 C2 0.2 0.2 0.0 Z clipped October 14, 2004 0.0 Z scaled 14 Mamdani fuzzy inference Composition: aggregation of clipped (or scaled) outputs of all rules into a single fuzzy set 1 1 C1 C2 1 0.5 C3 0.2 0.1 0 0.1 Z 0 z is C 1 (0.1)... membership function of the rule consequent A fuzzy singleton is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else October 14, 2004 19 Sugeno fuzzy inference Sugeno- and Mamdani-style fuzzy inference are similar The only difference is in the rule consequent Instead of a fuzzy set, Sugeno used a mathematical function... variable: IF AND THEN x is A y is B z is f(x, y) where x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; and f(x, y) is a mathematical function October 14, 2004 20 Sugeno fuzzy inference The zero-order Sugeno fuzzy model, in which the output of each fuzzy rule is constant, is most commonly used Here, the function f(x, y) = k and all consequent membership... the membership function is sliced, the clipped fuzzy set loses some information However, it is often preferred because it involves less complex and faster mathematics, and generates an aggregated output surface that is easier to defuzzify October 14, 2004 12 Mamdani fuzzy inference • Scaling: Offers a better approach for preserving the original shape of the fuzzy set The original membership function of... k3 Z ∑ z is k3 (0.5) 23 Sugeno fuzzy inference Defuzzification 0 z1 Z Crisp Output z1 Weighted average (WA): µ (k1) × k1 + µ (k 2) × k 2 + µ (k 3) × k 3 0.1× 20 + 0.2 × 50 + 0.5 × 80 WA = = = 65 µ (k1) + µ (k 2) + µ (k 3) 0.1 + 0.2 + 0.5 October 14, 2004 24 Mamdani or Sugeno? Mamdani method • widely accepted for capturing expert knowledge - it allows us to describe the expertise in more intuitive, more... 0.2 0 Z ∑ z is C 3 (0.5) 15 Mamdani fuzzy inference Defuzzification: conversion of fuzzy set produced by composition stage into a crisp value Several defuzzification methods exist, but probably the most popular one is the centroid technique It finds the centre of gravity (COG) of the aggregate set: b µ ∫ (x ) x dx A COG =ab µ ∫ (x ) dx A a October 14, 2004 16 Mamdani fuzzy inference Centre of gravity...Mamdani fuzzy inference 1 1 A3 1 B1 0.1 0.0 0 x1 0 X Rule 1: IF x is A3 (0.0) OR 1 y1 y is B1 (0.1) 1 A2 0 x1 y1 Rule 2: IF x is A2 (0.2) AND y is B2 (0.7) 1 0 A1 Rule 3: IF x is A1 (0.5) October 14, 2004 z is C1 (0.1) AND (min) Y 0.2 C1 C2 THEN C3 0 THEN Z z is C2 (0.2) C2 0 X C3 Z 1 0.5 C1 0.5 x1 C2 1 B2 0 C1 0 THEN 0.7 0.2 X Y OR (max) 0.1 C3 Z z is C3 (0.5) 11 Mamdani fuzzy inference Inference:... 14, 2004 21 Sugeno fuzzy inference Rule evaluation 1 1 A3 1 B1 0.1 0.0 0 x1 0 X Rule 1: IF x is A3 (0.0) OR 1 0 x1 Rule 2: IF x is A2 (0.2) 0 A1 y1 AND y is B2 (0.7) October 14, 2004 Z z is k1 (0.1) AND (min) Y 0.2 0 THEN k2 Z z is k2 (0.2) 0 X Rule 3: IF x is A1 (0.5) k1 1 0.5 0.5 x1 0 1 B2 0 0.1 THEN 0.7 0.2 X Y y is B1 (0.1) 1 A2 1 y1 OR (max) THEN k3 z is k3 (0.5) 22 Z Sugeno fuzzy inference Composition . Surrey Fuzzy Expert Systems Fuzzy Expert Systems CS364 Artificial Intelligence October 14, 2004 2  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy. Expert Systems October 14, 2004 3  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy Expert Systems October 14, 2004 4 The operation of a fuzzy. 2004 6  Introduction  Mamdani fuzzy inference  Sugeno fuzzy inference  Summary Fuzzy Expert Systems Fuzzy Expert Systems October 14, 2004 7 Mamdani fuzzy inference Mamdani fuzzy inference Example: