EE 368 Soft Computing Dr Unnikrishnan P C

EE 368 Soft Computing Dr. Unnikrishnan P. C. Professor, EEE

Module III q Hybrid Intelligent Systems q. Fuzzy Expert Systems

Hybrid Intelligent Systems: Neural expert systems and Neuro-fuzzy systems Introduction Neural expert systems Fuzzy expert Systems Neuro-Fuzzy systems ANFIS: Adaptive Neuro-Fuzzy Inference System 3

Fuzzy Rule-based Expert System 4

Fuzzy Rule-based Expert System 5

Fuzzy Rules • In 1973, Lotfi Zadeh published his second most influential paper. He suggested capturing human knowledge in fuzzy rules. • A fuzzy rule can be defined as a conditional statement in the form: IF x is A, THEN y is B where – x and y are linguistic variables; – A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively. – Antecedent (or condition): x is A – Consequent (or conclusion): y is B 6

Classical vs. Fuzzy Rules • Classical rule: Rule 1: Rule 2: IF speed is > 100 (km/h) IF speed is < 40 (km/h) THEN stopping_distance is > 100 m THEN stopping_distance is < 40 m • Fuzzy rule: Rule 1: IF speed is fast THEN stopping_distance is long Rule 2: IF speed is slow THEN stopping_distance is short • Fuzzy rules relate fuzzy sets. • In a fuzzy system, all rules fire partially. 7

Firing Fuzzy Rules IF height is tall THEN weight is heavy 8

Firing Fuzzy Rules • If the antecedent is true to some degree of membership, then the consequent is also true to that same degree. • This form of fuzzy inference is called monotonic selection. 9

Firing Fuzzy Rules • A fuzzy rule can have multiple antecedents, for example: IF AND THEN project_duration is long project_staffing is large project_funding is inadequate risk is high IF service is excellent OR food is delicious THEN tip is generous 10

Firing Fuzzy Rules • The consequent of a fuzzy rule can also include multiple parts, for instance: IF temperature is hot THEN hot_water is reduced; cold_water is increased • Solutions: Mamdani or Sugeno approaches 11

Fuzzy Inference Techniques • Mamdani – The most commonly used fuzzy inference technique – He built one of the first fuzzy systems to control a steam engine – He applied a set of fuzzy rules supplied by experienced human operators. – E. Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant” (Proc. IEE, Vol. 121, No. 12, pp. 1585 -1588, 1974) – E. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller”, (Int. J. of Man. Machine Studies, Vol. 7, No. 1, pp. 1 - 13, 1975) 12

Fuzzy Inference Techniques • Sugeno – The ‘Zadeh of Japan’ – Sugeno, Michio. ”Industrial applications of fuzzy control, ” Elsevier Science Inc. , 1985. 13

Mamdani Fuzzy Inference • Four steps: 1. Fuzzification of the input variables 2. Rule evaluation (inference) 3. Aggregation of the rule outputs (composition) 4. Defuzzification. 14

Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules: Rule: 1 IF x is A 3 IF project_fundingis adequate OR y is B 1 OR project_staffing is small THEN z is C 1 THEN risk is low Rule: 2 IF x is A 2 AND y is B 2 THEN z is C 2 Rule: 2 IF project_fundingis marginal AND project_staffing is large THEN risk is normal Rule: 3 IF x is A 1 THEN z is C 3 Rule: 3 IF project_fundingis inadequate THEN risk is high 15

Step 1: Fuzzification • Take the crisp inputs, x 1 and y 1 (project funding and project staffing; e. g. x 1=2 million, y 1: 10 persons), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. A 1: Inadequate, A 2: Marginal, A 3: Adequate B 1: Small, B 2: Large 16

Step 2: Rule Evaluation • Take the fuzzified inputs, (x=A 1) = 0. 5, (x=A 2) = 0. 2, (y=B 1) = 0. 1 and (y=B 2) = 0. 7, and apply them to the antecedents of the fuzzy rules. • If a given fuzzy rule has 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. (monotonic selection) 17

Step 2: Rule Evaluation 18

Step 2: Rule Evaluation • How the result of the antecedent evaluation can be applied to the membership function of the consequent? – Clipping (alpha-cut) • Cut the consequent membership function at the level of the antecedent truth. • losing some information. • it is often preferred because it involves less complex and faster mathematics – Scaling • offers a better approach for preserving the original shape of the fuzzy set. • Multiplying all its membership degrees by the truth value of the rule antecedent. • It loses less information 19

Step 2: Rule Evaluation clipping scaling 20

Step 3: Aggregation of the rule outputs • The process of unification of the outputs of all rules. • Combining with MAX operator 21

Step 4: Defuzzification • Input: the aggregate output fuzzy set • Output: a single number • The most popular method: – Centroid technique. – It finds the point where a vertical line would slice the aggregate set into two equal masses. – Mathematically, it’s the center of gravity (COG) 22

Step 4: Defuzzification • A reasonable estimate can be obtained by calculating it over a sample of points. 23

Step 4: Defuzzification 24

Mamdani Inference Technique 25

Sugeno Fuzzy Inference • In Mamdani-style inference, to find the centroid, an integration across a continuously varying function is required. no computationally efficient! • Michio Sugeno suggested to use a single spike, a singleton • Fuzzy Rules in zero-order Sugeno fuzzy model: IF x is A AND y is B THEN z is k where k is a constant. 26

Sugeno Rule Evaluation 27

Sugeno Aggregation of the Rule Outputs Rule 1: IF project_funding is adequate OR project_staffing is small, THEN risk is k 1 Rule 2: IF project_funding is marginal AND project_staffing is large, THEN risk is k 2 Rule 3: IF project_funding is inadequate, THEN risk is k 3 28

Sugeno Defuzzification Weighted Average (WA) Suppose: k 1=20, k 2=50, k 3=80 29

Sugeno Inference Technique 30

Mamdani or Sugeno? • Mamdani – widely accepted for capturing expert knowledge – more intuitive, more human-like manner – a substantial computational burden • Sugeno – computationally effective – works well with optimization and adaptive techniques – e. g. control problems, particularly for dynamic nonlinear systems. 31

Advantages and Problems of Fuzzy Logic • Advantages – general theory of uncertainty – wide applicability, many practical applications – natural use of vague and imprecise concepts • helpful for commonsense reasoning, explanation • Problems – membership functions can be difficult to find – multiple ways for combining evidence – problems with long inference chains 32