Chapter 13 (Continued) Fuzzy Expert Systems 1

Fuzzy Rule-based Expert System 2

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4 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: IFx is A, THENy 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

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

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

7 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.

8 Firing Fuzzy Rules A fuzzy rule can have multiple antecedents, for example: IFproject_duration is long ANDproject_staffing is large ANDproject_funding is inadequate THENrisk is high IFservice is excellent ORfood is delicious THENtip is generous The consequent of a fuzzy rule can also include multiple parts, for instance: IFtemperature is hot THENhot_water is reduced; cold_water is increased Solutions: Mamdani or Sugeno approaches

9 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 , 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 , 1975) Sugeno – The ‘Zadeh of Japan’ – Sugeno, Michio. ”Industrial applications of fuzzy control,” Elsevier Science Inc., 1985.

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

11 Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules:Rule: 1 IFx is A3IFproject_fundingis adequate ORy is B1ORproject_staffingis small THENz is C1THENriskis lowRule: 2 IFx is A2IFproject_fundingis marginal ANDy is B2ANDproject_staffingis large THENz is C2THENriskis normalRule: 3 IFx is A1IFproject_fundingis inadequate THEN z is C3THENriskis high

Step 1: Fuzzification 12 Take the crisp inputs, x1 and y1 (project funding and project staffing; e.g. x1=2million, y1:10 persons), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. A1: Inadequate, A2: Marginal, A3: Adequate B1: Small, B2: Large

13 Step 2: Rule Evaluation Take the fuzzified inputs,  (x=A1) = 0.5,  (x=A2) = 0.2,  (y=B1) = 0.1 and  (y=B2) = 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)

14 Step 2: Rule Evaluation

15 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

16 Step 2: Rule Evaluation clipping scaling

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

Step 4: Defuzzification 18 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)

Step 4: Defuzzification 19 A reasonable estimate can be obtained by calculating it over a sample of points.

Step 4: Defuzzification 20

Mamdani Inference Technique 21

22 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: IFx is A ANDy is B THENz is k where k is a constant.

23 Sugeno Rule Evaluation

24 Sugeno Aggregation of the Rule Outputs Rule 1: IF project_fundingis adequate OR project_staffing is small, THEN risk is k1 Rule 2: IF project_funding is marginal AND project_staffing is large, THEN risk is k2 Rule 3: IF project_funding is inadequate, THEN risk is k3

Sugeno Defuzzification 25 Weighted Average (WA) Suppose: k1=20, k2=50, k3=80

Sugeno Inference Technique 26

27 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.

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 28