1. Dr. Unnikrishnan P.C. Professor, EEE
EE368 Soft Computing
Dr. Unnikrishnan P.C.
Professor, EEE

2. Module III
Hybrid Intelligent Systems
Fuzzy Expert Systems

3. 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

4. Fuzzy Rule-based Expert System

5. Fuzzy Rule-based Expert System

6. 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

7. 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 > 100m THEN stopping_distance is < 40m
Fuzzy rule:
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.

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

9. 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.

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

11. 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

12. 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)

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

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

15. Mamdani Fuzzy Inference
We examine a simple two-input one-output problem that includes 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

16. Step 1: Fuzzification
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

17. 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)

18. Step 2: Rule Evaluation

19. 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

20. Step 2: Rule Evaluation
clipping
scaling

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

22. 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)

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

24. Step 4: Defuzzification

25. Mamdani Inference Technique

26. 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.

27. Sugeno Rule Evaluation

28. Sugeno Aggregation of the Rule Outputs
IF project_funding is 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

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

30. Sugeno Inference Technique

31. 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.

32. Advantages and Problems of Fuzzy Logic
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