fuzzy expert systems

• 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 a

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Demola Popoola Department of Computing University of Surrey

Fuzzy Expert Systems

CS364 Artificial Intelligence

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 Introduction

 Mamdani fuzzy inference

 Sugeno fuzzy inference

 Summary

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 Summary

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The operation of a fuzzy expert system depends on the execution of FOUR major tasks:

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• 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

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 Summary

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Mamdani fuzzy inference

Example: a simple two-input one-output problem with three rules

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Crisp Input y1

0.1 0.7 1

inputs x1 and y1 in appropriate fuzzy sets

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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

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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)]

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Inference: Two general methods of applying the result of the antecedent evaluation to the membership function of the consequent:

• Clipping (alpha–cut): This is the most common method It involves cutting the consequent

membership function at the level of the antecedent truth Since the top of 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

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• Scaling: Offers a better approach for preserving

the original shape of the fuzzy set The original

membership function of the rule consequent is

adjusted by multiplying all its membership degrees

by the truth value of the rule antecedent This

method, which generally loses less information, can

be very useful in fuzzy expert systems

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Z

C2

1.0

0.0 0.2

C2

clipped scaled

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outputs of all rules into a single fuzzy set

0.2 1

C

z is 2 (0.2)

0

0.5 1

C

z is 3 (0.5)

Z Z

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by composition stage into a crisp value

Several defuzzification methods exist, but probably the most popular one is the centroid technique It finds

( ) ( )

dx x x COG

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estimate is obtained by calculating it over a sample of points:

4 67 5

0 5 0 5 0 5 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0

5 0 ) 100 90

80 70 ( 2 0 ) 60 50 40 30 ( 1 0 ) 20 10 0 (

= +

+ +

+ + +

× +

+ + +

× +

+ + +

× +

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 Summary

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Sugeno fuzzy inference

Mamdani-style inference is, in general, not

computationally efficient This is because it involves finding the centroid of a two-dimensional shape by

integrating across a continuously varying function

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

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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 of the input variable:

IF x is A

AND y is B

THEN 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.

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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 functions are represented by singleton spikes:

IF x is A

AND y is B

THEN z is k

where k is a constant.

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Z

0 0.2 1

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0 2 0 1 0

80 5 0 50 2 0 20 1

0 )

3 ( )

2 ( )

1 (

3 )

3 ( 2

) 2 ( 1 )

1 (

= +

+

× +

×

= µ

+ µ

× µ

+

× µ

k

k k

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Mamdani or Sugeno?

Mamdani method

• widely accepted for capturing expert knowledge - it allows us to describe the expertise in more intuitive, more human-like manner

• entails a substantial computational burden

Sugeno method

• computationally effective and works well with

optimisation and adaptive techniques, which makes it very attractive in control problems, particularly for

dynamic nonlinear systems

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• 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 are computationally expensive but capture knowledge in intuitive, human-like

manner while Sugeno systems are more

computationally efficient but lose linguistic

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