Fuzzy Indra

7/28/2019 Fuzzy Indra

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Fuzzy Expert SystemsArtificialIntelligence


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Introduction

Mamdani fuzzy inference

Sugeno fuzzy inference

Summary

Fuzzy Expert Systems


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Introduction



Summary



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

Introduction

Fuzzification of input variables

Inference/rule evaluation

Composition/Aggregation

Defuzzification


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Introduction

Fuzzification: definition of fuzzy sets, anddetermination of the degree of membership of crisp

inputs in appropriate fuzzy sets.

Inference: evaluation of fuzzy rules to producean output for each rule.

Composition: aggregation or combination ofthe outputs of all rules.

Defuzzification:computation of crisp output


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Introduction



Summary



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Example:a simple two-input one-output problem withthree rules.

Rule: 1 Rule: 1IF xis A3 IF project_funding is adequateOR y is B1 OR project_staffing is small

THEN z is C1 THEN risk is low

Rule: 2 Rule: 2IF xis A2 IF project_funding is marginalAND y is B2 AND project_staffing is largeTHEN zis C2 THEN risk is normal

Rule: 3 Rule: 3IF xis A1 IF project_funding is inadequateTHEN zis C3 THEN risk is high


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Fuzzification:determine degree of membership of crispinputsx1 andy1 in appropriate fuzzy sets


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Inference:apply fuzzified inputs, (x=A1) = 0.5,(x=A2) = 0.2, (y=B1) = 0.1 and (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 membershipfunction.


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Inference:to evaluatei) the disjunction of rule antecedents, we use the OR

fuzzy operation, typically defined by the classical fuzzy

operation union:

AB(x) = max [A(x), B(x)]

ii) the conjunction of rule antecedents, we apply the

AND fuzzy operation intersection:

AB(x) = min [A(x), B(x)]


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Inference:Two general methods of applying theresult of the antecedent evaluation to the membership

function of the consequent:

Clipping (alphacut): 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 lesscomplex and faster mathematics, and generates an

aggregated output surface that is easier to defuzzify.


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Scaling: Offers a better approach for preserving theoriginal 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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clipped scaled


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Composition:aggregation of clipped (or scaled)outputs of all rules into a single fuzzy set.


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Defuzzification:conversion of fuzzy set produced bycomposition stage into a crisp value.

Several defuzzification methods exist, but probably the

most popular one is the centroid technique. It findsthe centre of gravity (COG) of the aggregate set:


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Centre of gravity (COG): In practice, a reasonableestimate is obtained by calculating it over a sample of

points:


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Introduction



Summary



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Mamdani-style inference is, in general, notcomputationally efficient. This is because it involves

finding the centroid of a two-dimensional shape by

integrating across a continuously varying function.

Michio Sugeno suggested the use of a single spike - a

singleton - as the 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 everywhereelse.


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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 zisf(x, y)

wherex,y andzare linguistic variables; A and B arefuzzy sets on universe of discourses X and Y,

respectively; andf(x, y) is a mathematical function.


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The zero-order Sugeno fuzzy model, in which the outputof each fuzzy rule is constant, is most commonly used.

Here, the functionf(x, y) = kand all consequent

membership functions are represented by singleton

spikes:

IF x is A

AND y is B

THEN zis k

where kis a constant.


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Rule evaluation


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Composition


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Defuzzification

Weighted average (WA):


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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 optimization

and adaptive techniques, which makes it very attractive incontrol problems, particularly for dynamic nonlinear

systems.


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Summary

The operation of a fuzzy expert system is in fourmajor 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 mannerwhile Sugeno systems are more computationally

efficient but lose linguistic interpretability.

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