fuzzy - basic concept

8/7/2019 Fuzzy - Basic Concept

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2011-03-08 VAN 1

Fuzzy Models

Models base on real human reasoning.Models can be

- linguistic

- simple (no number crunching),- comprehensible (no black boxes),

- fast in computing,- good in practice.


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2011-03-08 VAN 2

Fuzzy Systems

Fuzzy systems can cope with linguisticand imprecise entities of a model in acomputer environment.

Invented by Lotfi Zadeh at UC Berkeleyin the 1960s.

Stem from novel theories on fuzzy setsand fuzzy logic.


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2011-03-08 VAN 3

Applications: Control

Heavy industry

(Matsushita,Siemens, Stora-Enso,Metso)

Home appliances(Canon, Sony,Goldstar, Siemens)

Automobiles (Nissan,

Mitsubishi, Daimler-Chrysler, BMW,Volkswagen)

Space crafts (NASA)


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2011-03-08 VAN 4

Applications: Decision Making

Fuzzy scoring for mortgage applicants,

creditworthiness assessment,

fuzzy-enhanced score card for lease risk assessment,

risk profile analysis,

insurance fraud detection, cash supply optimization,

foreign exchange trading,

insider trading surveillance,

investor classification etc.

Source: FuzzyTech


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2011-03-08 VAN 5

Crisp and Fuzzy Sets

gradualchange

Escher


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2011-03-08 VAN 6

Crisp and Fuzzy MembershipFunctions

0

1

0 1 2 3 4 5 6 7 8 9 10E

MEMBERSHIP

0

1

0 1 2 3 4 5 6 7 8 9 10E

MEMBERSHIP

Five About five

{(x,(x)) | xE, (x)[0,1]},In which E is universe ofdicourse (reference set).


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2011-03-08 VAN 7

Typical Fuzzy Sets

Triangular,

Bell-shaped,

Trapezoidal.


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Basic Fuzzy Set Operations

0

1

0 1 2 3 4 5 6 7 8 9 10

E

MEMBER

SHIP

0

1

0 1 2 3 4 5 6 7 8 9 10

E

MEMBERS

HIP

0

1

0 1 2 3 4 5 6 7 8 9 10

E

MEMBERSHIP

Complement,

Intersection,

Union.


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2011-03-08 VAN 9

Linguisticvariables

Linguistic

values Syntacticrules

Vocabu-

lary

Universe of

discourse

Semantic

rules

Artificial

language in SC

Construction of FuzzyConstruction of Fuzzy

LanguageLanguage


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2011-03-08 VAN 10

male, negative, small negative, very high,

fairly old, not good, young or fairly young,slightly greater than, approximately equalwith

Precise and approximate linguistic values

and relations

Approximately x2+2y3+1, approximatelyx=y

Approximate numerical functions andrelations

X2+2y3+1, x=yPrecise numerical functions and relations

about 5, about 0.5, about [4.5,6], aboutfrom 4.5 to 6

Approximate numerical values andintervals

5, 0.5, [4.5,6]Precise numerical values and intervals

ExamplesType of Value

Table 3.1.1.1. Typical Values Used in SC Models.

Possible Values for Variables in Fuzzy LanguagePossible Values for Variables in Fuzzy Language


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2011-03-08 VAN 11

Figure. 3.1.1.2. Formation of LinguisticValues .

Very

oldOld

Fairly

oldNeutral

Fairly

youngYoung

Given a universe of discourse and variable, select two

appropriate primitive terms which are usually antonyms:

e.g. variable Age and set of ages.

Term:

young

Antonym:

old

Select other expressions which are modified according to the

primitive terms. The modifiers are adverbs. Use one of these terms as

a neutral value or central value, and the rest of the values should

usually be symmetrical with respect to the ne utral value: modifiers are

e.g. very,fairly, more or less, slightly and almost.

Very

young

linguist

icscal

e

linguist

icscal

e


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2011-03-08 VAN 12

Figure 3.1.1.4. Formation of Fuzzy Language Expressions.

Primitive terms

young, old

Modifiers

fairly, very etc.

Negation

not

Connectives

and, oretc.

Quantifiersall, most, some etc.

Expressionssome persons are very young


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2011-03-08 VAN 13

0

1

0 25 50 75 100

AGE

MEMBERS

HIP

0

1

0 25 50 75 100

AGE

MEMBERSHIP

.

Quantitative meaningsQuantitative meanings

of linguistic values areof linguistic values are

fuzzy sets.fuzzy sets.

E.g.E.g.

meaning of young is a fuzzymeaning of young is a fuzzy

set YOUNGset YOUNG

Fuzzy sets are denotedFuzzy sets are denoted

as functions,as functions,

membership functionsmembership functions

Crisp setCrisp set

YOUNGYOUNG


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2011-03-08 VAN 14

0

1

0 25 50 75 100

AGE

MEMBERS

HIP

0

1

0 25 50 75 100

AGE

MEMBERSHIP

.

Objects can also belongObjects can also belong

partially to a given fuzzypartially to a given fuzzy

set.set.

E.g., given fuzzy set YOUNG andE.g., given fuzzy set YOUNG and

ages of persons,ages of persons,

person aged 10: full membershipperson aged 10: full membershipperson aged 27: almost fullperson aged 27: almost full

person aged 35: smallperson aged 35: small

person aged 70: no membershipperson aged 70: no membershipDegrees of membershipDegrees of membership

are denoted as functions,are denoted as functions,

membership functionsmembership functions

horizontal axis:horizontal axis: values of ages, 0values of ages, 0--100100

vertical axis:vertical axis: degrees of membeship, 0degrees of membeship, 0--11


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2011-03-08 VAN 15

0

1

0 25 50 75 100

Tentative Fuzzy Sets Denoting Linguistic Values of AgeTentative Fuzzy Sets Denoting Linguistic Values of Age

young, fairly young, middle-aged, fairly old, old


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2011-03-08 VAN 16

Fuzzy sets: extension principle, etc.Quantifiers: all, most some, etc.

1.Set-theoretical operations of fuzzy sets:intersection, union, etc.

2.Fuzzy relations: order relation, etc.

Compound experessions: and, or, if-then etc.

Modified fuzzy sets: complement, etc.Negation: not

Fuzzy sets modified by translation: VERY YOUNGetc.

Modifiers: very, fairly, etc.

Fuzzy sets: YOUNG, OLDPrimitive terms: young, old

Fuzzy set-theoretical counterpart ("quantitativemeaning")

Expression

Correspondence between Linguistic Expressions and Set-theoretical Operations.


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2011-03-08 VAN 17

Inputs(precise or fuzzy)

Fuzzy rules

Reasoning system

Outputs(fuzzy)

Defuzzification(if necessary)

Final outputs(precise or fuzzy)

SC Model Construction withSC Model Construction with

Fuzzy ReasoningFuzzy Reasoning


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2011-03-08 VAN 18

Fuzzy Rule-Based Models

Types of fuzzy rules:1. If height is tall, then weight is fairly heavy.

2. If height is tall, then weight is 80 kg. (zero-order)

3. If height is tall, then weight is f(x). (first-order)

4. If height is tall and body is fat, then weight is _.

5. If height is tall or body is fat, then weight is _ andrisk of heart disease is _.

Rules have two parts: antecedent (if _) andconsequent (then _).


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2011-03-08 VAN 19

Example of Fuzzy Modeling when

Data Unavailable (MamdaniReasoning)

Problem: How much should I give tip in the restaurantin the USA according to given criteria? (=> multi-criteria decision-making)

No data, based on expertise.

Two criteria (inputs): quality of service (0-10)

quality of food (0-10)

Output: Tip (%).


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2011-03-08 VAN 20

Decision Model (Variables)

Quality of food

Quality of service

Tipping


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Linguistic Values of Variables

Service: poor, good, excellent.Food: rancid, delicious.

Tip: cheap, average, generous.


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2011-03-08 VAN 22

Example of Fuzzy Rules

1. If service is poor and food is rancid, then tip is cheap.

2. If service is good and food is delicious, then tip is

average.

3. If service is excellent or food is delicious, then tip is

generous.


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2011-03-08 VAN 23

Fuzzy Values and Model


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2011-03-08 VAN 24

Two Main Types of FuzzyReasoning

Mamdani (Mamdani-Assilian; no datarequired)

Takagi-Sugeno (-Kang; data required)

Matlab fuzzy logic toolbox

fuzzy - basic concept

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