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Pertemuan 21 MEMBERSHIP FUNCTION

Matakuliah : H0434/Jaringan Syaraf Tiruan

Tahun : 2005

Versi : 1

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Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• Menjelaskan konsep fungsi keanggotaan pada logika fuzzy.

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Outline Materi

• Pengertian Fungsi keanggotaan.

• Derajat keanggotaan.

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FUZZY LOGICFUZZY LOGIC

• Lotfi A. Zadeh“Fuzzy Sets”, Information and Control, Vol 8, pp.338-353,1965.

Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh, 1965).

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PROF. ZADEHPROF. ZADEH

Fuzzy theory should not be regarded

as a single theory, but rather a

methodology to generalize a specific

theory from being discrete, to being

more continuous

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WHAT IS FUZZY LOGICWHAT IS FUZZY LOGIC

Fuzzy logic is a superset of conventional (boolean) logic

An approach to uncertainty that combines real values [0,1] and logic operations

In fuzzy logic, it is possible to have partial truth values

Fuzzy logic is based on the ideas of fuzzy set theory and fuzzy set membership often found in language

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WHY USE FUZZY LOGIC ?WHY USE FUZZY LOGIC ?

An Alternative Design Methodology Which Is Simpler, And Faster • Fuzzy Logic reduces the design

development cycle • Fuzzy Logic simplifies design complexity • Fuzzy Logic improves time to market

A Better Alternative Solution To Non-Linear

Control • Fuzzy Logic improves control performance • Fuzzy Logic simplifies implementation • Fuzzy Logic reduces hardware costs

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WHEN USE FUZZY LOGIC WHEN USE FUZZY LOGIC

Where few numerical data exist and where only ambiguous or imprecise information maybe available.

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FUZZY SETFUZZY SET

In natural language, we commonly employ: classes of old people Expensive cars numbers much greater than 1

Unlike sharp boundary in crisp set, here boundaries seem vague

Transition from member to nonmember appears gradual rather than abrupt

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CRISP AND FUZZYCRISP AND FUZZY

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FUZZY SET AND MEMBERSHIP FUZZY SET AND MEMBERSHIP FUNCTIONFUNCTION

Universal Set X – always a crisp set. Crisp set assigns value {0,1} to

members in X Fuzzy set assigns value [0,1] to

members in X These values are called the

membership functions . Membership function of a fuzzy set A is

denoted by : A: X [0,1]A: [x1/1, x2/…, xn/n}

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HIMPUNAN HIMPUNAN CRISP DAN FUZZYCRISP DAN FUZZY

Himpunan kota yang dekat dengan Bogor

• A = { Jakarta, Sukabumi, Cibinong, Depok }

CRISP

• B = { (0.7 /Jakarta) , (0.6 /Sukabumi) , (0.9

/Cibinong) , (0.8/Depok) } FUZZY

Angka 0.6 – 0.9 menunjukkan tingkat

keanggotaan ( degree of membership )

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CONTOHCONTOH

30 oC30 oC

Grade ( )

0 0

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PanasDingin

CRISPCRISPFUZZYFUZZY

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TINGGI BADANTINGGI BADAN

0

1

0

1

150 160155

150 160155

0,5

tinggi

tinggi

sedang

sedang

CRISP

FUZZY

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SEASONSSEASONS

0

0.5

1

Time of the year

Mem

ber

ship

Spring Summer Autumn Winter

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AROUND 4AROUND 4

0 2 4 6 80

0.5

1

Measurements

Mem

ber

ship

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AGEAGE

0 20 40 60 80 100

0

0.5

1

Age

Mem

ber

ship old

more or less old

youngvery young

not very young

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MEMBERSHIP FUNCTIONMEMBERSHIP FUNCTION

0

0.5

1

(a) (d) (g) (j)

0

0.5

1

(b) (e) (h) (k)

-100 0 1000

0.5

1

(c)-100 0 100

(f)-100 0 100

(i)-100 0 100

(l)

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SET OPERATIONSET OPERATION

0 20 40 60 80 100

0

0.5

1

Mem

ber

ship

A B

A B A B A B

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SET OPERATIONSET OPERATION

A B

A B A B A

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LINGUISTIC VARIABLESLINGUISTIC VARIABLES

Linguistic variable is ”a variable whose

values are words or sentences in a natural

or artificial language”. Each linguistic

variable may be assigned one or more

linguistic values, which are in turn connected

to a numeric value through the mechanism

of membership functions.

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LINGUISTIC VARIABLESLINGUISTIC VARIABLES

Fuzzy linguistic terms often consist of two parts:

1) Fuzzy predicate : expensive, old, rare, dangerous, good, etc.

2) Fuzzy modifier: very, likely, almost impossible, extremely unlikely, etc.

The modifier is used to change the meaning of predicate and it can be grouped into the following two classes:

a) Fuzzy truth qualifier or fuzzy truth value: quite true, very true, more or less true, mostly false, etc.

b) Fuzzy quantifier: many, few, almost, all, usually, etc.

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FUZZY PREDICATEFUZZY PREDICATE

Fuzzy predicate

– If the set defining the predicates of individual is a fuzzy set, the predicate is called a fuzzy predicate

Example

– “z is expensive.”

– “w is young.”

– The terms “expensive” and “young” are fuzzy terms.

Therefore the sets “expensive(z)” and “young(w)” are fuzzy sets

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FUZZY PREDICATEFUZZY PREDICATE

When a fuzzy predicate “x is P” is given, we can interpret it in two ways :

• P(x) is a fuzzy set. The membership degree of x in the set P is defined by the membership function P(x)

• P(x) is the satisfactory degree of x for the property P.

Therefore, the truth value of the fuzzy predicate is defined by the membership function :

Truth value = P(x)

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FUZZY VARIABLESFUZZY VARIABLES

Variables whose states are defined by linguistic concepts like low, medium, high.

These linguistic concepts are fuzzy sets themselves.

Low HighVeryhigh

TemperatureMem

ber

ship

Trapezoidal membership functions

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FUZZY VARIABLESFUZZY VARIABLES

Usefulness of fuzzy sets depends on our capability to construct appropriate membership functions for various given concepts in various contexts.

Constructing meaningful membership functions is a difficult problem –GAs have been employed for this purpose.

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EXAMPLEEXAMPLE

if speed is interpreted as a linguistic variable, then its term set T (speed) could be T = { slow, moderate, fast, very slow, more or less fast, sligthly slow, ……..}.

where each term in T (speed) is characterized by a fuzzy set in a universe of discourse U = [0; 100]. We might interpret

• slow as “ a speed below about 40 km/h"

• moderate as “ a speed close to 55 km/h"

• fast as “ a speed above about 70 km/h"

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SPEEDSPEED

Values of linguistic variable speed.

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NORMALIZED NORMALIZED DOMAIN INPUTDOMAIN INPUT

• NB (Negative Big), NM (Negative Medium)

• NS (Negative Small), ZE (Zero)

• PS (Positive Small), PM (Positive Medium)

• PB (Positive Big)

A possible fuzzy partition of [-1; 1].

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MEMBERSHIP FUNCTIONMEMBERSHIP FUNCTION

0

PositiveLarge

0

NegativeMedium

0

PL

PMPSZERONSNMNL

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