bulanikmantik all

8/6/2019 Bulanikmantik All

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Ebru Sezer

[email protected]

ubat, 2009
mailto:[email protected]:[email protected]


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Bulank Mantk, 1965ylnda Ltfi Askerzade'ninyaynlad bir makalenin sonucu olumu bir mantkyapsdr.

Bulank mantn temeli bulank kme ve altkmelere dayanr. Klasik yaklamda bir varlk ya kmeninelemandr ya da

deildir. Matematiksel olarak ifade edildiinde varlk kme ile

olan yelik ilikisi bakmndan kmenin eleman olduunda(1) kmenin eleman olmad zaman (0) deerini alr.

Bulank mantk klasik kme gsteriminin geniletilmesidir.Bulank varlk kmesinde her bir varln yelik derecesivardr. Varlklarn yelik derecesi, [0,1] aralnda herhangibir deer olabilir ve yelik fonksiyonu (x) ile gsterilir .
http://tr.wikipedia.org/wiki/1961http://tr.wikipedia.org/wiki/Lotfi_Zadehhttp://tr.wikipedia.org/wiki/K%C3%BCmehttp://tr.wikipedia.org/wiki/K%C3%BCmehttp://tr.wikipedia.org/wiki/Lotfi_Zadehhttp://tr.wikipedia.org/wiki/1961


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There are two different context

Under Uncertainity (Belirsizlik durumunda)

ki deerli mantn, belirsizlik altnda sonu retebilmesi(dar balam)

Unsharp Boundaries (Keskin olamayan snrlar ) Keskin snrlar olamayan tm uygulama alanlarnda


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180cm 179cm

cm

Degree of

high

180160

Fuzzy1

cm

Degree of

high

180

Crisp1


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1.Fuzzy approach (hot, cold with some degree)

2.Crisp approach (hot or cold)


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0 70 78 F F70 78

1 1

yelik (Membership) yelik (Membership)


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Many decision-making and problem-solving tasks aretoo complex to be defined precisely

However, people succeed by using imprecise(kesinolamayan) knowledge

Fuzzy logic resembles human reasoning in its use ofapproximate information and uncertainty to generatedecisions.


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Consider:

Joe is tall -- what is tall?

Joe is very tall -- what does this differ from tall?

Natural language (like most other activities in life and

indeed the universe) is not easily translated into the

absolute terms of 0 and 1.

false true


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An approach to uncertainty that combines real values

[01] and logic operations (0, 1 arasi degerler aliyor,

mantiksal islemlere sokuluyor bu degerler)

Fuzzy logic is based on the ideas of fuzzy set theory

and fuzzy set membership often found in natural (e.g.,

spoken) language.


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Example:

Ann is 28, 0.8 in set Young

Bob is 35, 0.1 in set Young

Charlie is 23, 1.0 in set Young

Unlike statistics and probabilities, the degree

is not describing probabilities that the item

is in the set, but instead describes to whatextent the item is the set.(burdaki derece

olasiligi yansitmiyor, mertebeyi yansitiyor)


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Age25 40 55

Young Old1

Middle

0.5

DOM

Degree of

Membership

Fuzzy values

Fuzzy values have associated degrees of membership in the set.

0


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Dnya bulankAristo bir ey ya dorudur ya yanl

ikili (boolean) mantk

Ploton Doru iinde yanl, yanlta doruyuierebilirBulank (fuzzy mantk)

Zadeh, Tho closer one looks at a real worldproblems, the fuzzier becomes its solution (1973)


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Apartmannzdaki evli iftlerin says Apartmannzdaki mutlu iftlerinsays

Yaz mevsiminde 30zeri scaklk skl Yaz mevsiminin scakgnleri

Geliri 5000 ytl zeri ailelerin oran Zenginailelerinoran

YadaBen gzele gzel demem gzel Angelina olmaynca (keskin mantk)

Ben gzele gzel demem gzel Angelina gibi olmaynca (bulank mantk)

History 1961de Lotfi Zadeh ihtiyac tanmlad 1965de Lofti Zadeh makale yaynlad(California Berkeley) Ayn yl iddetle NSFde knand (belirsizlik savunulamaz!!!) lk 25 yln sonunda matematiin temelleri bulanklat

zdevinirler Diller Fonksiyonlar izgeler Algoritmalar/Programlar likiler MANTIK

1974, Mamdani ilk bulank mantk kontrolr (f.l. Controller) gelitirdi


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A traditional crisp set A fuzzy set


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Masaya oturdunuz iki bardak nnzde Birinci bardak %95 salkl ve besleyici zellikte olma ihtimali var kinci bardak 0.95 derece ile salkl ve besleyici bulank

kmesinin yesi

Hangisini seersiniz ?

Unutmaynz ilk bardak %5 ihtimalle tamamen zehirolabilir(Probability)

kinci bardak (szel deiken iki deer alyorsa salkl- salksz)

0.05 derece ile daha salksz ve besleyici deildir. Bir bakadeyile; 0.05 derece ile zehirli bulank kmesine aittir.(Possibility)


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Geleneksel matematik uygulamalarnn

zemedii/modelleyemedii gerekdnya sorunlarn zmek

Defined input matches defined condition andproduces expected conclusion, what can be

done if input is not defined, or has multiplevalues.

nsan bilgisayar etkileimini, insannifade ve anlamlandrma biimineevirebilmek

Recall data by using fuzzy expressions Poison ?


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Gnmzde bulank mantk otomobillerin viteskutularndan bulak makinelerine, elektronikdevrelerin ve yapay zekann karar vermealgoritmalarna kadar olduka kapsaml teknikuygulamalara sahip; hatta Tokyo metrosubulank metro temelli bilgisayar ve mhendisliksistemleriyle ilemektedir. Bilgisayar veenformatik bilimleri, kontrol sistemleri, karar

alma algoritmalar bulank mantn younolarak kullanld alanlar olarak beliriyor.
http://tr.wikipedia.org/wiki/Tokyohttp://tr.wikipedia.org/wiki/Tokyo


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Bulank mantk olaydaki phe ya da belirsizliinderecesini tanmlarken

ansolayn ihtimalini tanmlar(Gerekleme skldr aslnda)


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A : Deer aral (170 cm


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Bulank Kme (Fuzzy Set): Snrlar bulank

kme

Szel Deiken (Linguistic Variable): Deerinicelik ve nitelik bildiren szel ifadeler olabilendeiken

Mmknlk Dalm (Possibility Distribution):LV iin bulank kmeye atama yaplrkenalabilecei deerlerde snrlama

Bulank Kurallar (Fuzzyif-then rules): ikideerli karsamay genelletiren matksalforml iin, bilgi gsterim ve retim biimi


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Bulank mantkta bulank kmeler kadar

nemli bir dier kavramda szel deikenkavramdr.

Szel deiken scak veya souk gibikelimeler ve ifadelerle deer alabilen

deikenlerdir.Bir szel deikenin deerleri bulank kmeler

ile ifade edilir.

rnein oda scakl szel deikeniscak, souk ve ok scak ifadelerinialabilir. Bu ifadenin her biri ayr ayrbulank kmeler ile modellenir.


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[0, 1] aralnda yelik dereceleri

: membership function A: FSet, x : varlk A(x): Xin A kmesine yelik

derecesini reten ilev

Her kme iin deer aral deer aral(domain) vardr (cret kmesi iin (0, +sonsuz))

A fuzzy set is always defined in a contex(Zadeh, 1963)

TR iin uzun boy != JAP iin uzun boy


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AB, AUB, Ac

A = {x| 100k


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Temperature0

1

Medium

Mediumc

Ac

(x) =1 - A(x)

A

Ac


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Temperature0

1

AB(x) =min{A(x), B(x)}

A B


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1

0 Temperature

AUB(x) =max{A(x), B(x)}

A B


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Souk Hava Ak

Scak Hava Ak

Hedef Scaklk

Kark Hava Ak

V = 1 tamamyla souk hava

Vc= 0 tamamyla scak hava

0


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Otomatik Seim

Ykama Dngs

Ykama Sresi

amar Nicelii

amar Hassasiyeti

Makina


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How do we determine the exact shape ofthe ?

Three methods

Expert knowledge (80 sonlarna dek) From data

From performance

Most important property

gradual transition from outside to inside


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Age(suspect)= [21,27]

It generalizes the binary distinction betweenpossible and impossible to a matter of degreecalled possibility

Age(Suspect)(x) = young(x)Probability dist: indicates the likelihood that

the variable takes specific value in interval

Possibility dist: indicates the degree ofspecific value assigned varible to the definedset (olurluk)


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17 30292827262524232221201918 31

Age

Young


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1

80k 120k

high

There is no constraint on the design

Its strongly recommended that use parameterizable functions that can be

defined by a small number of parameters

reduces system complexitytuning of


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1

l p r l rlp rp

Trapezoid membership

function and itsparameters

Triangular membership

function and itsparameters

As a result Always use parameterizable membership functionDo not define it point by pointUse triangular or trepoziod, unless there is a good reasonIf you want learn mf using nn, choose a diffrentiable


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Generalization of a logic inference called modus

ponens (mantiksal cikarmanin genel adi)

Only introduction is here (To be continued)

R1:

IF the annual income of a person is greater than 120K

THEN the person is RICHAs annual in come 120K but Bs is 199.999A is RICH B is NOT RICH

In crisp logic condition of rule is supplied T or F values, so result is T or Ftoo.


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Great difference in If part

R2: IF the annual income a person is HighTHEN the person is Rich

R3: IF the annual income a person is High

AND(the credit report of the person is Clean

OR

the amount of credit requested is NOTtoo

High)

THEN recommend approving the credit


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Crisp Consequent

IF .THEN y = a Fuzzy Consequent

IF THEN y is A (with some degree)

Functional Consequent

IF x1 is A1 and x2 is A2and .. xn is An

THEN y = a0+ ai.xii=1

n

ans are coefficients


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T16C

Degree of match = 0.8

Degree of match = 0.2

Input

R4 : If the target temp. T is low Then set the voltage to V

R5: If the target temp. T is high Then set the voltage to Vc

Low, V ve Vc are fuzzy


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R6: If Laundary Quantity is Large and

Laundary Softness is Hard

Then washing cycle is Strong

R7: If Laundary Quantity is Normal and

Laundary Softness is NormalHard

Then washing cycle is Normal


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1

Hard Nor. HardNor. Soft

Soft

L. Softness

1

L. Quantity

Small Medium Large

1

Delicate Light Normal Strong

Washing Cyle


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3 basic + 1 optional stage

1. Fuzzy Matching: matching degree of

conditions

2. Inference: matching degree ofconsequences

3. Combination: combination of all

consequences

4. Defuzzification(optional): production a

crisp result


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R6: If Laundary Quantity is Heavy and

Laundary Softness is Hard

Then washing cycle is Strong

R7: If Laundary Quantity is Normal and

Laundary Softness is NormalHard

Then washing cycle is Normal


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3 basic + 1 optional stage

1. Fuzzy Matching: matching degree of

conditions2. Inference: matching degree of

consequences

3. Combination: combination of allconsequences

4. Defuzzification(optional): production a

crisp result


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Calculate degree to which the input data

match the condition of the fuzzy rules

IF (CONDITION ) Then ..

Heres the part calculated


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Low High

1

T

Input18C

1

T

Input18C

Degree of Match = 0.8

Degree of Match = 0.2



MatchingDegree(Input, R4) = Low(Input)

MatchingDegree(Input, R5) = High(Input)

There is no conjuction or disconjuction


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1

Hard Nor. HardNor. Soft

Soft

L. Softness

1

L. Quantity

Small Medium Large

1

Delicate Light Normal Strong

Washing Cyle


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L.QuantityL.Softness

Soft

Normal Soft

Normal Hard

Hard

Small Medium Large

Delicate

Light Normal

Strong

Light

Light

Light

Normal

Normal

Normal

Normal

Strong


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Input Input

L.Quantity

L.Quantity L.Softness

L.Softness

Heavy

Normal

Hard

Normal-hard

0.5

0.2

0.5

0.8

Degree ofMatch = 0.2

Degree ofMatch = 0.5

R6: If Laundary Quantity is Heavy and Laundary Softness is HardThen washing cycle is Strong

R7: If Laundary Quantity is Normal and Laundary Softness is NormalHardThen washing cycle is Normal


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General rule for conjunction

MatchingDegree = min{Ai1(x10),.,

Ain(xn0

)}General rule for disjunction

MatchingDegree = max{Ai1(x10),.,Ain(xn0)}


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Fuzzy inference step is invoked for each of

the relevant rules to produce a conclusion

based on their matching degree

How should the conclusion be produced ? Clipping method

Scaling method

Both of them are supress the membership

function of consequent


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1Degree of match = 0.8

y

1

y

Is this schema supports crisp inference ?

Fuzzy Consequent Inferred Conclusion


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1Degree of match = 0.8

y

1

y

Is this schema supports crisp inference ?

Fuzzy Consequent Inferred Conclusion


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Fuzzy rules are partially overlapping

Each input can triger more than one rule

Heres the combinations of their results

Superimpose all fuzzy conclusions about avariable


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Combined result


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Combined result


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Input Input

L.Quantity

L.Quantity L.Softness

L.Softness

Heavy

Normal

Hard

Normal-hard

0.5

0.2

0.5

0.8

Cycle

Strong

Cycle

Normal

Cycle


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Ebru Sezer

[email protected]


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Fuzzy Inference System (FIS) has many synonyms in

literature and most widely used forms are fuzzymodel, fuzzy rule based system or only fuzzysystem

FIS uses the collection of fuzzy rules which can bepopulated by using Boolean operators: and, or, not.

This collection is organized and stored in a rule

base with the purpose of expert thinking modeling.

As can be seen, fuzzy rules are the most importantpart of the FIS because of their purposes.


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The remaining parts of the FIS are

fuzzificatorfor convert crisp inputs to

linguistic values with some membership

degrees by using membership functions. In

other words, fuzzification is the assignmentof the crisp input values coming from real

world to the fuzzy sets with some degrees


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Fuzzy matcheris used to calculate matching

degree of rules antecedent parts and

produce rules evaluation results


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The FIS has an input space and it is divided into fuzzysets so it can be said that rules of the FIS are thesubset models of the input space. In other words,each rule or rule group is modeling a local behaviorof the system. Some inputs may fire more than onerule or rule group, because they belong more than

one fuzzy set. Normally, if the input takes place onthe boundary of any fuzzy set, it may be the memberof another fuzzy set with another degree. At thattime, two different rules (groups) are fired and twodifferent results representing two different localbehaviors of the system are produced. This situation

requires aggregating local behaviors and producingone result reflecting each local behavior according todegrees. As a result, the other part of the FIS iscalled aggregator.


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FIS girdi uzayina sahiptir ve bulanik

kumelerine bolunur. FIS icin girdi uzayinin


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The mission ofdefuzzifier is conversion of

fuzzy result coming from aggregator to crisp

value. If crisp value is required to go on

operation, defuzzifier is used.


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There are three types of fuzzy inference

system that have been widely employed:

the Mamdani Model

the Sugeno Model

the Tusokomoto Model

The divergent parts of the models are

consequent parts of the rules and normally

production of consequences, aggregation anddefuzzification.

The Mamdani fuzzy model is perhaps the

most employed model


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Constructing a fuzzy inference system can be carriedout as follows:

Define the problem

Choose type of fuzzy approach (inference system orrelations)

Determine input(s) and output(s) of the system

Define linguistic variables (if a FIS is constructing)

Determine fuzzy sets representing linguistic values foreach input and output

Choose membership functions

Adjust parameters of membership functions with the helpof experts

Design fuzzy rules by using domain knowledge with thehelp of experts

Evaluate the selected fuzzy approach in step 2

Tune system if it is needed


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Mamdani Fuzzy Model (Mamdani and Assilian, 1975)

was the first application of controlling a steamengine and boiler by using linguistic rules designedby experts

The Mamdani model uses the rules can beexampled as

If a is A1 and b is B1 then c is C1

If a is A2 and b is B2 then c is C2

In the Mamdani Fuzzy Model, min operator is usedfor production of local results. In other wordsfuzzy matcher uses min operator to concludeachievement degree of consequent. To aggregate

local consequents maxoperator is used


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Two Inputs (x, y) and one output (z)

Low High

1

0tX=0.32 Y=0.61

0.32

0.68

Low(x) = 0.68, High(x) = 0.32, Low(y) = 0.39, High(y) = 0.61

Crisp Inputs


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Rule 1: If x is low AND y is low Then z is high

Rule 2: If x is low AND y is high Then z is low

Rule 3: If x is high AND y is low Then z is low

Rule 4: If x is high AND y is high Then z is high


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Rule1: low(x)=0.68, low(y)=0.39 =>high(z)=MIN(0.68,0.39)=0.39

Rule2: low(x)=0.68, high(y)=0.61 =>

low(z)=MIN(0.68,0.61)=0.61

Rule3: high(x)=0.32, low(y)=0.39 =>low(z)=MIN(0.32,0.39)=0.32

Rule4: high(x)=0.32, high(y)=0.61 =>high(z)=MIN(0.32,0.61)=0.32

Rule strength


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Low High1

0

t

Low(z) = MAX(rule2, rule3) = MAX(0.61, 0.32) = 0.61

High(z) = MAX(rule1, rule4) = MAX(0.39, 0.32) = 0.39

0.61

0.39


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Center of Gravity

Low High1

0

0.61

0.39

tCrisp output

Max

Min

Max

Min

dttf

dtttf

C

)(

)(

Center of Gravity


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Assume that we need to evaluate studentapplicants based on their GPA and GRE scores.

For simplicity, let us have three categories for

each score [High (H), Medium (M), and Low(L)]

Let us assume that the decision should beExcellent (E), Very Good (VG), Good (G), Fair (F)or Poor (P)

An expert will associate the decisions to the GPAand GRE score. They are then Tabulated.


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mGRE = {mL ,mM ,mH }

mGRE


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mGPA

mGPA = {mL ,mM ,mH }


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F


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Transform the crisp antecedents into a vector of fuzzymembership values.

Assume a student with GRE=900 and GPA=3.6.Examining the membership function gives

mGRE = {mL = 0.8 ,mM = 0.2,mH = 0}

mGPA = {mL = 0 ,mM = 0.6,mH = 0.4}


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F


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Converts the output fuzzy numbers into a unique (crisp) number

Method: Add all weighted curves and find the center of mass

F


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Fuzzy set with the largest membership value is selected. Fuzzy decision: F = {B, F, G,VG, E}

F = {0.6, 0.4, 0.2, 0.2, 0} Final Decision (FD) = Bad Student If two decisions have same membership max, use the average of the

two.


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F


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Fuzzy logic system

Fuzzifier Defuzzifier

Inference Mechanism

IF- THEN Rules

input Fuzzy sets Output Fuzzy sets

Crispinput Crisp

output


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The Sugeno Fuzzy Model, also known as TSK Fuzzy Model, wasproposed by Takagi, Sugeno and Kang (Sugeno and Kang, 1988;Takagi and Sugeno, 1985). The Sugeno model uses the fuzzy rule

exampled asIf a is A and b is B then z = f(a, b)

where, A and B are fuzzy sets. As like in Mamdani differentvariations can be done in the antecedent part of the rule.

In the exemplary rule, z is the crisp output produced with thefunction of f(a, b). As can be seen, the Sugeno model does notuse defuzzificaion process; its consequent part is designed toproduce crisp result directly

The function used in consequent part is often the polynomialwhich uses input variables, but this is not an obligation. Newfunction forms can be designed to reflect the matching degree of

antecedent parts of the rule, for example z = f(a,b) may be z =a-b+1 In the Sugeno model, after production of all local consequences;

the aggregation process is implemented with either weightedaverage or weighted sum of them.


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Tsukomoto fuzzy model was proposed by

Tsukomoto (1979) and it uses the rules whichcan be exampled as

If a is A then b is B

where, A and B are fuzzy sets.

It is similar with the rule structure of theMamdani model. Accordingly, the Mamdani andthe Tsukomoto models are similar in theprocesses from fuzzification to production oflocal consequents.

However, the Tsokomoto model uses weightedaverages of local consequents instead of thedefuzzification process.


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A = {u | mA(u) } is called -cut.1A 2A and 1+A 2+A, when 12, which implies

that the set of all distinct -cuts (as well as strong -cuts) is always a nested family of crisp sets.

+A = {u | mA(u) > } is called strong -cut.

0+A = {u | mA(u) > 0} is called support of A.

1

A = {u | mA(u) = 1} is called core of A.When the core of A is not empty, A is called normal;

otherwise, it is called subnormal.

The largest value of A is called the height of A,denoted as h

A

.

The set of distinct values ofmA(u),u Uis called thelevel set of A and denoted as A.


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Core

1A

2A

2

1

1

mA(u)

u

hA

(A is normal)


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Theorem: (Decomposition theorem of fuzzy sets): For any A

F(X),A = [0,1] A

We now convert each of the -cuts to a special fuzzy set A

defined for each uA by the formula A = .mA(u). We obtainthe following results:

0.2A = 0.2/x1+0.2/x2+0.2/x3+0.2/x4+0.2/x5

0.4A = 0/x1+0.4/x2+0.4/x3+0.4/x4+0.4/x5

0.6A = 0/x1+0/x2+0.6/x3+0.6/x4+0.6/x5

0.8A = 0/x1+0/x2+0/x3+0.8/x4+0.8/x5

1A = 0/x1+0/x2+0/x3+0/x4+1/x5

The union of these five special fuzzy set is exactly the original

fuzzy set A, that is, A = 0.2A 0.4 A 0.6 A 0.8 A 1A

A = 0.2/x1 +0.4/x2+0.6/x3+0.8/x4+1/x5

A t f f t th t i d i d f l i l t


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Any property of fuzzy sets that is derieved from classical set

theory is called a cutworthy property.

Examples:

A = B iffmA(u) = mB(u), u U, similarly,

A = B iffA = B, [0,1]

A B iffA B, [0,1]

The convexityof fuzzy sets: A fuzzy set defined on the set of

real numbers (or more generally, on any n-dim Euclidean space)

is said to be convex iff all of its -cuts are convex in theclassical sense. For a fuzzy set to be convex the graph must

have just one peak.

convex non convex


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Basic operationsSet union:A B { u,mA B (u) | (u Au

B) m (A B) (u) = Max (mA(u), mB(u))}

Set intersection:AB {u,mA B (u)| (uA uB) m (A B) (u) = Min (mA(u), mB(u))}

Set equality:A = B {u,mA (u) | (u A u

B) mA(u) = mB(u)}


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Basic operationsSet Complement:= {u,mA (u) | (mA (u) =

(1- mA(u))}

Set containment:AB {u |u (uAuB)

mA(u) mB(u)Concentration:CON(A)={u,mCON(A) (u) (uA

mCON(A) (u)= (mA(u))2}

Dilation:DIL(A) = {u, mDIL(A) (u) | (u A

mDIL(A) (u) = (mA(u))1/2}


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veryA(x) = A(x)2

moreorlessA(x) = A(x)1/2

Moreover

AB(x)= A(x) * B(x)

AB(x)=A(x) + B(x)-A(x) * B(x)

Principles of duality1-t(x,y) = s(1-x,1-y)


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mMore-or-Less High u)=0.9

mHigh (u)= 0.80

mVery High (u)=0.3

u

Very High

High

More-or-Lessm

mvery A (u) mA (u) mMore-or-Less A (u)


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f: F(X) F(Y) f: A B,

B = f(A)

B = A(x) / f(x) B(y) = max x|y=f(x) A(x) for all y Y.


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The inverse function, f-1, is from F(Y) to F(X).f-1 : F(Y) F(X).

According to the extension principle, for any BF(Y),[f-1(B)] (x) = B(f(x)) = B(y), for all x X, where y= f(x).

Example: Employees ages and their salaries

Query: What is a young employees salary?

Answer: We use extension principle here. Let us have a function f: X Y,where X = {20,25,30,35,40,45,50,55,60,65} and

Y= {2.5, 3, 3.5, 4.0, 4.5, 5.0}

Age in years 20 25 30 35 40 45 50 55 60 65

Salary $ in K 2.5 2.5 3.0 3.5 3.5 4.0 4.0 4.5 4.5 5.0

Fi t t F l t th i f th t f t A f


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First step: Formulate the meaning of the concept young as a fuzzy set A ofgeneral form A = mA(x) / x for all x X. Assume that

Ayoung = 1/20 +1/25+0.8/30+0.6/35+0.4/40+0.2/45+0/50+0/55+0/60+0/65

Second step: Use the fuzzy set A and information in the table to determine an

appropriate fuzzy set B that captures the meaning of the linguistic expression

young employees salary.

This fuzzy set is dependent on A via function f which for each x in X assigns a

particular y = f(x) in Y. This dependency is expressed by the general formB(y) = max x|y=f(x) A(x) = max x|y=f(x)mA(x) / f(x)

B= mA(x) / f(x) = 1/f(20) + 1/f(25) + 0.8/f(30) + 0.6/f(35) + 0.4/f(40) + 0.2/f(45)+ 0/f(50) + 0/f(55) + 0/f(60) + 0/f(65)

= 1/2.5 + 1/2.5+ 0.8/3+ 0.6/3.5+ 0.4/3.5+ 0.2/4+ 0/4+ 0/4.5+ 0/4.5+ 0/5

Third step: B(y) = max x|y=f(x) A(x) = 1/2.5 +0.8/3+0.6/3.5+0.2/4+0/4.5+0/5,

which denotes the salary of young employes in the company.


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In order to develop computation with fuzzy sets, we need to takecrisp functions and fuzzify them. A principle for fuzzyfying crispfunctions is called the extension principle.

f: XY, where X and Y are crisp sets.

We say that the function is fuzzified when it is extended to act onfuzzy sets defined on X and Y. Formally, the fuzzified function, f, hasthe form

f: F(X) F(Y), where F(X) and F(Y) denote the fuzzy power set(the set of all fuzzy subsets) of X and Y, respectively.

To qualify as a fuzzified version of f, function f must conform to fwithin the extended domain F(X) and F(Y). This is guaranteed when a

principle is employed that is called an extension principle. Accordingto this principle,

B = f(A) is determined for any given fuzzy set AF(X) via the formulaB(y) = max x|y=f(x) A(x) for all y Y.

When the maximum does not exist, it is replaced with the supremum.


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POSSIBILITY

ANDNECESSITY


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Probability Dist: states the probability that the givenvariable takes a certain value

Istanbulda kilise bombaland.

Bombacnn yann(variable) 25 (certain value) olma ihtimali nedir?

nceki bombaclarn yan biliyormuyuz

Possibility Dist: states the possible value of the variableor the possibility that the variable takes a certain value

Istanbulda kilise bombaland.

Grg ahidi yann (variable), 20-25 aralnda olduunu syledi.Bombacnn 23(certain value) yanda olmas mmkn mdr ?,

Bombacnn19 yanda olmas mmkn mdr?

kiside ayn mmknlkte midir?

nceki bombaclarn yann durumla ilgisi nedir?)


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A possibility distribution, , maps a given domain ofdefinition into the interval [0,1]. We can view a possibility distribution as a mechanism for

interpreting factual statements involving fuzzy sets.

Example: the statement, Temperature is High, whereHigh is defined as mHigh : T [0,1], translates into apossibility distribution, (T) = mHigh(T).

For more complex statement, Temperature is High butnot too high translates into a possibility distribution in

terms of conjunction of the terms High and Not VeryHigh:

(T)=min(mHigh(T),mNotVeryHigh(T))=min[mHigh(T),(1-mHigh(T)

2)].


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Fuzzy logic offers an appealing alternative, such as

assigning the fuzzy set Young to the age of thesuspect. Thus, we obtain a distribution about thepossibility degree of the suspects age (e.g., thepossibility that the suspect is 19 is 0.7, while thepossibility of 21 - 28 is 1.0),

Age(suspect) (x) = mYoung (x),where denotes a possibility distribution of thesuspects age, and x is a variable representing apersons age.


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Pos(A|X) denotes theposibility of thecondition X is A given the possibilitydistribution X

Nec(A|X) denotes the necessity of thecondition X is A given the possibilitydistribution X.


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A x1

2 4 6 8 10

0.5

x deikenin almas mmkn olan deerlere bakldnda,

x deikeninin A olma mmknl nedir?x deikeninin A olma gereklilii nedir ?

x hangi deeri alrsa alsn, x Amdr?


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Possibility Dist: shows the possibilities of certainvalues can be assigned to x

Possibility Measure: shows the measure of

condition matching by using given possibilitydistrubution for x


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Jacks age is between 20 and 25 years old A person ages exceeds 22 years old

Is it possible for a person to be Jack ? YES Is it necessary for a person to be Jack ? NO

A : jack

X : Persons age


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Jacks age is between 20 and 25 years old A person ages is between 21 and 23 years

old

Is it possible for a person to be Jack ? YES

Is it necessary for a person to be Jack ?YES

A : jack

X : Persons age


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In Crisp Pos(AIx ) = {0,1}

Nec(AIx ) = {0,1}

In Fuzzy

Pos(AIx ) = [0,1]

Nec(AIx ) = [0,1]

A

Not A

B1

B2

B3

In which degree


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These observations can provide insights on the general

relationships between the two measures. The

relationships 1a and 1b can be generalized to

Nec (A|X) Pos(A|X)

The relationships 2a and 2b can be generalized to

1- Pos(A|X) = Nec(A|X).

Thus, one can automatically derive necessity measure

using a possibility measure.

In general, when we assign a fuzzy set A to a variable

X, the assignment results in a possibility distribution

of X, which is defined by As membership function:

X

(x) = mA

(x).


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The possibility measure for a variable X tosatisfy the condition X is A given a possibilitydistribution X is defined to be

Pos(A|X) = sup xiU [(mAX),

where denotes a fuzzy intersection (i.e., a

fuzzy conjunction) operator. A common choice of the fuzzy intersection

operator for calculating the possibilitymeasure is the min operator. Thus,

Pos(A|X) = supxiU [min (mA (xi), X(xi))]. It is easy to derive the corresponding formula

for the necessity measure:

Nec(A|X) = infxiU [max (mA (xi), 1-X(xi))].


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Example: Let the universe of discourse of a persons age be{10,15,20,25,30,35,40,45,50}, and

The age possibility distribution of a suspect (denoted J) be:

Age (J) = 0.2/15 + 0.5/20 + 1/25 + 0.8 /30

Suppose that the membership function for the linguistic term

Young is defined as a discrete fuzzy set as follows:Young = 1/10 + 1/15 + 1/20 + 0.8 / 25 + 0.4 /30 + 0.2 /35

Using the equation

Pos(mA|X) = Pos(mYoung |Age (J)) = sup xiU [(mYoungAge (J)]=

Pos(mYoung |Age (J)) =max {min(mYoung , Age (J))}= max {0.21, 0.51, 10.8, 0.80.4}= max {0.2, 0.5,0.8, 0.4}

Pos(mYoung |Age (J)) = 0.8

Example (cont ): Let the universe of discourse of a persons age be


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Example (cont.): Let the universe of discourse of a person s age be{10,15,20,25,30,35,40,45,50}, and

The age possibility distribution of a suspect (denoted J) be:

Age (J) = 0.2/15 + 0.5/20 + 1/25 + 0.8 /30Suppose that the membership function for the linguistic term Young is definedas a discrete fuzzy set as follows:

Young = 1/10 + 1/15 + 1/20 + 0.8 / 25 + 0.4 /30 + 0.2 /35

To calculate the necessity measure, we first calculate the complement of

the possibility distribution of a suspect Js age:1-Age(J) = 1/10+0.8/15+0.5/20+0/25+0.2/30+1/35+1/40+1/45+1/50 The necessity measure is obtained by

Nec(A|X)= infxiU [ mA(xi) 1-X(xi)]=Nec(mYoung |Age (J)) = infxiU[max(mYoung, 1- Age (J)]

Nec(mYoung|Age(J))=min{11,10.8,10.5,0.80,0.40.2,0.21,01,01,01}

= min {1, 1, 1, 0.8, 0.4, 1, 1,1, 1} = 0.4.

Therefore, the possibility that suspect J is young is 0.8, while the necessitythat he/she is young is 0.4.


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If the possibility distribution x is is a subsetof A, necessity measure of A given x must be1

The possibility measure is the height of the

intersection


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Problem hava scakl 15 derece civarlarnda (Keskindeer bilinmiyor) Hava scaklnn Cold olma mmknl ve gereklilii

nedir?

Havann Hot olma mmknl ve gereklilii nedir?

x(T)0 155 10 18 25 30 40 45

xColdTemperate

Hot


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[email protected]

2007
mailto:[email protected]:[email protected]


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Fuzzy relation generalizes the notion ofclassical relation

friend relation will classify all humansrelationships into either being friend or not

being friend Binary Rel: holds two object

Hold two or more object

n-ary relation: x student takes y couse at z

semester in year w

Bi l i d i bl h d i


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Binary relation on x and y variables whose domains areX and Y, can defined as a set of ordered pair

R = {(x,y) I x


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Fuzzy Relation IdealForm

Height = {150,155,160,165}

Weight = {50,55,60,65}

Matrix form of fuzzy relation

150

155

160

165

50 55 60 65

1

0.8

0.6

0.8

1

1

0.8 0.6 0.4

0.8 0.6

0.80.8

0.80.60.4

Height

Weight


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IdealForm(155cm, 50kg) = 0.8

Questions

What is the possibility distribution of specific

weight-height pairs, if we know the person is

ideal form

What is the possibility of one persons height ifwe know his weight and he is ideal form


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Aye is about 160cm tall and she is idealForm What is the the possibility that Ayes height is

155 cm given that she is about 160 cm tall

What is the possibility that Ayes weight is 50 kg

if she is about 160 cm tall (Answer can be givenby applying composition of fuzzy relation)


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[(possible-h (150) idealForm(150, 50))U(possible-h (155) idealForm(155, 50)) U

(possible-h (160) idealForm(160, 50)) U

(possible-h (165) idealForm(165, 50)) ]

Possible-w(50)


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U All wj [(possible-h (150) idealForm(150, wj))(possible-h (155) idealForm(155, wj)) U

(possible-h (160) idealForm(160, wj)) U

(possible-h (165) idealForm(165, wj)) ]Possible-w(wj)

All wj[Possible-w(wj) U hi Possible-h(hi) idealForm(hi, wj)]


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height(x)(hi) : The possibility degree for a personsheight to be hi

IdealForm((hi,wj) : The possibility degree for a idealFormperson two have a height hi and weight wj

weight(x)(wj) : Uhi(height(x)(hi)IdealForm((hi,wj))Y(x)(yi) : Uxi(xxiR(xi,yj))


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Max-min CompositionY(x)(yi) : max (xxi,R(xi,yj))

xi

Max-product Composition

Y(x)(yi) : max (xxi x R(xi,yj))xi


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She is about 160 cm, find the possibilitythat she is 50 kg weight if she isidealForm

About 160 cm = {0.2/150, 0.7/155, 1/160,0.7/165}

weight(50) = (0.21) U (0.70.8) U (10.6) U(0.70.4)

= 0.2 U 0.7 U 0.6 U 0.4= 0.7

If she is idealForm and she is about 160 cm, she isweight is 50 kg with 0.7 possibility degree


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Fuzzy Relation IdealForm

Height = {150,155,160,165}

Weight = {50,55,60,65}

Matrix form of fuzzy relation

150

155

160

165

50 55 60 65

1

0.8

0.6

0.8

1

1

0.8 0.6 0.4

0.8 0.6

0.80.8

0.80.60.4

Height

Weight


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Let R be a fuzzy relation and subset of U1xU

2. R

is the cylindrical ext. of R and its membership

function is defined as

R (U1,U2)= R(Ui1,Ui2)

150

155

160

165

50 55 60 65

0.2

0.7

1.0

0.7

1.0

0.7

0.2 0.2 0.2

0.7 0.7

1.01.0

0.70.7

Hei

ght

Weight

0.7

About 160 cm =


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y = f(x) = x2-6x+11f(4) = 16-24+11 = 3

f(around-4) = ?

around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6

f(around-4) =

0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6)

f(around-4) = 0.3/3+0.6/2+1/3+0.6/6+0.3/11

= 0.6/2+(0.3U1)/3+0.6/6+0.3/11

= 0.6/2+1/3+0.6/6+0.3/11


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FR associates a condition described usinglinguistic variables and fuzzy sets to a

conclusion

The main feature of reason using these

rules is its partial matching capability

Infer result even if its condition is partially

matched

FR Fuzzy mapping rule

Fuzzy implication rule


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There are two different kinds of fuzzy rules: Fuzzymapping rules and Fuzzy implication rules.

A fuzzy mapping rule describes an association;therefore, its fuzzy relation is constructed from theCartesian product of its antecedent fuzzy conditionand its consequent fuzzy condition.

A fuzzy implication rule, however, describes a

generalized logic implication; therefore, its fuzzyrelation needs to be constructed from the semanticsof a generalization to implication in multi-valuedlogic.

The difference between the semantics of fuzzy mapping rules and


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The difference between the semantics of fuzzy mapping rules andfuzzy implication rules can be seen from the difference in theirinference behavior. Even though these two types of rules behave the

same when their antecedents are satisfied, they behave differentlywhen their antecedents are not satisfied.

Example:

Implication rule (logic representation), Mapping rule (procedural

representation)

Given:x [1,3] y [7,8], stm:If x[1,3] Then y[7,8]Input: x=5 Variable value: x = 5

Infer: y is unkown (y [0,10] Execution result: no action

We interested in finding the functional relationship between a set


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g pof observable parameters and one or multiple parameter whosevalues we do not know

The needs to approximate a function of interest is often due toone ormore of the following reasons:

1) The mathematical structure of the function is not precisely

known.

2) The function is so complex that finding its precisemathematical form is either impossible or practicallyinfeasible due to its high cost.

3) Even if finding the function is not impractical, implementingthe function in its precise mathematical form in a product orservice may be too costly. This is particularly important forlow cost high volume products (e.g., automobiles, cameras,and many other consumer products).

Fuzzy rule-based function approximation is a


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y pppartition-based technique.

The partition-based approximation techniquesapproximate a function by partitioning the inputspace of the function and approximate the

function in each partitioned region separately(e.g.,piecewise linear approximation).

Because each fuzzy rule approximates a smallt f th f ti th ti f ti i


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segment of the function, the entire function isapproximated by a set of fuzzy mapping rules.

We refer to such a collection of fuzzy mapping rulesas fuzzy rule-based models or simply fuzzy models(describing a mapping (i.e., function) from a set ofinput variables to a set of output variables.)

Example: a fuzzy model of the stock market can be used to

predict future changes of the IMKB average.

A fuzzy control model of apetrochemical processcan be used to predict the future state of theprocess.

A fuzzy model can be defined as a model that is


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yobtained by fusing multiple local models that areassociated with fuzzy subspaces of the given inputspace.

The result of fusing multiple local models is usually afuzzy conclusion, which is converted to a crisp final

output through a defuzzification process.

The main difference between fuzzy and nonfuzzyrules for function approximation lies in theirinterpolative reasoning capability, which allows

the output of multiple fuzzy rules to be fused for agiven input.


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A fuzzy partition of a space is a collection of

fuzzy subspaces whose boundaries partiallyoverlap and whose union is the entire space. ,

Formally, a fuzzy partition of a space as acollection of fuzzy subspace A

iof S that

satisfies the following condition:mAi(x) = 1, x S.

That is, for any element of the space, itsmembership degree in all subspaces alwaysadds up to 1.


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We call a collection of fuzzy subspaces Ai of S a weak fuzzy partition

of S if and only if it satisfies the following condition:0< mAi(x) 1, x S.

The greater than 0 condition requires each element in the space Sto be covered by at least one fuzzy subspace in the partition.

The sum to 1 condition of a fuzzy partition can be relaxed to thesum to less or equal to 1 condition because the interpolativereasoning of fuzzy models includes a normalization step.

Research Note: It has been shown that mAi(x) = 1 is a desirable

property in a framework for analyzing the stability of fuzzy logiccontrollers.


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I i l li l b l d l (i h


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In particular, a nonlinear global model (i.e., whoseinput-output mapping function is not linear) can oftenbe approximated by a set of linear local models. Thiscan be understood by remembering the well-knownapproximation technique called piecewise linearapproximation, which approximates an arbitrarynonlinear function using segments of lines.

The following figure shows such an approximationtechnique, where dotted line indicates the functionbeing approximated.

y

x

Piecewise linear approximation has two major components:


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Piecewise linear approximation has two major components:

1. Partitioning the input space to crisp regions

2. Mapping each partitioned region to a linear local model.

The main difference between fuzzy modeling and piecewise linearapproximation is that the transition from one local subregion to aneighboring one is gradual rather than abrupt.

Generally, the mapping from a fuzzy subspace to a local model isrepresented as a fuzzy if-then rule in the form of:

Ifx is in FSi Then yj = LMi (x) where x and yj denote the vector of input variables and output

variable, respectively, FSi and LMi denote ith fuzzy subspace and the

corresponding local model, respectively.

The local model can be of three different types:


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The local model can be of three different types: 1. Crisp constant: This type of local model is

simply a crisp (nonvisual) constant. For example;If xi is Small Then y = 4.5

2. Fuzzy constant: A local model that is a fuzzyconstant (e.g., Small) belong to this type. For

example;If xi is Small Then y is Medium

3. Linear Model: this describes the output as alinear function of the input variables, such as:

If x1 is Small And x2 is Large Then y = 2x1 + 5x2+ 3.


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Fuzzy models use interpolative reasoning to fusemultiple local models into aglobal model.

The basic idea behind interpolative reasoning is

analogous to drawing a conclusion from a panel of

experts, each of whom is specialized in a subarea of theentire problem.

Each experts opinion is associated with a weight, whichreflects the degree to which the current situation is in

the experts specialized area.

These weighted opinions are combined to form an

overall opinion.


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In this analogy, an expert corresponds to a fuzzy if-then rule, the specialized subarea of the expert

corresponds to the fuzzy subspace associated with

the if-part of the rule.

The weight of an experts opinion is determined bythe degree to which the current situation belongs

to the subspace.

We may interpret a possibility distribution eitherthrough linguistic approximation or through


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through linguistic approximation, or throughdefuzzification.

The former gives a qualitative interpretation, while thelatter gives a quantitative summary and is morecommonly used in fuzzy logic applications, i.e.,industrial applications.

Given a possibility distribution of a fuzzy modelsoutput, defuzzification amounts to selecting a singlerepresentative value that captures the essentialmeaning of the given distribution. There are three

common defuzzification techniques: mean of maximum,center of area, and height.

Mean of Maximum (MOM): This calculates the average of those outputl th t h th high t ibilit d g


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values that have the highest possibility degrees.

Suppose y is A is a fuzzy conclusion to be fuzzified. We can expressthe MOM defuzzification method using the following formula:

MOM (A) = y*P y* / |P|Where P is the set of output values y with highest possibility degree inA.

If P is an interval, the result of MOM defuzzification is obviously themidpoint in that interval.

This technique does not take into account the overall shape of thepossibility distribution.

Center of Area (COA): This method (also referred to as


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Center of Area (COA):This method (also referred to asthe center-of-gravity, or centroid method) is the most

popular defuzzification technique. Unlike MOM, the COA method takes into account the

entire possibility distribution in calculating itsrepresentative point.

This method is similar to the formula for calculating the

center of gravity in physics, if we view mA(x) as thedensity of mass at x. If x is discrete, the fuzzification result of A is:

COA(A) = xmA(x) * x / xmA(x). The main disadvantage of the COA method is its high

computational cost. However, the calculation can besimplified for some fuzzy models.

A mathematical representation of fuzzy mapping rules: A fuzzy


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A mathematical representation of fuzzy mapping rules: A fuzzymapping rule imposes an elastic constraint on possible associations

between input and output variables.

It is elastic because a fuzzy rule can describe input-outputassociations that are somewhat possible (i.e., the gray areabetween totally possible and totally impossible).

The degree of possibility of an input-output association imposed bya rule R can be expressed as a possibility distribution, denoted byR.

Since a fuzzy relation is a general way for describing a possibilitydistribution, it is natural to use it to represent the possibilitydistribution imposed by a fuzzy rule.

How do you construct the fuzzy relation that represent


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How do you construct the fuzzy relation that representfuzzy mapping rules?

The answeris: Use the concept ofCartesian product!

A fuzzy mapping rule is represented mathematically asfuzzy relations formed by the Cartesian product of thevariables referred to in the rulesif-part and then-part.

For example, the mapping rule is:

IF x is A, THEN y is B,whichis mathematically represented as a fuzzy relation

R defined asmR(x,y)=mAB(x,y)=min{mA(x), mB(y)}.


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Example: Let us consider the following fuzzymapping rule from X to Y, whereX = {2,3,4,5,6,7,8,9} and Y = {1,2,3,4,5,6}

If x is Medium, Then y is Small

where Medium and Small are fuzzy subsets of Xand Y characterized by the followingmembership functions:

Medium 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 +0.5/8 + 0.2/9Small 1/1 + + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6

The fuzzy relation R representing the rule is the


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Cartesian product ofMedium and Small. If we use themin operator to construct the Cartesianproduct, we

have mR(x,y) = min{mMedium(x), mSmall(y)}.

The resulting fuzzy relation representing the rule is

0.1 0.1 0.1 0.1 0.1 0.1

0.3 0.3 0.3 0.3 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.1

0.5 0.5 0.5 0.5 0.3 0.1

0.2 0.2 0.2 0.2 0.2 0.1

R

Medium 0.1/2 + 0.3/3 +0.7/4 + 1/5 + 1/6 + 0.7/7 +0.5/8 + 0.2/9

Small 1/1 + + 0.9/3 +0.6/4 + 0.3/5 + 0.1/6


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The theoretical foundation of fuzzy mappingrules is afuzzy graph and a compositional ruleof inference.

A fuzzy graph can be conveniently described by

fuzzy rules in the form ofIf x is A Then y is B

Such a statement (or rule) generalizes thedependency relationship between variables in a

lookup table such asIf x is 5 Then y is 10

If x is 10 Then y is 14

A t f h d d i f f ti l


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A set of such dependencies form a functional

mapping from x to y. Generalizing point-to-point mappings to a

mapping from fuzzy sets to fuzzy setsintroduces two benefits.

1. We can reduce the total number of point-to-point rules required for approximating afunction

2. Using words in fuzzy rules makes it easier tocapture, understand, and communicate the

underlying human knowledge.

Let f* be a fuzzy graph described by a set of


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Let f be a fuzzy graph described by a set offuzzy mapping rules in the form of

If x is Aj Then y is Bj. The fuzzy graph can be expressed

mathematically asf* = j A j Bj

where A and B are two fuzzy subsets of X and Yrespectively.

A fuzzy graph f* from X to Y is union ofCartesian products involving linguistic input-output associations (i.e., pairs ifx is A

i

andy is Bi). The resulting fuzzy graph is basicallya fuzzy relation.


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A fuzzy graph describes a functional mapping

between a set of input linguistic variables and anoutput linguistic variable.

Example: If X is small Then Y is small.

If X is medium Then Y is large.

If X is large Then Y is small.Which form a fuzzy graph f*, where

f* = small small + medium large + large small In f*, + and denote, respectively, the disjunction

and Cartesian product. An expression of the form A B where A and B are words (fuzzy sets) is referred asa Cartesian granule.


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f (crisp function)

f* (fuzzy graph)

y

x

small medium large

large

small

Fuzzy Graph Approximation by a Disjunction of Cartesian Products

Th i f (i i t l ti i )


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The inference (i.e., interpolative reasoning)

of such a fuzzy rule-based model is based onthe compositional rule of inference.

The net effect is a possibility distributionover the domain of definition of the output

variable. In particular,B = A o f*

where f* represents the fuzzy graph of agiven fuzzy model, A is an input which canbe fuzzy or crisp, and B is the inferredoutput value before defuzzification.


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Using the definion of a compositional rule ofinference, we express this as

A o f* = ProjY (cyl-ext(A) f*)= ProjY [cyl-ext(A) (i AiBi)]= x X [cyl-ext(A) (i AiBi)]

where X and Y are the universe of discourse

of x and y respectively, and cyl-ext(A) is thecylindirical extension ofA to X Y.


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Example: Consider the following rule (again)

If x is Medium Then y is Small

Input data is: X is Small, where Small for x is defined

as

Small 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6 To find out the possible values of y, we compose the

possible values of x with the fuzzy relation T using the

sup-min composition:0.1 0.1 0.1 0.1 0.1 0.1

0.3 0.3 0.3 0.3 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.10.7 0.7 0.7 0.6 0.3 0.1

0.5 0.5 0.5 0.5 0.3 0.1

0.2 0.2 0.2 0.2 0.2 0.1

Small R = [1 0.9 0.6 0.3 0.1 0 0 0] o

= [0.6 0.6 0.6 0.6 0.3 0.1],

y = 0.6/1+0.6/2+0.6/3+0.6/4+0.3/5+0.1/6

as the result of the inference.


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In this example, we consider only one rule.

However, a fuzzy model for function

approximation is usually formed by a set of

fuzzy mapping rules. In such a case, the fuzzy relation of the entire

model (denoted FM) is constructed by forming

the union of fuzzy relations of individual rules:

mFM = mR1 mR2 mRn


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Bolum sonu


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There are three types of fuzzy rule-based models

for function approximation:1. The Mamdani model2. The Takagi-Sugeno-Kang (TSK) model,3. Koskos additive model (SAM)

The inference scheme of SAM is similar to thatof TSK model. Both of them use an inferenceanalogous to the weighted sum to aggregatethe conclusion of multiple rules into a final

conclusion. Therefore, we refer to these rule models as

additive rule models.


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The Mamdani Model:

One of the most widely used fuzzy models in practice is the Mamdanimodel, which consists of the following linguistic rules that describe amapping from U1 U2 Ur to W.

Ri: If x1 is Ai1 and and xris AirThen y is Ci

where xj is (j = 1,2,..r) are the input variables, y is the output variable,and A

ijand C

iare fuzzy sets for x

jand y respectively.

Given inputs of the form: x1 is A1 , x2 is A2 x r is Ar where A1 ,A2 Arare fuzzy subsets of U1, U2, ,Ur (e.g., fuzzy numbers), the contributionof rule Ri to a Mamdani models output is a fuzzy set whose membershipfunction is computed by

mCi(y) = (i1i2ir) mCi (y)

where mCi(y) is the matching degree of rule Ri, and where ij is thematching degree between xj and Ris condition about xj.

ij = sup xj (mAj(xj) mAij (xj) )


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and denotes the min operator. This is the clipping inference method.

The final output of the model is the aggregation of outputs from all

rules using the max operator.

mC (y) = max(mC1(y), mC2(y),..., mCm(y))

Notice that the output C is a fuzzy set. This output can be defuzzified

into a crisp output using one of the defuzzification techniques.

The Mamdani model can be derived from the following operators:

1) Sup-min composition

2)Min for Cartesian product

3) Min for conjunctive conditions in rules

4) Max for aggregating multiple rules


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One of the main advantages of the TSK model is that

it can approximate a function using fewer rules. In contrast, the Mamdani model combines inference

results of rules using superimposition, not addition.Hence nonadditive rule model.

The Mamdani and SAM use rules whose consequent

part is a fuzzy set (uses a fuzzy constant as its ruleslocal model).

The TSK model uses a rule whose then part is a linearmodel (uses a linear local model).

The fundamental difference between the Mamdani

and SAM lies in the choice of composition,conjunction, and disjunction operators in theirreasoning (inference mechanism).


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Any logic system has two major components:1. a formal language for constructing statements about

the world,

2. a set of inference mechanisms for inferringadditional statements about the world from those

already given. Fuzzy logic is the most commonly used

reasoning scheme in applications of fuzzylogic (narrow sense).

The subject is complicated by the fact thatthere isnt a unique definition of fuzzyimplications.


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An important goal of fuzzy logic is to be ableto make reasonable inference even when thecondition of an implication rule is partiallysatisfied.

This capability is sometimes referred to asapproximate reasoning. This is achieved infuzzy logic by two related techniques:

1. representing the meaning of a fuzzy implication ruleusing a fuzzy relation, and

2. obtaining an inferred conclusion by applying thecompositional rule of inference to the fuzzyimplication relation.

Fuzzy rule-based inference is a generalization of a


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logical reasoning scheme (inference) called modus

ponens (MP) and modus tollens (MT).

It combines the conclusion of multiple fuzzy rules in

a manner similar to linear interpolation. For

example:

Rule: If a persons IQ is high Then the person is smartFact: Jacks IQ is high

Infer: Jack is smart.

Rule: If a persons IQ is high Then the person is smartFact: Jack is not smart

Infer: Jacks IQ is not high.


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First, these inferences insist on perfectmatching.

However, common sense reasoning suggest thatwe can infer Jack is more or less smart whenthe Jacks IQ is more or less high is given.

Secondly, these inferences cannot handleuncertainty.

For instance, if Jack told us his IQ is high butcannot provide documents supporting the claim,

we may be somewhat uncertain about the claim. Under such a circumstance, however, ordinary

logic cannot reason about the uncertainty.


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These limitations motivated L.A. Zadeh to develop a

reasoning scheme that generalizes classical logic so

that

It can conduct common-sense reasoning underpartial matching, and

It can reason about the certainty degree of a

statement

In particular, logic implications are generalized to

allow partial matching.

Rule: A persons IQ is high the person is smart

Fact: Jacks IQ is somewhat high

Infer: Jack is somewhat smart


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The second limitation of logic (i.e., inability to deal

with uncertainty) has motivated another extension toclassical logic: multivaluedlogic.

Since fuzzy logic also generalizes the truth-values inclassical logic beyond true and false, it is related to

multivaluedlogic. However, fuzzy logic differs from multivalued logic

in that it also addresses the first limitation of logic(i.e., restricted to perfect matching) by usinglinguistic variables in its antecedent.

Consequently, the statement in the antecedentdescribes an elastic condition that can be partiallysatisfied.


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Other approaches for reasoning under

uncertainty include

1. Bayesian probabilistic inference,

2. Dempster-Shafer theory,

3. nonmonotonic logic. Fuzzy logic, among these, is unique in that it

addresses both the uncertainty management

problem and the partial matching issue.

d l l f


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Let us consider an implication involving fuzzy sets

(i.e., fuzzy implication):(x is A) (y is B)

where A and B are fuzzy subsets of U and V,respectively.

This implication also specifies the possibility ofvarious point-to-point implications.

The possibilities are a matter of degree. Therefore,the meaning of the fuzzy implication can berepresented by an implication relation R defined as

Rl(xi,yj) = l ((x = xi) (y = yj)) Where l denotes the possibility distribution

imposed by the implication.

I f l i hi ibili di ib i i


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In fuzzy logic, this possibility distribution is

constructed from the truth values of the instantiatedimplications obtained by replacing variables in theimplication (i.e., x and y) with pairs of their possiblevalues (i.e., xi and yj):

((x = xi)

(y = yj)) = t ((xi is A)

(yj is B))where t denotes the truth value of a proposition.

For the convenience of our discussion, we refer tothe truth values as i and j as follows:

t(xi

is A) = i

t(yj is B) = jt((xi is A) (yj is B)) = I(i,j)

we call the function I an implication function.

There is not a unique definition for


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There is not a unique definition for

implication function. Different implication functions lead to

different fuzzy implication relations. V

arious definitions of implication functionshave been developed from both the fuzzylogic and multivalued logic researchcommunities.

However, all of them at least satisfy thefollowing rules:

I(0, j) = 1I(i, 1) = 1

Given a possibility distribution of the variable X and


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Given a possibility distribution of the variable X and

the implication possibility from X to Y, we infer thepossibility distribution of Y.

Given: X = xi is possible AND

X = xi Y = yj is possible

Infer: Y = yj is possible

More generally, we have

Given: (X= xi ) = a AND(X = x

i

Y = yj

) = b

Infer: (Y = yj ) a bWhere is a fuzzy conjunction operator.

Wh i f j ti t


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Where is a fuzzy conjunction operator.

When different values of X imply an identical

value of Y say yj with potential varying

possibility degrees, these inferred

possibilities about Y = yj need to be combinedusing fuzzy disjunction.

Hence, the complete formula for computing

the inferred possibility distribution of Y is

(Y = yj ) = xi ((X= xi) ((X = xi Y = yj )))

which is the compositional rule of inference.

Even though both fuzzy implication and fuzzy


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Even though both fuzzy implication and fuzzy

mapping rules use the compositional rule ofinference to compute their inference results, theirusage differ in two ways.

First, the compositional rule of inference is appliedto individual implication rules, while composition isapplied to a set of fuzzy mapping rules thatapproximate a functional mapping.

Second, the fuzzy relation of a fuzzy mapping rule isa Cartesian product of the rules antecedent and its

consequent part. An entry in the fuzzy implicationrelation, however, is the possibility that a particularinput value implies a particular output value.


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Fuzzy Implication Rules Fuzzy Mapping Rules

Purpose Generalizes implications for

handling imprecision

Approximate functional mappings

Desired Inference Generalizes modus ponens and

modus tollens

Forward only

Application Diagnostics, high-level decision

making

Control, system modeling, and signal

processing

Related Disciplines Classical logic, multivalued logic

(other extended logic systems)

System ID, piecewise linear

interpolation, neural networks

Typical Design Approach Designed individually Designed as a rule set

Suitable Problem Domains Domains with continuous and

discrete variables

Continuous nonlinear domains

Criteria of fuzzy Implications:


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Criteria of fuzzy Implications:

The criteria of desired inference involvingfuzzy implication results can be grouped intosix:

1. The basic criterion of modus ponens

2. The generalized criterion of modus ponensinvolving hedges,

3. The mismatch criterion

4. The basic criterion of modus tolens

5. The generalized criterion of modus tolensinvolving hedges, and

6. The chaining criterion of implications

1 The basic criterion of modus ponens


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1. The basic criterion of modus ponens

The basic criterion of modus ponens

Given: x is A y is B

x is A

Infer: y is B


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Fuzzy Rule associates a conditiondescribed using linguistic variables and

fuzzy sets to a conclusion

The main feature of reason using these

rules is its partial matching capability Infer result even if its condition is partially

matched

Fuzzy Rules

Fuzzy mapping rule

Fuzzy implication rule

We interested in finding the functional relationship between a setof observable parameters and one or multiple parameter whosevalues we do not know


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values we do ot ow

The needs to approximate a function of interest is often due toone ormore of the following reasons:

1) The mathematical structure of the function is not preciselyknown.

2) The function is so complex that finding its precisemathematical form is either impossible or practicallyinfeasible due to its high cost.

3) Even if finding the function is not impractical, implementingthe function in its precise mathematical form in a product orservice may be too costly. This is particularly important forlow cost high volume products (e.g., automobiles, cameras,and many other consumer products).

Fuzzy rule-based function approximation is apartition-based technique.


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The partition-based approximation techniquesapproximate a function by partitioning the inputspace of the function and approximate thefunction in each partitioned region separately(e.g.,piecewise linear approximation).

Because each fuzzy rule approximates a smallsegment of the function, the entire function isapproximated by a set of fuzzy mapping rules.


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approximated by a set of fuzzy mapping rules.

We refer to such a collection of fuzzy mapping rulesas fuzzy rule-based models or simply fuzzy models(describing a mapping (i.e., function) from a set ofinput variables to a set of output variables.)

Example: a fuzzy model of the stock market can be used to

predict future changes of the IMKB average. A fuzzy control model of apetrochemical process

can be used to predict the future state of theprocess.

A fuzzy model can be defined as a model that isobtained by fusing multiple local models that areassociated with fuzzy subspaces of the given input


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associated with fuzzy subspaces of the given input

space.

The result of fusing multiple local models is usually afuzzy conclusion, which is converted to a crisp finaloutput through a defuzzification process.

The main difference between fuzzy and nonfuzzyrules for function approximation lies in theirinterpolative reasoning capability, which allowsthe output of multiple fuzzy rules to be fused for a

given input.


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The four major concepts in fuzzy rule-basedmodels thus are as follows:

1. Fuzzy partition,

2. Mapping of fuzzy subregion to local models,

3. Fusion of multiple local models,4. Defuzzification.

A fuzzy partition of a space is a collection offuzzy subspaces whose boundaries partially


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fuzzy subspaces whose boundaries partiallyoverlap and whose union is the entire space. ,

Formally, a fuzzy partition of a space as acollection of fuzzy subspace Ai of S that

satisfies the following condition:mAi(x) = 1, x S.

That is, for any element of the space, its

membership degree in all subspaces alwaysadds up to 1.

We call a collection of fuzzy subspaces Ai of S a weak fuzzy partitionf S if d l if i i fi h f ll i di i


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iof S if and only if it satisfies the following condition:

0< mAi(x) 1, x S.

The greater than 0 condition requires each element in the space Sto be covered by at least one fuzzy subspace in the partition.

The sum to 1 condition of a fuzzy partition can be relaxed to thesum to less or equal to 1 condition because the interpolativereasoning of fuzzy models includes a normalization step.

Research Note: It has been shown that mAi(x) = 1 is a desirableproperty in a framework for analyzing the stability of fuzzy logiccontrollers.

A mathematical representation of fuzzy mapping rules: A fuzzymapping rule imposes an elastic constraint on possible associations


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pp g p p

between input and output variables.

It is elastic because a fuzzy rule can describe input-outputassociations that are somewhat possible (i.e., the gray areabetween totally possible and totally impossible).

The degree of possibility of an input-output association imposed bya rule R can be expressed as a possibility distribution, denoted byR.

Since a fuzzy relation is a general way for describing a possibilitydistribution, it is natural to use it to represent the possibilitydistribution imposed by a fuzzy rule.

How do you construct the fuzzy relation that representfuzzy mapping rules?


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f y pp g

The answeris: Use the concept ofCartesian product!

A fuzzy mapping rule is represented mathematically asfuzzy relations formed by the Cartesian product of thevariables referred to in the rulesif-part and then-part.

For example, the mapping rule is:

IF x is A, THEN y is B,whichis mathematically represented as a fuzzy relation

R defined asmR(x,y)=mAB(x,y)=min{mA(x), mB(y)}.

Example: Let us consider the following fuzzy


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Example: Let us consider the following fuzzy

mapping rule from X to Y, whereX = {2,3,4,5,6,7,8,9} and Y = {1,2,3,4,5,6}

If x is Medium, Then y is Small

where Medium and Small are fuzzy subsets of Xand Y characterized by the followingmembership functions:

Medium 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 +0.5/8 + 0.2/9Small 1/1 + + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6


Cartesian product of Medium and Small If we use the


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Cartesian product ofMedium and Small. If we use the

min operator to construct the Cartesianproduct, we

have mR(x,y) = min{mMedium(x), mSmall(y)}.

The resulting fuzzy relation representing the rule is

0.1 0.1 0.1 0.1 0.1 0.1

0.3 0.3 0.3 0.3 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.10.7 0.7 0.7 0.6 0.3 0.1

0.5 0.5 0.5 0.5 0.3 0.1

0.2 0.2 0.2 0.2 0.2 0.1

R

Medium 0.1/2 + 0.3/3 +0.7/4 + 1/5 + 1/6 + 0.7/7 +0.5/8 + 0.2/9

Small 1/1 + + 0.9/3 +0.6/4 + 0.3/5 + 0.1/6



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If x is Medium Then y is Small

Input data is: X is Small, where Small for x is defined

as

Small 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6

To find out the possible values of y, we compose the

possible values of x with the fuzzy relation T using the

sup-min composition:0.1 0.1 0.1 0.1 0.1 0.1

0.3 0.3 0.3 0.3 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

1 1 0.9 0.6 0.3 0.1

0.7 0.7 0.7 0.6 0.3 0.10.5 0.5 0.5 0.5 0.3 0.1

0.2 0.2 0.2 0.2 0.2 0.1

Small R = [1 0.9 0.6 0.3 0.1 0 0 0] o

= [0.6 0.6 0.6 0.6 0.3 0.1],

y = 0.6/1+0.6/2+0.6/3+0.6/4+0.3/5+0.1/6

as the result of the inference.


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Up to now, you learn If x is A then y is B Find the condition matching degree

Map to conclusion with some matching degree

Now, you know (A)x and (B)y, find thetruth value of (x is A y is B), t(x is A y is B) = ?

We interested in finding the truth value of

relationship between a set of observableparameters and one or multipleparameter whose values we know

You need to know truth value of relation not


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You need to know truth value of relation, notthe conclusion

You will use the relation to go on process,not only conclusion

Methods Zadehs arithmetic fuzzy implication Standart sequence fuzzy implication

Godelian sequence fuzzy implication

Goguens fuzzy implication

bulanikmantik all

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