Type of Uncertainty1. vagueness: associated with the difficulty

of making sharp or precise distinctions in the world. the concept of fuzzy set provides a basic

mathematical framework for dealing with vagueness.

2. ambiguity: associated with one-to-many relations. the concept of fuzzy measure provides a

general framework for dealing ambiguity.

3. types of ambiguity:a. nonspecificity in evidence: connected with the size

of the subsets that are designated by a fuzzy measure as prospective locations of the element in questions. The larger the subsets, the less specific the characterization.

b. dissonance in evidence: exhibited by disjoint subsets of X that are designated by the given fuzzy measure as prospective locations of the element of concern.

c. confusion in evidence: associated with the number of subsets of X that are designated by a fuzzy measure as prospective locations of the element under consideration and that do not overlap or overlap only partially.

Imprecision的不同面目

1. Measurement imprecision/inaccuracy

2. Intrinsic imprecision: associated with a description of the properties of a phenomenon and not with our measurement of the properties using some external device.

3. Sometimes both are conflict.

Ambiguity

1. Semantic ambiguity:

“Do I turn left here?” Ans. Right!

“The food is hot.”

Not fuzzy! It deals with the interpretation of disjoint and discrete concepts.

Fuzzy! Attach an universal discourse and define the membership of each element to one of possible interpretation.

Ambiguity (cont.)

2. Visual (perceptual) ambiguity:

Möbius strip;

GUI: the use of color, icons, dialogues can introduce ambiguous visual stimuli.

3. Structural ambiguity: existing in the relationships between components of a system can reflect a high-level of ambiguity.

multiple inheritance in an OO systems

Undecidability

1. The ability to discriminate between different states of an event.

2. Generally stems from the properties of the model itself, not from any lack of knowledge on our part.

3. Nekker cube

4. Kaniza Square

5. Half-full v.s. Half-empty glass

Measure of FuzzinessRXPf )(~:.1

set. crispa is A ifonly and if0f(A):f1Axioma.

f(B).f(A) thenB,A If:f2Axiomb.

and if valuemaximum theassumes f(A):f3Axiomc.

fuzzy.maximally is A ifonly

Xx

AAAA xxxxAf )])(1[log)](1[)(log)(()(.2 22

INTRODUCTION

]).1,0([::setsfuzzy valued-interval.7

sets.fuzzy of level high6.

sets.fuzzy of ehigher typ5.

[0,1]). P(X):(g measurefuzzy set fuzzy 4.

lattice) :(L L. X:μ :setfuzzy -L3.

[0,1]. X: :function membership2.

set. Fuzzy set Crisp.1

A

A

A

PX

─

──

─

─

─

Basic characteristics of fuzzy sets

1. Vertical dimension: height, normalization (maximum form: at least one element with 1.0 membership and one element with membership 0).

2. Horizontal dimension: universal discourse, support sets, alpha cuts.

3. Representation schemes: membership function, ordered set of pairs, polynomial-like (integral-like).

BASIC CONCEPTS

A ofset level thein allfor ,)A(

by defined is function membership whoseN on defined number)(fuzzy set fuzzy a :ycardinalitfuzzy .7

).(A:ycardinalitscalar 6.

.continuous piecewise is function membership whose Ron definedset fuzzy normalized andconvex a :numberfuzzy .5

}.)(|{A :cut-4.

grade. membership possible maximum theattains elements its of oneleast at :setfuzzy Normalized3.

set. that in elementany by attained grade membershiplargest the:Height2.

0}.(x)μ|X{x A supp :Support1.

A~

A

Xx

A

A

x

xXx

).f(x/)f(x/)f(x/

)///f(f(A)

,///A

:principle extension13.

)](),(min[)( :onintersecti12.

)](),(max[)( :union.11

).(1)( :complement10.

B.A )()( :eequivalenc9.

subset.proper B.A )()( :subset.8

n2211

2211

2211

BABA

BABA

AA

BA

BA

n

nn

nn

xxx

xxx

xxx

xxx

xx

xx

xx

─

─

Types of Membership Functions

1. Linear type: linear/step functions. Approximating an unknown or poorly understood concept that is not a fuzzy number. (often expressed as shouldered sets)

2. Triangular type:

often used to model process control systems.

Types of Membership Functions (cont.)• when you decompose a variable into fuzzy

sets, the amount of overlap must vary between 10% to 50%.

• In modeling dynamic systems, it can approximate their behaviors to nearly any degree of precision.

cx

cxbbcxc

bxaabax

ax

cbaxTri

0

)/()(

)/()(

0

),,;(

Types of Membership Functions (cont.)

3. Trapezoidal Type:

dx

dxccdxd

cxb

bxaabax

ax

dcbaxTra

0

)/()(

1

)/()(

0

),,,;(

Types of Membership Functions (cont.)4. Sigmoid/Logistic Type:

* modeling population dynamics where the sampling of individual values approximates a continuous random variable (e.g., 硬碟MTBF,…)

* frequency (proportional) representation: usually, most, always.

cx

cxbaccx

bxaacax

ax

cbaxS

1

))/()((21

)/()(2

0

),,;( 2

2

Types of Membership Functions (cont.)1. Fuzzy numbers and around representation: around,

close to, few, some.

2. PI, Beta, Gaussian fuzzy sets (bell-shaped): the slope and width of the bell curves indicate the degree of compactness associated with the fuzzy number.

3. PI curves: is not asymptotic. Zero point is at a discrete and specified point.

xbcbccxS

xcbcbcxScbx

),2/,;(1

),2/,;(),;(

Types of Membership Functions (cont.)4. Beta curves: more tightly compacted than PI.

5. Similar to beta curves, but the slope goes to zero very quickly with a very short tail.

6. Irregularly shaped and arbitrary fuzzy sets: (domain value, membership) pairs linear interpolation.

2)(

12

),;(

))/)((1(),;(xckeckxG

bcxbcxB

FUZZY LOGIC

etc. unlikely, extremely ,impossiblealmost likely, :modifierfuzzy

etc. false,mostly less,or more , very true true,quite : valuehfuzzy trut etc. usually, all, almost, few, many, :quantifierfuzzy etc. dangerous, rare, old, expensive, :predicatefuzzy 6.

language. natural on is logicfuzzy of focusprimary The5.

tool.principle theas set theory fuzzy using npropositio imprecise withresoning eapproximat

for foundation provide tois logicfuzzy of goal ultimate The4.

.set theoryfuzzy original the toisomorphic is )(L logic cz Lukasiewistandard.3

)(L logic zLukasiewic.2

)}.1/()1(,),1/(2),1/(1,0{T :logic value-N.1

1

n

n nnnn

7. Linguistic hedges: linguistic terms that can be used to modify fuzzy sets, such as very, extremely, quite, etc.8. In general, fuzzy quantifiers are represented in fuzzy logic by fuzzy numbers.

Hedges1. Modify the shape of a fuzzy set’s surface

causing a change in the membership function. become a new fuzzy set.

2. Represented by a linguistic-like construct. (intensification: very, extremely. dilation: somewhat, rather, quite)

3. Play the same role in fuzzy production rules that adjectives and adverbs play in English sentences.

4. Without mathematical theory.5. Multiple hedges can be applied to a single fuzzy

region.

Approximation hedges1. Not only broaden or restrict existing bell-

shaped fuzzy regions, but convert scalar values into bell-shaped fuzzy regions.

2. Basic approximation hedges: about, around, near, roughly; all produce the same PI curve.

3. Decide the spread of PI curve: essential issue.a. implicitly determined from context,b. explicitly domain statement.

Approximation Hedges (cont.)4. About: creates a space that is proportional to the

height and width of the generated fuzzy space. (slightly broadens the fuzzy region)

5. In the vicinity: constructs a very wide fuzzy region much broader than the region created by “about.” If several closely allied sets share the same underlying domain, this approximation may lead to an unforeseen degree of ambiguity.

6. Close to (neighboring): narrowly approximated regions.

Restricting a fuzzy region1. Modify the shape of directional or bell-shaped fuzz

y regions.

2. Below: can only be applied to a fuzzy membership function that increases as the domain moves from left to right.

3. Above: can only be applied to a fuzzy membership function that declines as the domain moves from left to right.

4. Can also be applied to unhedged scalars produces a classical step function space.

Restricting a fuzzy region (cont.)

5. To prevent this rather crisp representation of what should intuitively be a fuzzy space, the scalar should be approximated as the restriction hedge is applied; such as

above around 40.

below close to our direct-costs.

Intensifying and diluting fuzzy regions

1. Intensification: very, extremely.

2. Dilution: somewhat, quite, rather, sort of.

3. Very:

domain independent: rescales the fuzzy set surface but maintains the normalization space for the set.

domain dependent:

)()( 2very xx aa

)()(

)()(

very

),1(very

xx

xx

kaa

kaMina

Intensifying and diluting fuzzy regions

4. Concentrator hedge in Zadeh:

5. Somewhat (rather, quite): dilute the force of the membership function in a particular fuzzy region.

a. effect: to increase the candidate space within the fuzzy region with a characteristic function relationship of

little : slightly; :

[1,4]; valuepracticed :

extremely; : ;:

1.31.7

n

32

very

)()( xx asomewhata

Intensifying and diluting fuzzy regions

b. domain independent:

c. context dependent: k provides a context sensitive adjusting control on the strength of the dilator.

)()( 5.0 xx aasomewhat

)()(

)()(

) (1, max

) (0, max

xx

xx

kaasomewhat

kaasomewhat

Intensifying and diluting fuzzy regions

d. generalization of dilator hedge

e. power function should not exceed 0.7 to avoid too close to base set to effectively be applied in the model.f. very and somewhat are the only naturally idempotent members of the hedge set, and are the only commutative hedges. somewhat very tall = very somewhat tall

)()(

[1,8].in isn range valuepracticed

1.n where);()(

7.0reatly

1

il

xx

xx

aag

naad

Contrast Intensification and Diffusion1. Change the nature of fuzzy regions by either

making the region less fuzzy (intensification) or more fuzzy (diffusion).

2. The idea behind contrast hedging is related to the concept of fuzzy entropy and intrinsic ambiguity.

3. At the extremes of the fuzzy set, membership is less ambiguous the fuzziness is minimized.

4. The values that are centered around the midpoint in the fuzzy truth function, are in an area of maximum fuzziness.

Contrast Intensification and Diffusion5. Positively hedge (absolutely, definitely):

a. changes the fuzzy surface by raising all the truth function values above 0.5 and decreasing all the truth function values below 0.5.

b. reduce the overall fuzziness of the region.

else. ))(1(-1

5.0)( if )()(

point. inflectionan as acts 0.5

else. ))(1(2-1

5.0)( if )(2)(

int

2

2 int

xn

xxnx

x

xxx

na

anaa

a

aaa

Contrast Intensification and Diffusion6. Generally hedge: diffusion modifier

a. changes the fuzzy surface by reducing all the truth function values above 0.5 and increasing all the truth function values below 0.5.

b. increase the overall fuzziness of the region.

c. allow us to make decisions based on whether or not a truth value is broadly within the fuzzy set or definitely within the fuzzy set.

else. ))(1(.50-1

5.0)( if )(5.0)(5.0

5.0 int

x

xxx

a

aaa

OPERATIONS ON FUZZY SETS

* Possibility Theory* Fuzzy Complement1. axiomatic skeleton [Axiom c1]: (boundary conditions)

c(0)=1 and c(1)=0, that is, c behaves as the ordinary complement for crisp sets.

[Axiom c2]: For all a,b [0,1], if a < b, thenc(a) c(b), that is, c is monotonic nonincreasing.≧

E.g.,threshold-type complements

t.afor 0t,afor 1c(a)

),0(,)1()(c :class Yager

).,1(),1/()1()(c :class Sugeno E.g.,

[0,1].a allfor a c(c(a)) that means which,involutive is c :c4] [Axiom

a).cos(1 1/2 c(a) E.g.,

function. continuousa is c :c3] [Axiom

tsrequiremen additional.2

/1

aa

whereaaa

e.associativ is i is, that c));i(b,i(a, c)b),i(i(a, :i4 Axiom

monotonic. is i is, that );b,ai(b)i(a, then;bb and aa If :i3 Axiom

e.commutativ is i is, that a);i(b, b)i(a,:i2 Axiom

).conditions(boundary sets crsip with onintersecti classical theas behaves i is, that

0; i(0,0) i(1,0) i(0,1) 1; i(1,1):i1 Axiom

:skeleton Axiomatic .3

)].(i),(i[ii 2.

[0,1].[0,1] [0,1]:i 1.

onIntersecti Fuzzy

BABA

xx

performed. onintersecti

theof strengh of degree theindicates1/w (3)

b).min(a,i , w when(2)

i5. throughi1 Axioms satisfies (1)

])b)-(1a)-((1min[1,-1b)(a,i :clss Yager 5.

.idempotent is i is, that a;a)i(a, :i6 Axiom

function. continuousa is i :i5 Axiom

axioms additional .4

w

1/www

w

e.associativ is u is, that c));u(b,u(a, c)b),u(u(a, :u4 Axiom

monotonic. is u is, that );b,a(b)u(a, then;bb and aa If :u3 Axiom

e.commutativ is u is, that a);u(b, b)u(a,:u2 Axiom

).conditions(boundary sets crsip withunion classical theas behaves u is, that

1; u(1,1) u(1,0) u(0,1) 0; u(0,0):u1 Axiom

:skeleton Axiomatic .3

)].(),(u[ 2.

[0,1].[0,1] [0,1]:u 1.

ON Fuzzy UNI

BABA

u

xx

.u theinpresent

eabilityinterchang of degree theindicates1/w (3)

b).max(a,u , w when(2)

u5. throughu1 Axioms satisfies (1)

])b(amin[1,b)(a,u :clss Yager 5.

.idempotent is u is, that a;a)u(a, :u6 Axiom

function. continuousa is u :u5 Axiom

axioms additional .4

w

w

1/www

w

.idempotent and continuous are that operators onintersecti and unionfuzzy only theconstitute min andmax 4.

on).intersecti(weakest i functions of boundupper theand union) (strongest u functions of boundlower

thely,respective represent, operations min andmax 3.

otherwise. 0,1a whenb,1b a when

b)(a,i

b),min(a,b)i(a,b)(a,i 2.

otherwise. 10,a whenb0,b a when

b)(a,u

),,(ub)u(a,b)max(a, 1.

Operations of nsCombinatio

min

min

max

max

ba

5. Fuzzy set operations of union, intersection, and continuous complement that satisfy the law of excluded middle and the law of contradiction are not idempotent or distributive.

6. Given two of the three operation u, i, c, it is sometimes desirable to determine the third operation in such a way that Demorgan’s laws are satisfied.

c(b)).i(c(a),b))c(u(a,

arguments. its all in ingnondecreas monotonic is h is, that ),Ni|h(b)Ni|h(a

then,Ni allfor ba if ],1,0[b and ]1,0[a where),Ni|(b and )Ni|(apair any For [h2]

)conditions(boundary 1,1)h(1,1, and 0,0)h(0,0, [h1]

skelton Axiomatic .2

)).(,),(),(h()(

],1,0[[0,1]:h

set. singlea produce tocombined are setsfuzzy several by which operations :defination 1.

Operations nAggregatio General

nini

niii

inini

AnA2A1A

n

xxxx

.)n/)aaa(()a,,a,a(h

.operations

max and min thebetween interval entire thecovers that operations averaging of class one :means dgeneralize .5

).a,,a,max(a)a,,a,h(a)a,,a,min(a

for which operations naggregatio :operations averaging .4

.N on p npermutatioany for

)Ni|h(a)Ni|h(a is,that arguments, its all in function symmetrica is h [h4]

function. continuousa is h [h3]

axioms additional 3.

/1n21n21

n21n21n21

n

np(i)ni

1w)aw()w,,w,w;a,,a,a(h

:means dgeneralize weighted.6

)aaa(n1

.h :mean arithmetic (3)

)a1

a1

a1

n/( .h :mean harmonic (2)

)aa(a .h :mean geometric (1)

n

1ii

n

1i

/1

iin21n21a

n211

n21

1-

1/n

n210

a

TABLE 2.2 SOME CLASSES OF FUZZY SET UNIONS AND INTERSECTIONS

Reference Fuzzy Unions Fuzzy IntersectionsRange of Parameter

Schweizer & Sklar [1961]

Hamacher [1978]

Frank [1979]

Yager [1980]

Dubois & Prade [1980]

Dombi [1982]

),( pppp ba /1)]1)1()1(,0max[1 ppp ba /1)1,0max[

ab

abba

)1(1

)2(

))(1( abba

ab

),0(

),0( s

),0( w

)1,0(

),0(

]1

)1)(1(1[log1

11

s

ss ba

s ]1

)1)(1(1[log

s

ss ba

s

])(,1min[ /1 www ba ]))1()1((,1min[1 /1 www ba

),1,1max(

)1,,min(

ba

baabba

),,max( ba

ab

/1])11

()11

[(1

1

ba

/1])11

()11

[(1

1

ba

set Y. of

support the toequal is range the:relation onto 6.

X.set ofsupport the to

equal is domain the:relation specified completely 5.

1.h(R) relation normal (1)

y).(x,maxmaxh(R) :height 4.

Y. yeachfor y),(x,max)y(

,Xx somefor Ry)(x,Y,y|yY) R(X,ran :range 3.

X.x eachfor y),(x,max)(

,Y ysomefor Ry)(x,X,x|xY) R(X,dom :domain 2.

diagram sagittal 1.

lations Bunary Re

RXxYy

RXx Rran

RYy Rdom

x

)].,(),,(min[ max),( :ncompositio min-max (2)

R).(QPRQ)(P ,PQQ)(P

P,QQ P(1)

Z)Q(Y,Y)P(X,Z) R(X,:ncompositio .10

).,(),( ,),(|),()YX,( R:relation inverse .9

).,(),(,U R:form resolution 8.

one.-to-one (2)

one,-to-many (1) R.in onceexactly

appears domain theofmember each :(function) mapping .7

QPYy

QP

1-1-1-

R

1-

1-

zyyxzx

yxxyRyxxy

yxyxR

R

RR

)].,(),,(min[),,(

,Qz)(y, and Py)(x,|z)y,(x,Z)Q(Y,Y) P(X,

:Join l Relationa11.

)].,(),([max),(

ncompositioproduct -max (3)

QPQP

QPQP

zyyxzyx

zyyxzxYy

FUZZY RELATIONS

unbiased. (3)

.projection its of one from relationa deriving incity nonspecifi themaximizes (2)

.projection given the withcompatible ishat trelationfuzzy specific)(least largest theproduces (1)

y.x that suchx eachfor )()(

Y]-X[R :extension cylindric .4

).(max)(

Y][R :n Projectio3.

relationsfuzzy - L2.

.membership of degrees varyinghavemay )x,,x,(x tuples where,X,,X,X sets crisp ofproduct Cartesian theon definedset fuzzy a : Def..1

Y]-X[R

][

n21n21

yx

xy

R

RyxYR

unbiased. (2)

s.projection theall withcompatible isat product th Cartesian

extended theon defined relationfuzzy largest the(1)

).(min)(

.extensions cylindric their of onintersecti set theby taking sprojcetion its of serveral

from tedreconstruc is relationa :closure cylindric .5

][Rcyl ixx

ii YXRIi

members. possiblefewest thehas and X), R(X,contains

e, transitivis that relation the: Rclosure e transitiv5.

)].,(),,(min[max),( :e transitiv4.

).,(),( :symmetric 3.

1.e0 wheree,),( :yreflexivit-e 2.

X.x allfor 1),( :reflexive 1.

X) R(X,Set, Singlea on lations Binary Re

T

RR

R

R

R

zyyxzx

xyyx

xx

xx

RYy

R

tree.parition nested are which,partitions

ofset a of consists |)(R(R) .5

)).(R( partitiona forms and relation equivalent an is cut R- each thenrealtion, similaritya is form, resolutiona by drepresente relation,fuzzy a If 4.

disjoint).generally not (are classes similaritydegree specific some tohave X of elements that thesimilarity

of degree heconsider t tomay wish weely,Alternativ 3.

degree. specific some other to each tosimilar"" are members whosesets crisp into elements

group toconsidered be can relation similarity the2.

e. transitivand symmetric, reflexive, is that relationbinary fuzzy a :relation similarity 1.

RelationsSimilarity and ce Equivalen

relations. similarity

are relationsity compatibil of closures e transitivThe 6.

X. of partitions

form covers- complete all for which relations

itycompatibil of cases special are relations similarity .5

.Ayx, allfor ),( thatsuch X of Asubset a :classity compatibil- 4.

cover. )( complete class,ity compatibil )maximal( class,ity compatibil 3.

symmetric. and reflexive is that relationbinary fuzzy a :relation Proximity2.

relation. )(toleranceity compatibila calledusually is symmetric and reflexive is that relationbinary A 1.

RelationsToleranceor ity Compatibil

R

yx

.0),( and 0)( :boundupper least (1) .A) U(R,:boundupper Fuzzy .7

).,()(:)( Rclass g Dominatin.6

).,()(:)( Rclass g Dominatin5.

.refinement of levels increasing produce that cuts- of seriesa by taking orderings partial crisp of series

a into resolved be can ordering partialfuzzy Any 4.

sitivity.fuzzy tran of form someunder e transitivand ric,antisymmet reflexive, is

thatrelationbinary fuzzy a :ordering partial Fuzzy 3.

ordering. partial connecteda :orderinglinear 2.

e. transitivric,antisymmet reflexive, :ordering Partial1.

Orderings

RA)U(R,

][

][[x]

][[x]

yxxR

xyyy

yxyy

xAx

RxR

RxR

e. transitivand ric,antisymmet ive,antireflex is that ordering an :orderingstrict Fuzzy 10.

y.x allfor 0),( or 0),(either whichin gpreorderinfuzzy a

ric.antisymmet except orderinglinear fuzzy a of properties

theall satisfying ordering an :orderingk Fuzzy wea9.

e. transitivand reflexive is that relationfuzzy a :gpreorderin Fuzzy .8

R

xyyx

R

blocks. ofpair eachfor ),y,(y)b,(amax :relationsfuzzy for (2)

Y.ofelement same the to themmaps h ifonly and if partition the

ofblock same the tobelong Xx,x elements twoany that such Xset theon partitiona imposes h (1)

.Yy, yallfor R,))(yh),(yimplies(h Q)y,(y

.Xx,x allfor Q,))h(x),ximplies(h( R)x,(x :smhomomorphi strong .2

].)h(x),h(x[)x,( :relationsfuzzy for (1)

.Xx,x allfor Q,))h(x),(h(x implies R)x,(x if Q)(Y, toR)(X, from YX :h :smhomomorphi 1.

Morphisms

21QjiRji,

21

n

2121-

11-

21

212121

21Q21R

212121

x

equal. are Q and Rrelations theand itself toX maps sm.endomorphi

an and misomorphis an both is h :smautomorphi 5.

X. Yand YX :h :smendomorphi 4.

Q. in Rof properties theall preserves that set Y theof elements into X

set theof elements of relabelingdirect or iona translat onto.

and one to-one specified, completely is h if and Q)(Y, toR)(X, from smhomomorphia is YX :h :misomorphis 3.

.similarityor ordering as such sets, in properties desirable preserve that systems of tionssimplifica of

kinds variousperformingfor useful are functions these(5) R.of tionsimplificaa represents Q relation thethen

Q),(Y, toR)(X, from exists smhomomorphi stronga if (4) X). R(X,relation

theof properties theof some preserves Y)Q(Y, relation the thenQ),(Y, toR)(X, from exists smhomomorphia if (3)

Fuzzy Propositions

1. A proposition whose truth value is a matter of degree. (in the range of [0,1])

2. Unconditional and unqualified propositions:

p: ν is F (temperature is high)

T(p) = F(v) (= rF(v))

3. p: ν(i) is F (ν: IV)

Unconditional and Qualified Propositions

1. p: ν is F is S (truth-qualified)

2. p: Pro{ν is F}is P (probability-qualified)

3. Both S (fuzzy truth qualifier) and P (fuzzy probability qualifier) are presented by fuzzy sets on [0,1].

4. T(p) = S(F(v))

5. ν is G

Unconditional and Qualified Propositions (cont.)

• Unqualified propositions are special truth-qualified propositions, in which the truth qualifier S is assumed to be true. (S(F(v)) =F(v))

• Pro{ν is F} =

• T(p) = P( )• p: Pro{temperature t (at given place and time) is aro

und 75oF} is likely.

Vv

vFvf )()(

Vv

vFvf )()(

Conditional and Unqualified Propositions

• p: If χ is A, then y is B

• <χ, y> is R• R(x,y) = δ[A(x), B(y)] (fuzzy implication)

Conditional and Unqualified Propositions (cont.)

• p: If χ is A, then y is B is S

• p: Pro{χ is A|y is B} is P