6 Fuzzy Relations - UNICAMPgomide/courses/IA861/transp/FSE_Chap6.pdf · 6.2 Fuzzy relations ... 6.5...

6 Fuzzy Relations

Fuzzy Systems EngineeringToward Human-Centric Computing

6.1 The concept of relations

6.2 Fuzzy relations

6.3 Properties of fuzzy relations

6.4 Operations on fuzzy relations

6.5 Cartesian product, projections, and cylindrical extension

6.6 Reconstruction of fuzzy relations

6.7 Binary fuzzy relations

Contents

Pedrycz and Gomide, FSE 2007

6.1 The concept of relations


Relation

di

wj

X{ d1, d2,...,di,...dn}

Y{ w1, w2,...,ji,...wm}

R= {(di, wj) | di∈X, wj∈Y}

Docs Keywords


0 2 4 6 80

2

4

6

8

x

y

(a) Relation "equal to"

02

46

8

0

2

4

6

80

0.2

0.4

0.6

0.8

1

x

(b) Characteris tic function of "equal to"

y

X = Y = {2, 4, 6, 8} equal to R= {(2,2), (4,4), (6,6), (8,8)}

=

1000

0100

0010

0001

R

y

x

y

x

R(x,y)

Relation R : X×Y → {0,1}


≤≤

=otherwise0

1and1if1)(

|y||x|y,xR

=+=

otherwise0

if1)(

222 ryxy,xR

Circle Square

x

yR(x,y)

1

x

yR(x,y)

1

Examples


6.2 Fuzzy relations


Fuzzy relation R : X×Y → [0,1]

=

1080

010

0180

6001

.

.

.

R

D = { dfs, dnf, dns, dgf}

W = { wf, wn, wg}

Docs

Keywords

R : D×W → [0,1]

dfs

dnf,

dns

dgf

wf wn wg

Example

1

23

4

1

2

30

0.2

0.4

0.6

0.8

1

documents

(a) Membership function of R

keywords


0)( >α

α−−= ,

|yx|expy,xRe

Example

x approximately equal to y

X = Y = [0,4]

α = 1


6.3 Properties of fuzzy relations


Domain

Codomain

)(sup)(dom y,xRxRy Y∈

=

)(sup)(cod y,xRyRx X∈

=



Representation of fuzzy relations

U]10[ ,RR

∈ααα=

)]}([{minsup)(]10[

y,xR,y,xR,

α=∈α

Representation theorem


Equality

Inclusion

Fuzzy relations P,Q : X×Y → [0,1]

P(x,y) = Q(x,y) ∀(x,y) ∈ X×Y

P(x,y) ≤ Q(x,y) ∀(x,y) ∈ X×Y


6.4 Operations on fuzzy relations


Fuzzy relations P,Q : X×Y → [0,1]

Union: R= P ∪ Q

Intersection: R= P ∪ Q

R(x,y) = P(x,y) s Q(x,y) ∀(x,y) ∈ X×Y ( s is a t-conorm)

R(x,y) = P(x,y) t Q(x,y) ∀(x,y) ∈ X×Y ( t is a t-norm)



Standard complement: R

Transpose: RT

R(x,y) = 1–R(x,y) ∀(x,y) ∈ X×Y

RT(y,x) = R(x,y) ∀(x,y) ∈ X×Y


6.5 Cartesian product,projections,and cylindricalextension of fuzzy sets


Cartesian product

A1, A2, ..., An fuzzy sets on X1, X2, ..., Xn

R= A1× A2 × ... × An

R(x1, x2,...,xn) = min {A1(x1), A2(x2), ..., An(xn)} ∀(xi,yi) ∈ Xi×Yi

Generalization

R(x1, x2,...,xn) = A1(x1) t A2(x2) t ...t An(xn) ∀(xi,yi) ∈ Xi×Yi

t = t-norm


R(x,y) = min {A(x), B(y)} R(x,y) = A(x)B(y)

A(x) = exp[-2(x – 5)2]

B(y) = exp[-2(y – 5)2]

Examples

R = A×B


Projections of fuzzy relations

R: X1× X2 × ... × Xn → [0, 1]

X = Xi× Xj × ... × Xk

)(sup)()( 2121 nx,...,x,x

nkji x,...,x,xRx,...,x,xRojPrx,...,x,xRvut

== XX

I = { i, j, ..., k}, J = { t, u, ..., v}, I∪J = N, I∩J = ∅

N = {1,2,...n}


R(x, y) = exp{–α[(x – 4)2 + (y – 5)2]}, α = 1

Example

)(sup)(Proj)( y,xRy,xRxRy

== XX

)(sup)(Proj)( y,xRy,xRyRx

Y == Y


Example

R: X × Y →[0, 1] , X = {1, 2, 3}, Y = {1, 2, 3, 4, 5}

=9030806080

9020018060

2050806001

)(

.....

.....

.....

y,xR

1

2

3

1

2

3

4

50

0.2

0.4

0.6

0.8

1

x

Relation R and its projections on X and Y

y

RRxRy

Ο∇∆

R

Rx

Ry

RX = [1.0, 1.0, 0.9]

RY = [1.0, 0.8, 1.0, 0.5, 0.9]


Cylindrical extension

cylA(x,y) = A(x) , ∀x ∈ X


cylA R

cylA∩RcylA∪R


6.6 Reconstruction of fuzzyrelations


Reconstruction using Cartesian product

ProjXR × ProjYR ⊇ R

0 5 100

10

20

x

y

(b) Contours of Rm on X and Y

0 5 100

10

20

x

y

(e) Contours of the Cartesian Produtc of ProjxRm and ProjyRm

Rnoninteractive


0 5 100

10

20

x

y

(e) Contours of the Cartesian Produtc of ProjxRp and ProjyRp

0 5 100

10

20

x

y

(b) Contours of Rp on X and Y

Rinteractive


6.7 Binary fuzzy relations


Binary fuzzy relation R : X×X → [0,1]

Features

(a) Reflexivity

R(x,x) = 1

R(x,x) ⊇ I

I = Identity

R(x,x) ≥ ε ε-reflexive

max {R(x,y), R(y,x)} ≤ R(x,x) locally reflexive

x


(b) Symmetry

R(x,y) = R(y,x) ∀∈×

RT = R

(c) Transitivity

sup z∈X { R(x, z) t R(z, y)} ≤ R(x, y) ∀x, y, z ∈X

x

y

x

z

yz’

z’’


Transitive closure

trans(R) = R= R∪ R2 ∪..... ∪Rn

R2 = RoR ........ Rp = RoRp –1

RoR(x,y) = maxz{ R(x,z) t R(z,y)}

If R is reflexive, then I ⊆ R ⊆ R2 ⊆... ⊆ Rn–1 = Rn

I = identity


procedure TRANSITIVE-CLOSUR-W (R) returns transitive fuzzy relation

static: fuzzy relation R = [r ij]

for i = 1:n dofor j = 1:n do

for k = 1:n dor jk ← max (r jk, r ji t r ik)

return R

Floyd-Warshall procedure to find trans(R)


Equivalence relations

R is an equivalence relation if it is

– reflexive

– symmetric

– transitive

Equivalence class

Ax = {y ∈ X | R(x,y) = 1}

X/R = family of all equivalence classes of R (partition of X)

R : X×X → {0,1}

equivalence relationsgeneralize the idea ofequality


R is a similarity relation if it is

– reflexive

– symmetric

– transitive

Equivalence class

P(R) = {X/Rα | α ∈ [0, 1]}

Nested partitions: if α > β then X/Rα finer than X/Rβ

R : X×X → [0,1]

Similarity relations


Example

=

01505000

50019000

50900100

0000180

0008001

...

...

...

..

..

R

=

=

=

10000

001100

010100

00010

00001

10000

01100

01100

00011

00011

11100

11100

11100

00011

00011

908050

.

.R,R,R ...


=

=

=

10000

001100

010100

00010

00001

10000

01100

01100

00011

00011

11100

11100

11100

00011

00011

908050

.

.R,R,R ...

Partition tree induced by similarity relation R

c,d,e a,b

a,b c,d e

a b c,d e

a b c d e

α=0.8

α=0.9

α=1.0

α=0.5


Compatibility relations

R is a compatibility relation if it is

– reflexive

– symmetric

α -Compatibility class: A ⊂ X such that

R(x,y) = 1 ∀ x,y ∈ A

Do not necessarily induce partitions

R : X×X → {0,1}


Proximity relations

R is a proximity relation if it is

– reflexive

– symmetric

Compatibility class: A ⊂ X such that

R(x,y) = 1 ∀ x,y ∈ A

Do not necessarily induce partitions

R : X×X → [0,1]

=

015040060

50017000

407001600

00600170

6007001

....

...

....

...

...

R


6 Fuzzy Relations - UNICAMPgomide/courses/IA861/transp/FSE_Chap6.pdf · 6.2 Fuzzy relations ... 6.5...

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Transcript of 6 Fuzzy Relations - UNICAMPgomide/courses/IA861/transp/FSE_Chap6.pdf · 6.2 Fuzzy relations ... 6.5...