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
Pedrycz and Gomide, FSE 2007
Relation
di
wj
X{ d1, d2,...,di,...dn}
Y{ w1, w2,...,ji,...wm}
R= {(di, wj) | di∈X, wj∈Y}
Docs Keywords
Pedrycz and Gomide, FSE 2007
0 2 4 6 80
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x
y
(a) Relation "equal to"
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8
0
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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}
Pedrycz and Gomide, FSE 2007
≤≤
=otherwise0
1and1if1)(
|y||x|y,xR
=+=
otherwise0
if1)(
222 ryxy,xR
Circle Square
x
yR(x,y)
1
x
yR(x,y)
1
Examples
Pedrycz and Gomide, FSE 2007
6.2 Fuzzy relations
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
0)( >α
α−−= ,
|yx|expy,xRe
Example
x approximately equal to y
X = Y = [0,4]
α = 1
Pedrycz and Gomide, FSE 2007
6.3 Properties of fuzzy relations
Pedrycz and Gomide, FSE 2007
Domain
Codomain
)(sup)(dom y,xRxRy Y∈
=
)(sup)(cod y,xRyRx X∈
=
Fuzzy relation R : X×Y → [0,1]
Pedrycz and Gomide, FSE 2007
Representation of fuzzy relations
U]10[ ,RR
∈ααα=
)]}([{minsup)(]10[
y,xR,y,xR,
α=∈α
Representation theorem
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
6.4 Operations on fuzzy relations
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
Fuzzy relation R : X×Y → [0,1]
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
Pedrycz and Gomide, FSE 2007
6.5 Cartesian product,projections,and cylindricalextension of fuzzy sets
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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
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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}
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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
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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]
Pedrycz and Gomide, FSE 2007
Cylindrical extension
cylA(x,y) = A(x) , ∀x ∈ X
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cylA R
cylA∩RcylA∪R
Pedrycz and Gomide, FSE 2007
6.6 Reconstruction of fuzzyrelations
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Reconstruction using Cartesian product
ProjXR × ProjYR ⊇ R
0 5 100
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x
y
(b) Contours of Rm on X and Y
0 5 100
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20
x
y
(e) Contours of the Cartesian Produtc of ProjxRm and ProjyRm
Rnoninteractive
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
6.7 Binary fuzzy relations
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
(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’’
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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 ...
Pedrycz and Gomide, FSE 2007
=
=
=
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
Pedrycz and Gomide, FSE 2007
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}
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007