931102fuzzy set theory chap07.ppt1 Fuzzy relations Fuzzy sets defined on universal sets which are...
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Transcript of 931102fuzzy set theory chap07.ppt1 Fuzzy relations Fuzzy sets defined on universal sets which are...
931102 fuzzy set theory chap07.ppt 1
Fuzzy relations
Fuzzy sets defined on universal sets which are Cartesian products.
<x, y>
921028 fuzzy set theory chap07.ppt 2
Example (information retrieval)
DocumentsD
Key termsT
Binary fuzzy relationR
Membership degree R(d,t): the degree of relevance of document d and the key term t
921028 fuzzy set theory chap07.ppt 3
Example (equality relation)
)10(max)(c
y||x-, x,yE
y is approximately equal to x
921028 fuzzy set theory chap07.ppt 4
Representations of fuzzy relation
• List of ordered pairs with their membership grades• Matrices• Mappings• Directed graphs
921028 fuzzy set theory chap07.ppt 5
Matrices
Fuzzy relation R on XY
X={x1, x2 , …, xn}, Y={y1, y2 , …, ym}
rij=R(xi , yj) is the membership degree of pair (xi , yj)
921028 fuzzy set theory chap07.ppt 6
Example (very far from)
Example: X={Beijing, Chicago, London, Moscow, New York, Paris, Sydney, Tokyo}
Very far fromVery far from
X
921028 fuzzy set theory chap07.ppt 7
Mappings
The visual representations
On finite Cartesian products
Document D = {d1, d2 , …, d5}
Key terms T = {t1, t2 , t3, t4 }
921028 fuzzy set theory chap07.ppt 8
Directed graphs
921028 fuzzy set theory chap07.ppt 9
Inverse operation
• Given fuzzy binary relation RXY– Inverse R-1YX
– R-1(y,x) = R(x,y)
– (R-1) -1 = RThe transpose of R
921028 fuzzy set theory chap07.ppt 10
The composition of fuzzy relations P and Q
• Given fuzzy relations PXY, QYZ– Composition on P and Q = P Q = R XZ
– The membership degree of a chain <x, y, z> is determined by the degree of the weaker of the two links, <x, y> and <y, z> .
– R(x, z) = (P Q )(x, z) = max yY min [ P(x, y ), Q(y, z )]
921028 fuzzy set theory chap07.ppt 11
Example (composition of fuzzy relations)
X = {a,b,c}
Y = {1,2,3,4}
Z = {A,B,C}
921028 fuzzy set theory chap07.ppt 12
Example (matrix composition)
P XY Q YZ R XZ
921028 fuzzy set theory chap07.ppt 13
Fuzzy equivalence relations and compatibility relations
• Equivalence: Any fuzzy relation R that satisfies reflexive, symmetric and transitive properties.
– R is Reflexive R(x, x) = 1 for all x X.
– R is symmetric R(x, y) = R(y, x) for all x, y X.
– R is transitive R(x, z) max yY min [ R(x, y ), R(y, z )] all x, z X.
• Compatibility: satisfy reflexive and symmetric
921028 fuzzy set theory chap07.ppt 14
Example (fuzzy equivalence)
921028 fuzzy set theory chap07.ppt 15
Example(fuzzy compatibility)
R(1,4)<max {min[R(1,2),R(2,4)], min[R(1,5),R(5,4)]}
921028 fuzzy set theory chap07.ppt 16
Transitive closure
• The transitive closure of R is the smallest fuzzy relation that is transitive and contains R.
• interactive algorithm to find transitive closure RT
of R 1. Compute R’ = R(R R)
2. If R’R, rename R’ as R and go to step 1. Otherwise, R’= RT.
921028 fuzzy set theory chap07.ppt 17
Evaluate the algorithm
equal
Firstiteration
seconditeration
921028 fuzzy set theory chap07.ppt 18
Example (transitive closure)
Transitive closure of R
921028 fuzzy set theory chap07.ppt 19
Fuzzy Partial ordering• Fuzzy relations that satisfy reflexive, transitive
and anti-symmetric, denoted by xy
– R on X is anti-symmetric R(x,y)>0 and R(y, x)>0 imply that x=y for any x,yX.
921028 fuzzy set theory chap07.ppt 20
Example (fuzzy partial ordering)
921028 fuzzy set theory chap07.ppt 21
Projections example
S ={s1, s2, s3}
D ={d1, d2, d3 , d4 , d5}
Projection on S
Projection on D
),( (s)Q maxDd
1 dsQ
),( (s)Q maxSs
2 dsQ
symptoms
diseases
921028 fuzzy set theory chap07.ppt 22
Projections
Given an n-dimensional fuzzy relation R on X =X1 X2 … Xn and any subset P(any chosen dimensions) of X, the projection RP of R on P for each p P is
Where
),( )(R maxP
P ppRpp
P is the remaining dimensions
921028 fuzzy set theory chap07.ppt 23
Cylindric extension
Given an n-dimensional fuzzy relation R on X =X1 X2 … Xn , any (n+k)-dimensional relation whose projection into the n dimensions of R yields R is called an extension of R.
An extension of R with respect to Y =Y1 Y2 …
Yh is call the cylindric extension EYR of R into YEYR(x,y) = R(x)
For all x X and y Y.
921028 fuzzy set theory chap07.ppt 24
Example (Cylindric extension)
921028 fuzzy set theory chap07.ppt 25
Example of cylindric closure
• cylindric closure : the intersection of the cylindric extensions of R is the original relation R.