lecture 31 fuzzy set theory (3)
DESCRIPTION
Lecture 31 Fuzzy Set Theory (3). Outline. Fuzzy Relation Composition and an Example Fuzzy Reasoning. Fuzzy Relation Composition. Let R be a fuzzy relation in X Y, and S be a fuzzy relation in Y Z. The Max-Min composition of R and S, R o S, is a fuzzy relation in X Z such that - PowerPoint PPT PresentationTRANSCRIPT
Intro. ANN & Fuzzy Systems
Lecture 31 Fuzzy Set Theory (3)
(C) 2001-2003 by Yu Hen Hu 2
Intro. ANN & Fuzzy Systems
Outline
• Fuzzy Relation Composition and an Example• Fuzzy Reasoning
(C) 2001-2003 by Yu Hen Hu 3
Intro. ANN & Fuzzy Systems
Fuzzy Relation Composition
• Let R be a fuzzy relation in X Y, and S be a fuzzy relation in Y Z.
• The Max-Min composition of R and S, RoS, is a fuzzy relation in X Z such that
RoS µRoS(x,z) = {µR(x,y) µS(y,z) }
= Max. {Min. {µR(x,y), µS(y,z)}}/(x,z)
• The Max-Product Composition of R and S, RoS, is a fuzzy relation in X Z such that
RoS µRoS(x,z) = {µR(x,y) µS(y,z) }
= Max. {µR(x,y) µS(y,z)}/(x,z)
(C) 2001-2003 by Yu Hen Hu 4
Intro. ANN & Fuzzy Systems
Fuzzy Composition Example
• Let the two relations R and S be, respectively:
• The goal is to compute RoS using both Max-min and Max-product composition rules.
R y1
y2
y3
S z1
z2
x1
0.4 0.6 0 y1
0.5 0.8
x2
0.9 1 0.1 y2
0.1 1
y1
0 0.6
(C) 2001-2003 by Yu Hen Hu 5
Intro. ANN & Fuzzy Systems
MAX-MIN Composition
RoS =
max{min(0.4,0.5), min(0.6, 0.1), min(0, 0)}
= max{ 0.4, 0.1, 0} = 0.4
max{min(0.4,0.8), min(0.6, 1), min(0, 0.6)}
= max{ 0.4, 0.6, 0} = 0.6
max{min(0.9,0.5), min(1, 0.1), min(0.1, 0)}
= max{ 0.5, 0.1, 0} = 0.5
max{min(0.9,0.8), min(1, 1), min(0.1, 0.6)}
= max{ 0.8, 1, 0.1} = 1
15.0
6.04.0
6.00
11.0
8.05.0
1.019.0
06.04.0
(C) 2001-2003 by Yu Hen Hu 6
Intro. ANN & Fuzzy Systems
MAX-PRODUCT Composition
RoS =
145.0
6.006.0
6.00
11.0
8.05.0
1.019.0
06.04.0
max{0.40.5, 0.60.1, 00} = max{0.02,0.06,0} = 0.06
max{0.40.8, 0.61, 00.6} = max{0.32, 0.6, 0} = 0.6
max{0.90.5, 10.1, 0.10} = max{0.45, 0.1, 0} = 0.45
max{0.90.8, 11, 0.10.6} = max{0.72, 1, 0.06} = 1
(C) 2001-2003 by Yu Hen Hu 7
Intro. ANN & Fuzzy Systems
Fuzzy Reasoning
• Comparing crisp logic inference and fuzzy logic inference
Translation –
Age(Mary) = 22
(Age(Dana),Age(Mary)) = Age(Dana)–Age(Mary) = 3
Age(Dana) = Age(Mary) + 3 = 22 + 3 = 25
Crisplogic
Mary is 22 years oldDana is 3 years older than Mary .Dana is (22 + 3) years old
(C) 2001-2003 by Yu Hen Hu 8
Intro. ANN & Fuzzy Systems
Fuzzy Reasoning
Fuzzylogic
Mary is YoungDana is much older than Mary .Dana is (Young o Much_older)
Translation –
Age(Mary) = Young (Young is a fuzzy set)
(Age(Dana),Age(Mary)) = Much_older (a relation)
Age(Dana) = Young o Much_older
– a composite relation!
(C) 2001-2003 by Yu Hen Hu 9
Intro. ANN & Fuzzy Systems
Fuzzy Reasoning (cont'd)
• µAge(Dana)(x) = {µyoung(y) µmuch_older(x,y) }
The universe of discourse (support) is "Age" which may be quantified into several overlapping fuzzy (sub)sets: Young, Mid-age, Old with the following definitions:
Age
Young Mid-age Old
20 35 50
µ(Age)
5
(C) 2001-2003 by Yu Hen Hu 10
Intro. ANN & Fuzzy Systems
Fuzzy Reasoning (cont'd)
• Much_older is a relation which is defined as:
µmuch_older(x,y) = ,
.0
200)(20
1,201
yx
yxyx
yx
010
2030
40 y
x10
2030
40
µ (x,y)much_older
(C) 2001-2003 by Yu Hen Hu 11
Intro. ANN & Fuzzy Systems
Reasoning ExampleFor each fixed x, find
µAge(Dana)(x) = max(min(µyoung(y),µmuch_older(x,y)):
0
0.2
0.4
0.6
0.8
0 5 10 15 20 25 30 35 40 45
1.0
x
µ (x)Age(Dana)