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Chapter 2, Neuro-Fuzzy and Soft Computing
by J.-S. Roger Jang
Chapter 2, Neuro-Fuzzy and Soft Computing
by J.-S. Roger Jang
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
2
Fuzzy Sets: OutlineFuzzy Sets: Outline
IntroductionBasic definitions and terminologySet-theoretic operationsMF formulation and parameterization
• MFs of one and two dimensions• Derivatives of parameterized MFs
More on fuzzy union, intersection, and complement• Fuzzy complement• Fuzzy intersection and union• Parameterized T-norm and T-conorm
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
3
Fuzzy SetsFuzzy Sets
Sets with fuzzy boundaries
A = Set of tall people
Heights5’10’’
1.0
Crisp set A
Membershipfunction
Heights5’10’’ 6’2’’
.5
.9
Fuzzy set A1.0
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
4
Membership Functions (MFs)Membership Functions (MFs)
Characteristics of MFs:• Subjective measures• Not probability functions
MFs
Heights5’10’’
.5
.8
.1
“tall” in Asia
“tall” in the US
“tall” in NBA
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
5
Fuzzy SetsFuzzy Sets
Formal definition:A fuzzy set A in X is expressed as a set of ordered
pairs:
A x x x XA= ∈{( , ( ))| }µ
Universe oruniverse of discourse
Fuzzy set Membershipfunction
(MF)
A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
6
Fuzzy Sets with Discrete UniversesFuzzy Sets with Discrete UniversesFuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and nonordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
Fuzzy set A = “sensible number of children”X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
7
Fuzzy Sets with Cont. UniversesFuzzy Sets with Cont. Universes
Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)B = {(x, µB(x)) | x in X}
µ B x x( ) =
+−
1
1 5010
2
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
8
Alternative NotationAlternative Notation
A fuzzy set A can be alternatively denoted as follows:
A x xAx X
i ii
=∈
∑ µ ( ) /
A x xAX
= ∫ µ ( ) /
X is discrete
X is continuous
Note that Σ and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
9
Fuzzy PartitionFuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
lingmf.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
10
More DefinitionsMore Definitions
SupportCoreNormalityCrossover pointsFuzzy singletonα-cut, strong α-cut
ConvexityFuzzy numbersBandwidthSymmetricityOpen left or right, closed
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
11
MF TerminologyMF Terminology
MF
X
.5
1
0 Core
Crossover points
Support
α - cut
α
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
12
Convexity of Fuzzy SetsConvexity of Fuzzy Sets
A fuzzy set A is convex if for any λ in [0, 1],µ λ λ µ µA A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21+ − ≥
Alternatively, A is convex is all its α-cuts are convex.
convexmf.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
13
Set-Theoretic OperationsSet-Theoretic Operations
Subset:
Complement:
Union:
Intersection:
A B A B⊆ ⇔ ≤µ µ
C A B x x x x xc A B A B= ∪ ⇔ = = ∨µ µ µ µ µ( ) max( ( ), ( )) ( ) ( )
C A B x x x x xc A B A B= ∩ ⇔ = = ∧µ µ µ µ µ( ) min( ( ), ( )) ( ) ( )
A X A x xA A= − ⇔ = −µ µ( ) ( )1
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
14
Set-Theoretic OperationsSet-Theoretic Operations
subset.m
fuzsetop.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
15
MF FormulationMF Formulation
Triangular MF: trim f x a b cx a
b a
c x
c b( ; , , ) max m in , ,=
−−
−−
0
Trapezoidal MF: trapm f x a b c dx a
b a
d x
d c( ; , , , ) m ax m in , , ,=
−−
−−
1 0
Generalized bell MF: b
acx
cbaxgbellmf 2
1
1),,;(−
+
=
Gaussian MF:2
21
),;(
−
−= σσ
cx
ecxgaussmf
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
17
MF FormulationMF Formulation
Sigmoidal MF: )(11),;( cxae
caxsigmf −−+=
Extensions:
Abs. differenceof two sig. MF
Productof two sig. MF
disp_sig.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
18
MF FormulationMF Formulation
L-R MF:LR x c
Fc x
x c
Fx c
x c
L
R
( ; , , )
,
,
α βα
β
=
−
<
−
≥
Example: F x xL ( ) max( , )= −0 1 2 F x xR ( ) exp( )= − 3
difflr.m
c=65a=60b=10
c=25a=10b=40
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
19
Cylindrical ExtensionCylindrical Extension
Base set A Cylindrical Ext. of A
cyl_ext.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
20
2D MF Projection2D MF Projection
Two-dimensionalMF
Projectiononto X
Projectiononto Y
µ R x y( , ) µµ
A
y R
xx y
( )max ( , )
= µµ
B
x R
yx y
( )max ( , )
=
project.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
22
Fuzzy ComplementFuzzy Complement
General requirements:• Boundary: N(0)=1 and N(1) = 0• Monotonicity: N(a) > N(b) if a < b• Involution: N(N(a) = a
Two types of fuzzy complements:• Sugeno’s complement:
• Yager’s complement:
N a asas( ) =
−+11
N a aww w( ) ( ) /= −1 1
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
23
Fuzzy ComplementFuzzy Complement
negation.m
N a asas( ) =
−+11
N a aww w( ) ( ) /= −1 1
Sugeno’s complement: Yager’s complement:
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
24
Fuzzy Intersection: T-normFuzzy Intersection: T-norm
Basic requirements:• Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a• Monotonicity: T(a, b) < T(c, d) if a < c and b < d• Commutativity: T(a, b) = T(b, a)• Associativity: T(a, T(b, c)) = T(T(a, b), c)
Four examples (page 37):• Minimum: Tm(a, b)• Algebraic product: Ta(a, b)• Bounded product: Tb(a, b)• Drastic product: Td(a, b)
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
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T-norm OperatorT-norm Operator
Minimum:Tm(a, b)
Algebraicproduct:Ta(a, b)
Boundedproduct:Tb(a, b)
Drasticproduct:Td(a, b)
tnorm.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
26
Fuzzy Union: T-conorm or S-normFuzzy Union: T-conorm or S-norm
Basic requirements:• Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a• Monotonicity: S(a, b) < S(c, d) if a < c and b < d• Commutativity: S(a, b) = S(b, a)• Associativity: S(a, S(b, c)) = S(S(a, b), c)
Four examples (page 38):• Maximum: Sm(a, b)• Algebraic sum: Sa(a, b)• Bounded sum: Sb(a, b)• Drastic sum: Sd(a, b)
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
27
T-conorm or S-normT-conorm or S-norm
tconorm.m
Maximum:Sm(a, b)
Algebraicsum:
Sa(a, b)
Boundedsum:
Sb(a, b)
Drasticsum:
Sd(a, b)
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
28
Generalized DeMorgan’s LawGeneralized DeMorgan’s Law
T-norms and T-conorms are duals which support the generalization of DeMorgan’s law:
• T(a, b) = N(S(N(a), N(b)))• S(a, b) = N(T(N(a), N(b)))
Tm(a, b)Ta(a, b)Tb(a, b)Td(a, b)
Sm(a, b)Sa(a, b)Sb(a, b)Sd(a, b)
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
29
Parameterized T-norm and S-normParameterized T-norm and S-norm
Parameterized T-norms and dual T-conormshave been proposed by several researchers:
• Yager• Schweizer and Sklar• Dubois and Prade• Hamacher• Frank• Sugeno• Dombi
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
30
What is the Difference Between Probabilistic and Fuzzy Methods? ExampleWhat is the Difference Between Probabilistic and Fuzzy Methods? Example
Bottles filled with liquid in a desert – Before one tastes