Fuzzy Sets
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
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Fuzzy 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
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Fuzzy 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
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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
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Fuzzy Sets
Formal definition:A fuzzy set A in X is expressed as a set of ordered
pairs:
A x x x XA {( , ( ))| }
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).
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Fuzzy 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)}
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Fuzzy Sets with Cont. Universes
Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)B = {(x, B(x)) | x in X}
B x x( )
1
1 5010
2
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Alternative Notation
A fuzzy set A can be alternatively denoted as follows:
A x xAx X
i ii
( ) /
A x xAX
( ) /
X is discrete
X is continuous
Note that and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.
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Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
lingmf.m
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Set-Theoretic Operations
Subset:
Complement:
Union:
Intersection:
A B A B
C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )
C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )
A X A x xA A ( ) ( )1
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Set-Theoretic Operations
subset.m
fuzsetop.m
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MF Formulation
Triangular MF: trimf x a b cx ab a
c xc b( ; , , ) max min , ,
0
Trapezoidal MF: trapmf x a b c dx ab a
d xd c( ; , , , ) max min , , ,
1 0
Generalized bell MF: gbellmf x a b cx cb
b( ; , , )
1
12
Gaussian MF: gaussmf x a b c ex c
( ; , , )
12
2
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MF Formulation
disp_mf.m
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MF Formulation
Sigmoidal MF: sigmf x a b c e a x c( ; , , ) ( )
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Extensions:
Abs. differenceof two sig. MF
Productof two sig. MF
disp_sig.m
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MF FormulationL-R MF:
LR x cF
c xx c
Fx c
x c
L
R
( ; , , ),
,
Example: F x xL ( ) max( , ) 0 1 2 F x xR ( ) exp( ) 3
difflr.m
c=65a=60b=10
c=25a=10b=40
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Mamdani Fuzzy Models
Graphics representation:A1 B1
A2 B2
T-norm
X
X
Y
Y
w1
w2
C1
C2
Z
Z
C’Z
X Yx is 4.5 y is 56.8 z is zCOA
