Fuzzy Logic &
Approximate Reasoning
Fuzzy Logic & Approximate
Reasoning
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Fuzzy Sets
Fuzzy Logic & Approximate
Reasoning
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References
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Journal:
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Conferences:
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IEEE Trans. on Fuzzy Systems.
Fuzzy Sets and Systems.
Journal of Intelligent & Fuzzy Systems
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
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IEEE Conference on Fuzzy Systems.
IFSA World Congress.
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Books and Papers:
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Z.Chi et al, Fuzzy Algorithms with applications to Image Processing and Pattern Recognition, World
Scientific, 1996.
S. N. Sivanandam, Introduction to Fuzzy Logic using MATLAB, Springer, 2007.
J.M. Mendel, Fuzzy Logic Systems for Engineering: A toturial, IEEE, 1995.
W. Siler, FUZZY EXPERT SYSTEMS AND FUZZY REASONING, John Wiley Sons, 2005
G. Klir, Uncertainty and Informations, John Wiley Sons, 2006.
L.A. Zadeh, Fuzzy sets, Information and control, 8, 338-365, 1965.
L.A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I, II, III,
Information Science 8, 1975
L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy
logic, Fuzzy Sets and Systems 90(1997), 111-127.
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http://www.type2fuzzylogic.org/
IEEE Computational Intelligence Societỵ http://ieee-cis.org/
International Fuzzy Systems Association http://www.isc.meiji.ac.jp/~ifsatkym/
Fuzzy Logic & Approximate
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J.M. Mendel http://sipi.usc.edu/~mendel
Reasoning
Contents
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Fuzzy sets.
Fuzzy Relations and Fuzzy reasoning
Fuzzy Inference Systems
Fuzzy Clustering
Fuzzy Expert Systems
Applications: Image Processing, Robotics,
Control...
Fuzzy Logic & Approximate
Reasoning
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Fuzzy Sets: Outline
• Introduction: History, Current Level and Further Development of
Fuzzy Logic Technologies in the U.S., Japan, and Europe
• Basic definitions and terminology
• Set-theoretic operations
• MF 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
• Fuzzy Number
• Fuzzy Relations
Fuzzy Logic & Approximate
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History, State of the Art, and
Future Development
1965
Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in
Electrical Engineering, U.C. Berkeley, Sets the Foundation of
the “Fuzzy Set Theory”
1970
First Application of Fuzzy Logic in Control Engineering
(Europe)
1975
Introduction of Fuzzy Logic in Japan
1980
Empirical Verification of Fuzzy Logic in Europe
1985
Broad Application of Fuzzy Logic in Japan
1990
Broad Application of Fuzzy Logic in Europe
1995
Broad Application of Fuzzy Logic in the U.S.
1998
Type-2 Fuzzy Systems
2000
Fuzzy Logic Becomes a Standard Technology and Is Also
Applied in Data and Sensor Signal Analysis. Application of
Fuzzy Logic in Business and Finance.
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Types of Uncertainty and the
Modeling of Uncertainty
Stochastic Uncertainty:
 The Probability of Hitting the Target Is 0.8
Lexical Uncertainty:
 "Tall Men", "Hot Days", or "Stable Currencies"
 We Will Probably Have a Successful Business Year.
 The Experience of Expert A Shows That B Is Likely to
Occur. However, Expert C Is Convinced This Is Not True.
Most Words and Evaluations We Use in Our Daily Reasoning Are
Not Clearly Defined in a Mathematical Manner. This Allows
Humans to Reason on an Abstract Level!
Fuzzy Logic & Approximate
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Possible Sources of Uncertainty
and Imprecision
• There are many sources of uncertainty facing
any control system in dynamic real world
unstructured environments and real world
applications; some sources of these
uncertainties are as follows:
– Uncertainties in the inputs of the system due
to:
• The sensors measurements being affected by high noise
levels from various sources such a electromagnetic and radio
frequency interference, vibration, etc.
• The input sensors being affected by the conditions of
observation (i.e. their characteristics can be changed by the
environmental conditions such as wind, sunshine, humidity,
rain, etc.).
Fuzzy Logic & Approximate
Reasoning
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Possible Sources of Uncertainty
and Imprecision
• Other sources of Uncertainties include:
– Uncertainties in control outputs which can
result from the change of the actuators
characteristics due to wear and tear or due to
environmental changes.
– Linguistic uncertainties as words mean
different things to different people.
– Uncertainties associated with the change in
the operation conditions due to varying load and
environment conditions.
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Fuzzy Set Theory
Conventional (Boolean) Set Theory:
38.7°C
38°C
40.1°C
39.3°C
41.4°C
Fuzzy Set Theory:
42°C
“Strong Fever”
38.7°C
38°C
37.2°C
40.1°C
39.3°C
“More-or-Less” Rather Than “Either-Or” !
37.2°C
Fuzzy Logic & Approximate
Reasoning
41.4°C
42°C
“Strong Fever”
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Fuzzy Sets
• Sets with fuzzy
boundaries
A = Set of tall people
Fuzzy Logic & Approximate
Reasoning
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Membership Functions (MFs)
• Characteristics of MFs:
– Subjective measures
– Not probability functions
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Fuzzy Sets
• A fuzzy set A is characterized by a member set
function (MF), A, mapping the elements of A to
the unit interval [0, 1].
• Formal definition:
A fuzzy set A in X is expressed as a set of ordered
pairs:
A  {( x,  A ( x ))| x  X }
Fuzzy set
Membership
function
(MF)
Fuzzy Logic & Approximate
Reasoning
Universe or
universe of discourse
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Fuzzy Sets with Discrete Universes
• Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and non-ordered)
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)}
Fuzzy Logic & Approximate
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Fuzzy Sets with Cont. Universes
• Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and non-ordered)
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)}
1
 B ( x) 
2
 x  50 
1 

 10 
Fuzzy Logic & Approximate
Reasoning
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Alternative Notation
• A fuzzy set A can be alternatively denoted as
follows:
A
X is discrete

A
( xi ) / xi
A
(x) / x
xi X
A
X is continuous

X
Note that S and integral signs stand for the
union of membership grades; “/” stands for a
marker and does not
imply division.
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Reasoning
Fuzzy Partition
• Fuzzy partitions formed by the linguistic
values “young”, “middle aged”, and “old”:
Fuzzy Logic & Approximate
Reasoning
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Linguistic Variables
• A linguistic variable is a variable whose values are not numbers but words
or sentences in a natural or artificial language (Zadeh, 1975a, p. 201)
• Linguistic variable is characterized by [c , T(c), U], in which c : name of the
variable, T(c) : the term set of c , universe of discourse U
A Linguistic Variable
Defines a Concept of Our
Everyday Language!
Fuzzy Logic & Approximate
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Fuzzy Hedges
• Suppose you had already defined a fuzzy set to describe
a hot temperature.
• Fuzzy set should be modified to represent the hedges
"Very" and "Fairly“: very hot or fairly hot.
Fuzzy Logic & Approximate
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MF Terminology
Fuzzy Logic & Approximate
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Convexity of Fuzzy Sets
• A fuzzy set A is convex if for any l in [0, 1],
A ( x1  ( 1   ) x2 )  min( A ( x1 ), A ( x2 ))
Alternatively, A is convex if all its a-cuts are
convex.
Fuzzy Logic & Approximate
Reasoning
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Set-Theoretic Operations
• Subset:
A  B  A  B
• Complement:
A  X  A  A ( x )  1  A ( x )
• Union:
C  A  B  c ( x )  max( A ( x ), B ( x ))  A ( x ) B ( x )
• Intersection:
C  A  B  c ( x )  min( A ( x ), B ( x ))  A ( x ) B ( x )
Fuzzy Logic & Approximate
Reasoning
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Set-Theoretic Operations
Fuzzy Logic & Approximate
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MF Formulation
• Triangular MF:
Trapezoidal MF:
Gaussian MF:

x a c  x 
,
 , 0
b a c b 
trimf ( x ; a , b , c )  max min 


d  x 
x a
, 1,
 , 0
b a
d c  
trapmf ( x ; a , b , c , d )  max min

gaussmf ( x ; a , b , c )  e
Generalized bell MF:
gbellmf ( x ; a , b , c ) 
Fuzzy Logic & Approximate
Reasoning
1  x c 
 

2  
2
1
x c
1
b
2b
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MF Formulation
Fuzzy Logic & Approximate
Reasoning
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MF Formulation
• Sigmoidal MF:
sigmf ( x ; a , b , c ) 
1
1  e a ( x c )
Extensions:
Abs. difference
of two sig. MF
Product
of two sig. MF
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Fuzzy 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:
1 a
N s (a ) 
1  sa
– Yager’s complement:
Nw (a )  (1  a )
Fuzzy Logic & Approximate
Reasoning
w 1/ w
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Fuzzy Complement
Sugeno’s complement:
1 a
N s (a ) 
1  sa
Yager’s complement:
Fuzzy Logic & Approximate
Reasoning
N w (a )  (1  a w )1/ w
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Fuzzy 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:
– Minimum: Tm(a, b) = min{a, b}.
– Algebraic product: Ta(a, b) = a.b
– Bounded product: Tb(a, b) = max{0, a+b-1}
– Drastic product:
Td(a,
b)
Fuzzy
Logic & Approximate
Reasoning
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T-norm Operator
Minimum:
Tm(a, b)
Algebraic
product:
Ta(a, b)
Bounded
product:
Tb(a, b)
Fuzzy Logic & Approximate
Reasoning
Drastic
product:
Td(a, b)
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Fuzzy 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:
– Maximum: Sm(a, b) = max{a, b}
– Algebraic sum: Sa(a, b) = a+b-a.b
– Bounded sum: Sb(a, b) = min{a+b, 1}.
– Drastic sum: Sd(a,
b)& =
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T-conorm or S-norm
Maximum:
Sm(a, b)
Algebraic
sum:
Sa(a, b)
Bounded
sum:
Sb(a, b)
Fuzzy Logic & Approximate
Reasoning
Drastic
sum:
Sd(a, b)
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Fuzzy Number
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A fuzzy number A must possess the
following three properties:
1. A must must be a normal fuzzy set,
2. The alpha levels A(a ) must be closed for
every a (0,1] ,
3. The support of A, A(0) , must be
bounded.
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Membership function
Fuzzy Number
 z, z 
is the suport of
z
1
z1 is the modal value
a’
[ z]a   za , za  is an a-level of
a
z, a
(0,1]
a < a'  [z]a'  [z]a
z
za z1
za
z
Fuzzy Logic & Approximate
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Fuzzy Relations
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A fuzzy relation  is a 2 D MF:
 :{ ((x, y), (x, y)) | (x, y)  X  Y}
Examples:
 x is close to y (x & y are real numbers)
 x depends on y (x & y are events)
 x and y look alike (x & y are persons or objects)
 Let X = Y = IR+
and R(x,y) = “y is much greater than x”
The MF of this fuzzy relation can be subjectively defined as:
 yx
, if y  x

x

y

2
 R ( x, y )  
 0
, if y  x

if X = {3,4,5} & Y = {3,4,5,6,7}
Fuzzy Logic & Approximate
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Extension Principle
The image of a fuzzy set A on X
A  A (x1 ) / x1  A (x2 ) / x2   A (xn ) / xn
under f(.) is a fuzzy set B:
B  B (x1 ) / y1  B (x2 ) / y 2   B (xn ) / yn
where yi = f(xi), i = 1 to n
If f(.) is a many-to-one mapping, then
 B ( y )  max  A ( x)
1
Fuzzy Logic & Approximate
x

f
(y)
Reasoning
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Example
– Application of the extension principle to fuzzy sets
with discrete universes
Let A = 0.1 / -2+0.4 / -1+0.8 / 0+0.9 / 1+0.3 / 2
and f(x) = x2 – 3
Applying the extension principle, we obtain:
B = 0.1 / 1+0.4 / -2+0.8 / -3+0.9 / -2+0.3 /1
= 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / 1
= 0.8 / -3+0.9 / -2+0.3 / 1
where “V” represents the “max” operator, Same
reasoning for continuous universes
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Transition From Type-1 to
Type-2 Fuzzy Sets
• Blur the boundaries of a T1
FS
• Possibility assigned—could
be non-uniform
• Clean things up
• Choose uniform
possibilities—interval type-2
FS
Fuzzy Logic & Approximate
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Interval Type-2 Fuzzy Sets:
Terminology-1
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Fuzzy Logic & Approximate
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Questions
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