Fuzzy logic in approximate Reasoning

Fuzzy Logic & Approximate
Reasoning
1
Fuzzy Logic &
Approximate Reasoning

Reasoning
2
Fuzzy Sets

Reasoning
3
References
• Journal:
– IEEE Trans. on Fuzzy Systems.
– Fuzzy Sets and Systems.
– Journal of Intelligent & Fuzzy Systems
– International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
– ...
• Conferences:
– IEEE Conference on Fuzzy Systems.
– IFSA World Congress.
– ...
• Books and Papers:
– 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.
– ......
– http://www.type2fuzzylogic.org/
– http://ieee-cis.org/
– International Fuzzy Systems Association http://www.isc.meiji.ac.jp/~ifsatkym/
– J.M. Mendel http://sipi.usc.edu/~mendel

Reasoning
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Contents
• Fuzzy sets.
• Fuzzy Relations and Fuzzy reasoning
• Fuzzy Inference Systems
• Fuzzy Clustering
• Fuzzy Expert Systems
• Applications: Image Processing, Robotics,
Control...

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

Reasoning
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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.

Reasoning
7
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 AreMost Words and Evaluations We Use in Our Daily Reasoning Are
Not Clearly Defined in a Mathematical Manner. This AllowsNot Clearly Defined in a Mathematical Manner. This Allows
Humans to Reason on an Abstract Level!Humans to Reason on an Abstract Level!

Reasoning
8
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.).

Reasoning
9
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.

Reasoning
10
Fuzzy Set Theory
Conventional (Boolean) Set Theory:
“Strong Fever”
40.1°C
42°C
41.4°C
39.3°C
38.7°C38.7°C
37.2°C37.2°C
38°C38°C
Fuzzy Set Theory:
40.1°C
42°C
41.4°C
39.3°C
38.7°C
37.2°C
38°C
““More-or-Less” Rather Than “Either-Or” !More-or-Less” Rather Than “Either-Or” !
“Strong Fever”

Reasoning
11
Fuzzy Sets
• Sets with fuzzy
boundaries
A = Set of tall people

Reasoning
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Membership Functions (MFs)
• Characteristics of MFs:
– Subjective measures
– Not probability functions

Reasoning
13
Fuzzy Sets
• A fuzzy set A is characterized by a member set
function (MF), µA, 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:
Membership
function
(MF)
A x x x XA= ∈{( , ( ))| }µ
Universe or
universe of discourseFuzzy set
Universe or
universe of discourse
Universe or
universe of discourse

Reasoning
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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)}

Reasoning
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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)}
2B
10
50x
1
1
)x(





 −
+
=µ

Reasoning
16
Alternative Notation
• A fuzzy set A can be alternatively denoted as
follows:
A x xA
x X
i i
i
=
∈
∑ µ ( ) /
A x xA
X
= ∫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.

Reasoning
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Fuzzy Partition
• Fuzzy partitions formed by the linguistic
values “young”, “middle aged”, and “old”:

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 [χ , T(χ), U], in which χ : name of
the variable, T(χ) : the term set of χ , universe of discourse U
A Linguistic VariableA Linguistic Variable
Defines a Concept of OurDefines a Concept of Our
Everyday Language!Everyday Language!

Reasoning
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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.

Reasoning
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MF Terminology

Reasoning
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Convexity of Fuzzy Sets
µ λ λ µ µA A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21+ − ≥
Alternatively, A is convex if all its α-cuts are
convex.
• A fuzzy set A is convex if for any l in [0, 1],

Reasoning
22
Set-Theoretic Operations
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
• Subset:
• Complement:
• Union:
• Intersection:

Reasoning
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Set-Theoretic Operations

Reasoning
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MF Formulation
• Triangular MF: trimf x a b c
x a
b a
c x
c b
( ; , , ) max min , ,=
−
−
−
−











0
Trapezoidal MF:
Generalized bell MF: gbellmf x a b c
x c
b
b( ; , , ) =
+
−
1
1
2
Gaussian MF: gaussmf x a b c e
x c
( ; , , ) =
−
−





1
2
2
σ
trapmf x a b c d
x a
b a
d x
d c
( ; , , , ) max min , , ,=
−
−
−
−











1 0

Reasoning
25
MF Formulation

Reasoning
26
MF Formulation
• Sigmoidal
MF:
sigmf x a b c
e a x c( ; , , ) ( )=
+ − −
1
1
Extensions:
Abs. difference
of two sig. MF
Product
of two sig. MF

Reasoning
27
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:
– Yager’s complement:
N a
a
sa
s( ) =
−
+
1
1
N a aw
w w
( ) ( ) /
= −1 1

Reasoning
28
Fuzzy Complement
N a
a
sa
s( ) =
−
+
1
1
N a aw
w w
( ) ( ) /
= −1 1
Sugeno’s complement: Yager’s complement:

Reasoning
29
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)

Reasoning
30
T-norm Operator
Minimum:
Tm(a, b)
Algebraic
product:
Ta(a, b)
Bounded
product:
Tb(a, b)
Drastic
product:
Td(a, b)

Reasoning
31
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) =

Reasoning
32
T-conorm or S-norm
Maximum:
Sm(a, b)
Algebraic
sum:
Sa(a, b)
Bounded
sum:
Sb(a, b)
Drastic
sum:
Sd(a, b)

Reasoning
33
Fuzzy Number
• A fuzzy number A must possess the
following three properties:
• 1. A must must be a normal fuzzy set,
• 2. The alpha levels must be closed for
every ,
• 3. The support of A, , must be
bounded.
)(αA
]1,0(∈α
)0( +A

Reasoning
34
Fuzzy Number
1
Membershipfunction
is the suport of
z1
is the modal value
is an α-level of , α (0,1]α
'
< ' [ ] [ ]z zα α
α α ⇔ ⊂
( ),z z− +
z+1zzα
− zα
+
z−
,[ ] z zz − + ≡
 α  α α
z%
α’
z% ∈

Reasoning
35
Fuzzy Relations
• A fuzzy relation ℜ is a 2 D MF:
ℜ :{ ((x, y), µℜ(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:
if X = {3,4,5} & Y = {3,4,5,6,7}





≤
>
++
−
=µ
xyif,0
xyif,
2yx
xy
)y,x(R

Reasoning
36
Extension Principle
The image of a fuzzy set A on X
nnA22A11A
x/)x(x/)x(x/)x(A µ++µ+µ= 
under f(.) is a fuzzy set B:
nnB22B11B
y/)x(y/)x(y/)x(B µ++µ+µ= 
where yi = f(xi), i = 1 to n
If f(.) is a many-to-one mapping, then
)x(max)y( A
)y(fx
B 1
µ=µ −
=

Reasoning
37
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

Reasoning
38
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

Reasoning
39
Interval Type-2 Fuzzy Sets:
Terminology-1

Reasoning
40

Reasoning
41
Questions

Fuzzy logic in approximate Reasoning

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