# Introduction to Fuzzy Set Theory Weldon A. Lodwick - PowerPoint

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```					                  Introduction to Fuzzy Set Theory
Weldon A. Lodwick

OBJECTIVES
1. To introduce fuzzy sets and how they are used
2. To define some types of uncertainty and study what
methods are used to with each of the types.
3. To define fuzzy numbers, fuzzy logic and how they are
used
4. To study methods of how fuzzy sets can be constructed
5. To see how fuzzy set theory is used and applied in cluster
analysis

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                    2003                      1
OUTLINE

I.      INTRODUCTION – Lecture 1
A. Why fuzzy sets
1. Data/complexity reduction
2. Control and fuzzy logic
3. Pattern recognition and cluster analysis
4. Decision making
B. Types of uncertainty
1. Deterministic, interval, probability
2. Fuzzy set theory, possibility theory
C. Examples – Tejo river, landcover/use, surfaces

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II.       BASICS – Lecture 2

A. Definitions
1. Sets – classical sets, fuzzy sets, rough sets, fuzzy interval
sets, type-2 fuzzy sets
2. Fuzzy numbers
B. Operations on fuzzy sets
1. Union
2. Intersection
3. Complement

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BASICS (continued)

C. Operations on fuzzy numbers
1. Arithmetic
2. Relations, equations
3. Fuzzy functions and the extension principle

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III. FUZZY LOGIC – Lecture 3
A. Introduction
B. Fuzzy propositions
C. Fuzzy hedges
D. Composition, calculating outputs
E. Defuzzification/action
IV.     FUZZY SET METHODS Cluster analysis – Lecture 4

I. INTRODUCTION: Math Clinic Fall
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I. INTRODUCTION – Lecture 1

   Fuzzy sets are sets that have gradations of belonging
EXAMPLES:
Green
BIG
Near
   Classical sets, either an element belongs or it does not
EXAMPLES:
Set of integers – a real number is an integer or not
You are either in an airplane or not
Your bank account is x dollars and y cents

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A. Why fuzzy sets?
- Modeling with uncertainty requires more than probability theory
- There are problems where boundaries are gradual
EXAMPLES:
What is the boundary of the USA? Is the boundary a
mathematical curve? What is the area of USA? Is the area a
real number?
Where does a tumor begin in the transition?
What is the habitat of rabbits in 20km radius from here?
What is the depth of the ocean 30 km east of Myrtle Beach?
1. Data reduction – driving a car, computing with language
2. Control and fuzzy logic
a. Appliances, automatic gear shifting in a car
b. Subway system in Sendai, Japan (control outperformed
humans in giving smoother rides)
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Temperature control in NASA space shuttles
IF x AND y THEN z is A
IF x IS Y THEN z is A … etc.
If the temperature is hot and increasing very fast then air
conditioner fan is set to very fast and air conditioner
temperature is coldest. There are four types of propositions
we will study later.
3. Pattern recognition, cluster analysis
- A bank that issues credit cards wants to discover
whether or not it is stolen or being illegally used prior to a
customer reporting it missing
- Given a cat-scan determine the organs and their
position; given a satellite imagine, classify the land/cover use
- Given a mobile telephone, send the signal to/from a
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4. Decision making
- Locate mobile telephone receptors/transmitters to
optimally cover a given area
- Locate recycling bins to optimally cover UCD
- Position a satellite to cover the most number of mobile
phone users
- Deliver sufficient radiation to a tumor to kill the
cancerous cells while at that same time sparing healthy
cells (maximize dosage up to a limit at the tumor while
minimizing dosage at all other cells)
- Design a product in the following way: I want the
product to be very light, very strong, last a very long time
and the cost of production is the cheapest.

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Introduction

B. Types of Uncertainty
1. Deterministic – the difference between a known real number
value and its approximation is a real number (a single number).
Here one has error. For example, if we know the answer x must
be the square root of 2 and we have an approximation y, then the
error is x-y (or if you wish, y-x).
2. Interval – uncertainty is an interval. For example, measuring pi
using Archimedes’ approach.
3. Probabilistic – uncertainty is a probability distribution function
4. Fuzzy – uncertainty is a fuzzy membership function
5. Possibilistic - uncertainty is a possibility distribution function,
generated by nested sets (monotone)
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Types of sets (figure from Klir&Yuan)

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Introduction (figure from Klir&Yuan)

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Error, uncertainty - information/data is often imprecise,
incoherent, incomplete
DEFINITION: The error is the difference between the
exact value (a real number) and a value at hand (an
approximation). As such, when one talks about error, one
presupposes that there exists a “true” (real number)
value. The precision is the maximum number of digits
that are used to measure an approximation. It is the
property of the instrument that is being used to measure
or calculate the (exact) value. When a subset is being
used to measure/calculate, it corresponds to subset that
can no longer be subdivided. It depends on the
granularity of the input/output pairs (object/value pairs)
or the resolution being used.
EXAMPLE – satellite imagery at 1meter by 1 meter
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DEFINITION: Accuracy is the number of correct digits in an
approximation. For example, a gps reading is (x,y) +/- …
DEFINITION: Item of information – is a quadruple
(attribute, object, value, confidence) (definition is from
Attribute: a function that attaches value to the object; for
example: area, position, color; it’s the recipe that tells us
how to obtain an output (value) from an input (object)
Object: the entity (domain or input); for example, Sicily for
area or my shirt for color or room 4.2 for temperature.
Value: the assignment or output of the attribute; for
example 211,417.6 sq. km. for Sicily or green for shirt
Confidence: reliability of the information

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AMBIGUITY: a one to many relationship; for example, she
is tall, he is handsome. There are a variety of alternatives
1. Non-specificity: Suppose one has a heart blockage
and is prescribed a treatment. In this case “treatment”
is a non-specificity in that it can be an angioplasty,
medication, surgery (to name three alternatives)
2. Dissonance/contradiction: One physician says to
operate and another says go to Myrtle Beach.

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VAGUENESS – lack of sharp distinction or boundaries, our
ability to discriminate between different states of an
event, undecidability (is a glass half full/empty)

SET THEORY                                PROBABILITY

POSSIBILITY
THEORY

FUZZY SET                   DEMPSTER/SHAFER
THEORY                         THEORY

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EXAMPLES
Cidalia Fonte will go over in more detail the ideas introduced here at a later time.
Example 1. Tejo River
- The problem
The dimension of water bodies, and consequently their position, is subject to
variation over time, especially in regions which are frequently flooded or
subject to tidal variations, creating considerable uncertainty in positioning
these geographical entities. River Tejo is an example, since frequent floods
occur in several places along its bed. The region near the village of
Constância, where rivers Tejo and Zezere meet, was the chosen for this
example.
A fuzzy geographical entity corresponding to rivers Tejo and Zezere is considered
a fuzzy set. To generate this fuzzy entity, the membership function has to
be constructed. This was done using a Digital Elevation Model of the
region, created from the contours of the 1:25 000 map of the Army
Geographical Institute of Portugal and information regarding the daily
means of the river water level registered in the hydrometric station of
Almourol, located in the vicinity, from 1982 to 1990. The variation of the
water level during these year are on the next slide:

I. INTRODUCTION: Math Clinic Fall
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Example 1 (figures from Cidalia Fonte & Lodwick)

meters above the 20m level
12
10
8
6
4
The membership function of                                          2
0
points to the fuzzy set is

1982
1983
1984
1984
1985
1986
1987
1988
1989
1989
1990
-2
given by:  x, y  f zx, y
T          100
f(z)
100%
80%
60%
40%
20%
0%
20 21 22 23 24 25 26 28 29 30
altitude z

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Example (figure from Cidalia Fonte & Lodwick)

The river limits represented on the map

Line corresponding to the maximum water
level registered during the considered period

 T( x , y )

1        y

Line corresponding to the region always
submerged during the considered period                               x

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Example 2 – Landcover/use
(figures from Cidalia Fonte & Lodwick)

Water regions
a)                    b)
Vegetation
Bareland

c)                    d)

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Example 2 – Landcover/use continued

 Bareland  x, y                   Water regions  x, y 

  x, y   1
  x, y   0.75
  x, y   0.5
  x, y   0.25
  x, y   0

Vegetation  x, y 

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GIS - Display
y

 forest  x, y  1
0

x
y

b)

 grass  x, y  1
0

y   x

 wet regions  x, y  1                                            c)
0

x

a)

I. INTRODUCTION: Math Clinic Fall
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Example 3 – Surface modeling

3. Surface models
- The problem: Given a set of reading of the bottom of the
ocean whose values are uncertain, generate a surface
that explicitly incorporates this uncertainty mathematically
and visually - The approach: Consistent fuzzy surfaces

- Here with just introduce the associated ideas

I. INTRODUCTION: Math Clinic Fall
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Imprecision in Points: Fuzzy Points (figures from Jorge dos Santos)

2D                                          3D

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Transformation of real-valued functions to fuzzy
functions

Instead of a real-valued function z  f ( x) or z  f ( x, y) let’s now
            
consider a fuzzy function z  f ( x) or z  f (x, y) where every element

x or (x,y) is associated with a fuzzy number z .

Statement of the Interpolation Problem


Knowing the values {zi} of a fuzzy function over a finite set of points
{xi} or {(xi,yi)}, interpolate over the domain in question to obtain a
(nested) set of surfaces that represent the uncertainty in the data.
.
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Computing surfaces

Given a data set of fuzzy numbers:
~ 1 d fuzzy triangular a / b / c
z
~( x)  N ~ L ( x)
p        zi i
i 1
~( x)]  N z ( )L ( x)
[p   i           i
i 1

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Computing surfaces – Example

~  0.5 /1.5 / 2, ~  0.75/1/1.5
z1                z2

L1( x)  x  2, L2 ( x)  3x 1
x 1 L1(1) 1 2  3, L2 (1)  3*11 2
 ~(1)  3~  2~2 1.5 / 4.5 / 6 1.5 / 2 / 3
p        z1     z
 3/6.5/9
[ ~(1)]  0  [3, 9]
p
[ ~(1)]0.5  [4.75, 7.75]
p
[ ~(1)]1  [6.5, 6.5]
p

I. INTRODUCTION: Math Clinic Fall
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Consistent Fuzzy Surfaces (curves)

The surfaces (curves) are defined enforcing the
following properties:
1. The surfaces are defined analytically via the fuzzy
functions; that is, model directly the uncertainty
              
using fuzzy functions z  f ( x) or z  f (x, y)
2. All fuzzy surfaces maintain the characteristics of
the generating method. That is, if splines are
being used then all generated fuzzy surfaces have
the continuity and smoothness conditions
associated with the splines being used.

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                  2003                      28
Fuzzy Interpolating Polynomial -                          
p( x) (figure from
Jorge dos Santos & Lodwick)

Utilizing alpha-levels to obtain fuzzy polynomials, we have:
 p(x)   p ( x), p ( x) 


 
        

     zR : z p (x), d   z  
d        i    i 


z2

p ( x)

z1


z2

p ( x)   z1


x1       x       x2

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                           2003                                    29
2-D Example (from Jorge dos Santos & Lodwick)
60

50

40
z   30

20

10

0

-10
-50                     0                50
x          100                 150               200

xi        0        15     25         50         90           121    143          165   200
zi-     19.5       14.9   5.8        -3.9       39.0         22.3   32.1     29.4      2.5
zi1     20.0       15.0   6.0        -4.0       40.0         23.0   33.0     30.0      3.0
zi+      20.3       15.6   6.3        -4.2       41.2         23.7   34.0     30.1      3.2

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                           2003                                                   30
Fuzzy Curves (figures from Jorge dos Santos & Lodwick)
50
P. Lagrange
20

0        15

10
-50

5

-100
0           20   10 40   60 15    80        20
100       120      25 140   160 30   180        200
60

40
Spline linear

20

0

-20
0         20     40    60       80        100       120        140   160      180        200

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                          2003                                            31
Fuzzy Curves (figures from Jorge dos Santos & Lodwick)

50
Cubic Spline

40

30
z
20

10

0

-10
0         20    40    60     80      100            120   140   160   180   200
x
50
Cubic Spline
Consistent

40

30
z    20

10

0

-10
0     20    40   60      80      100           120   140   160   180   200
x
I. INTRODUCTION: Math Clinic Fall
August 12, 2003                            2003                                           32
Details of the Consistent Fuzzy Cubic Spline
(figures from Jorge dos Santos & Lodwick)

33

32

31
z
30

29

28

27

155           160                     165          170
x

I. INTRODUCTION: Math Clinic Fall
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3-D Example (from Jorge dos Santos & Lodwick)

200

180

160

140

120
y
100

80

60

40

20

0
-50      0    50           100           150     200   250
x

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                   2003                                34
Another Representation/View of the Fuzzy Points
(figure from Jorge dos Santos & Lodwick)

35

30
z 25
20

15                                                                                 200
10
150
5
-50
100
0
50
x
y     100
50

150
200   0

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                            2003                                               35
Fuzzy Surface via Triangulation (figure
from Jorge dos Santos & Lodwick)

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                     2003                    36
Fuzzy Surfaces via Linear Splines
(figure from Jorge dos Santos & Lodwick)

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                     2003                    37
Fuzzy Surfaces via Cubic Splines
(figure from Jorge dos Santos & Lodwick)

I. INTRODUCTION: Math Clinic Fall
August 12, 2003                    2003                    38

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