August 12, 2003 I. INTRODUCTION: Math Clinic Fall 20031 Introduction to Fuzzy Set Theory Weldon A. Lodwick OBJECTIVES 1. To introduce fuzzy sets and how.

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

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Introduction to Fuzzy Set Theory Weldon A. Lodwick

OBJECTIVES1. To introduce fuzzy sets and how they are used2. 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 used4. To study methods of how fuzzy sets can be

constructed5. To see how fuzzy set theory is used and applied in

cluster analysis

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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 3A. IntroductionB. Fuzzy propositionsC. Fuzzy hedgesD. Composition, calculating outputs

E. Defuzzification/action

IV. FUZZY SET METHODS Cluster analysis – Lecture 4

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I. INTRODUCTION – Lecture 1

Fuzzy sets are sets that have gradations of belongingEXAMPLES: 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 particular receiver to/from the telephone

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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 Dubois&Prade, Possibility Theory)

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 alternatives1. 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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EXAMPLESCidalia Fonte will go over in more detail the ideas introduced here at a later time.Example 1. Tejo River - The problemThe 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:

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Example 1 (figures from Cidalia Fonte & Lodwick)

The membership function of points to the fuzzy set is given by:

100,, yxzfyx

T

-202468

1012

1982

1983

1984

1984

1985

1986

1987

1988

1989

1989

1990

met

ers

abov

e th

e 20

m l

evel

0%20%40%60%80%

100%

20 21 22 23 24 25 26 28 29 30altitude z

f(z)

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Example (figure from Cidalia Fonte & Lodwick)

x

y

T ( x , y )

1

Line corresponding to the maximum waterlevel registered during the considered period

The river limits represented on the map

Line corresponding to the region alwayssubmerged during the considered period

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

a) b)Water regions

Vegetation

Bareland

d)c)

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Example 2 – Landcover/use continued

, 1

, 0 . 7 5

, 0 . 5

, 0 . 2 5

, 0

x y

x y

x y

x y

x y

,B a r e l a n d x y ,W a t e r r e g i o n s x y

,V e g e t a t i o n x y

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GIS - Display

10

,forest x y

y

x

10

,grass x y

x

y

10 ,wet regions x y

x

y

a)

b)

c)

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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

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2D 3D

Imprecision in Points: Fuzzy Points (figures from Jorge dos Santos)

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

Transformation of real-valued functions to fuzzy functions

( )z f x( )z f x

Knowing the values { } 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.

.

Statement of the Interpolation Problem

( , )z f x y( , )z f x yz

iz

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Computing surfaces

Given a data set of fuzzy numbers:

N

iii

z

N

iii

xLxp

xLzxp

cbatriangularfuzzydz

1

)(

1

)()](~[

)(~)(~// 1~

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Computing surfaces – Example

].56 ,5.6[)]1(~[

].757 ,75.4[)]1(~[

]9 ,3[)]1(~[3/6.5/9

3/2/5.16/5.4/5.1~2~3)1(~211*3)1( ,321)1(1

13)( ,2)(

5.1/1/75.0~ ,2/5.1/5.0~

1

5.0

0

21

21

21

21

p

p

p

zzp

LLxxxLxxL

zz

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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 or

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.

Consistent Fuzzy Surfaces (curves)

( , )z f x y( )z f x

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Fuzzy Interpolating Polynomial - (figure from Jorge dos Santos & Lodwick)

( )p x

Utilizing alpha-levels to obtain fuzzy polynomials, we have:

( ) ( ), ( ) : ( ), d i ip x p x p x z R z p x d z

2z

1z

1z

2z

1x

2xx

( )p x

( )p x

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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

2-D Example (from Jorge dos Santos & Lodwick)

-50 0 50 100 150 200-10

0

10

20

30

40

50

60

x

z

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Fuzzy Curves (figures from Jorge dos Santos & Lodwick)

Spline linear

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

-50

0

50P. L

agrange

10 15 20 25 30

5

10

15

20

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

0

20

40

60

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Fuzzy Curves (figures from Jorge dos Santos & Lodwick)

Cubic S

pline

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

0

10

20

30

40

50

x

z

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

0

10

20

30

40

50

x

z

Consistent

Cubic S

pline

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Details of the Consistent Fuzzy Cubic Spline (figures from Jorge dos Santos & Lodwick)

x

z

155 160 165 170

27

28

29

30

31

32

33

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3-D Example (from Jorge dos Santos & Lodwick)

-50 0 50 100 150 200 2500

20

40

60

80

100

120

140

160

180

200

x

y

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Another Representation/View of the Fuzzy Points(figure from Jorge dos Santos & Lodwick)

-50

0

50

100

150

200 0

50

100

150

200

5

10

15

20

25

30

35

z

yx

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Fuzzy Surface via Triangulation (figure from Jorge dos Santos & Lodwick)

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Fuzzy Surfaces via Linear Splines (figure from Jorge dos Santos & Lodwick)

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Fuzzy Surfaces via Cubic Splines (figure from Jorge dos Santos & Lodwick)