Fuzzy Sets and Rough Sets - Nanjing University · 2016. 1. 9. · Fuzzy Sets and Rough Sets !...

INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015

Fuzzy Sets and Rough Sets

n  Introduction �n  History and definition �

n  Fuzzy Sets �n  Membership function �n  Fuzzy set operations�

n  Rough Sets�n  Approximation �n  Reduction �


“Fuzzifica(on is a kind of scien(fic permisiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard work.”

R. E. Kalman, 1972


Set �

Fuzzy Set�

Rough Set�

(collections) of various objects of interest �

“Number of things of the same kind, that belong together because they are similar or complementary to each other.”�The Oxford English Dictionary �

Set Theory: George Cantor (1893) �

an element can belong to a set to a degree k (0 ≤ k ≤ 1) �

completely new, elegant approach to vagueness�

Fuzzy Set theory: Lotfi Zadeh(1965) �

imprecision is expressed by a boundary region of a set�

another approach to vagueness �

Rough Set Theory: Zdzisaw Pawlak(1982) �

Introduction


Lotfi Zadeh


Introduction

n  Early computer science�•  Not good at solving real problems�•  The computer was unable to make accurate inferences�•  Could not tell what would happen, give some

preconditions�•  Computer always seemed to need more information �


Lotfi Zadeh

n  “Fuzzy Sets” paper published in 1965 �n  Comprehensive - contains everything needed to implement

FL �n  Key concept is that of membership values: �extent to which an object meets vague or imprecise properties�n  Membership function: membership values over domain of

interest�n  Fuzzy set operations�n  Awarded the IEEE Medal of Honor in 1995 �


History for fuzzy sets and system

n  First fuzzy control system, work done in 1973 with Assilian (1975) �

n  Developed for boiler-engine steam plant�

n  24 fuzzy rules�

n  Developed in a few days�

n  Laboratory-based �

n  Served as proof-of-concept�


Early European Researchers

Hans Zimmerman, Univ. of Aachen �• Founded first European FL working group in 1975 �• First Editor of Fuzzy Sets and Systems�• First President of Int’l. Fuzzy Systems Association ��Didier Dubois and Henri Prade in France�• Charter members of European working group �• Developed families of operators�• Co-authored a textbook (1980) �


Early U. S. Researchers

K. S. Fu (Purdue) and Azriel Rosenfeld (U. Md.) (1965-75) ��Enrique Ruspini at SRI�• Theoretical FL foundations�• Developed fuzzy clustering ��James Bezdek, Univ. of West Florida�• Developed fuzzy pattern recognition algorithms�• Proved fuzzy c-means clustering algorithm�• Combined fuzzy logic and neural networks�• Chaired 1st Fuzz/IEEE Conf. in 1992 and others�• President of IEEE NNC 1997-1999 �


The Dark Age

• Lasted most of 1980s�

• Funding dried up, in US especially�

$“...Fuzzy logic is based on fuzzy thinking. It fails to distinguish　between the issues specifically addressed by the traditional "methods of logic, definition and statistical decision-making...” �

" " " "- J. Konieki (1991) in AI Expert��• Symbolics ruled: “fuzzy” label amounted to the ‘kiss of death’�


Michio Sugeno

• Secretary of Terano’s FL working group, est. in 1972 �

• 1974 Ph.D. dissertation: fuzzy measures theory�

• Worked in UK�

• First commercial application of FL in Japan: control system for water purification plant (1983) ��


Other Japanese Developements

n  1st consumer product: shower head using FL circuitry to control temperature (1987) �

n  Fuzzy control system for Sendai subway (1987) �

n  2d annual IFSA conference in Tokyo was turning point for FL (1987) �

n  Laboratory for Int’l. Fuzzy Engineering Research (LIFE) founded in Yokohama with Terano as Director, Sugeno as Leading Advisor in 1989. �


Fuzzy Systems Theory and Paradigms

n  Variation on 2-valued logic that makes analysis and control of real (non-linear) systems possible�

n  Crisp “first order” logic is insufficient for many applications because $almost all human reasoning is imprecise�

n  fuzzy sets, approximate reasoning, and fuzzy logic issues and applications�


Fuzziness is not probability

•  Probability is used, for example, in weather forecasting �

•  Probability is a number between 0 and 1 that is the

certainty that an　event will occur �

•  The event occurrence is usually 0 or 1 in crisp logic, but fuzziness　says that it happens to some degree�

•  Fuzziness is more than probability; probability is a subset

of fuzziness�

•  Probability is only valid for future/unknown events�

•  Fuzzy set membership continues after the event �


Probability

•  Probability is based on a closed world model in which it is�$assumed that everything is known �

�•  Probability is based on frequency; Bayesian on subjectivity�

• 　Probability requires independence of variables�

• 　In probability, absence of a fact implies knowledge��


Fuzzy Sets


Fuzzy Set Membership

• In fuzzy logic, set membership occurs by degree�• Set membership values are between 0 and 1 �• We can now reason by degree, and apply logical operations to fuzzy sets��We usually write �

or, the membership value of x in the fuzzy set A is m, where

. mxA =)(µ

10 ≤≤ m


Fuzzy Set Membership Functions

•  Fuzzy sets have “shapes”: the membership values plotted versus�$the variable�

�•  Fuzzy membership function: the shape of the fuzzy set over the�

$range of the numeric variable�$Can be any shape, including arbitrary or irregular �$Is normalized to values between 0 and 1 �$Often uses triangular approximations to save computation time�


Fuzzy Sets Are Membership Functions

from Bezdek


Representations of Membership Functions

⎭⎬⎫

⎩⎨⎧ ++++=

⎭⎬⎫

⎩⎨⎧ ++=

900

805.

701

605.

500

15.21

95.150.

75.10

Warm

TAMPBP

( ) ( ) 50/80_

2−−= pPRICEFAIR epµ


Two Types of Fuzzy Membership Function


Equality of Fuzzy Sets

•  In traditional logic, sets containing the same members are equal: �${A,B,C} = {A,B,C}�

�•  In fuzzy logic, however, two sets are equal if and only if all�

$elements have identical membership values: �$ ${.1/A,.6/B,.8C} = {.1/A,.6/B,.8/C}�


Fuzzy Union

•  In traditional logic, all elements in either (or both) set(s) �$are included �

�•  In fuzzy logic, union is the maximum set membership value�

$�( ) ( ) ( )If m x and m x then m xA B A B= = =∪07 09 09. . .


Fuzzy Relations and Operators


Summary: FUZZY SETS Membership function and Fuzzy set operations


Tom is rather tall, but Judy is short. �

�

If you are tall, than you are quite likely heavy. �

Examples on fuzzy concepts


•  The description of a human characteristic such as

healthy. �

•  The classification of patients as depressed. �

•  The classification of certain objects as large. �

•  The classification of people by age such as old. �

•  A rule for driving such as “if an obstacle is close, then

brake immediately”. �

Examples on fuzzy concepts


Concept and set

intension (内涵)：attributes of the object�

concept �

　　　　extension (外延)：all of the objects defined by

the concept（set）

G. Cantor (1887)

{ | ( )}A a P a=


If a fuzzy concept can be rigidly described by Cantor’s

notion of sets or the bivalent (true/false or two-valued)…


模糊概念能否用Cantor集合来刻画？秃头悖论⼀一位已经谢顶的⽼老教授与他的学⽣生争论他是否为秃头问题。教授：我是秃头吗？学⽣生：您的头顶上已经没有多少头发，确实应该说是。教授：你秀发稠密，绝对不算秃头，问你，如果你头上脱落了⼀一根头发之后，

能说变成了秃头了吗？学⽣生：我减少⼀一根头发之后，当然不会变成秃头。教授：好了，总结我们的讨论，得出下⾯面的命题：‘如果⼀一个⼈人不是秃头，那

么他减少⼀一根头发仍不是秃头’，你说对吗？学⽣生：对！教授：我年轻时代也和你⼀一样⼀一头秀发，当时没有⼈人说我秃头，后来随着年龄

的增⾼高，头发⼀一根根减少到今天的样⼦子。但每掉⼀一根头发，根据我们刚才的命题，我都不应该称为秃头，这样经有限次头发的减少，⽤用这⼀一命题有限次，结论是：‘我今天仍不是秃头’。


Postulate: If a man with n (a nature number) hairs is

baldheaded, then so is a man with n+1 hairs.

Baldhead Paradox：Every man is baldheaded.

Cause： due to the use of bivalent logic for inference,

whereas in fact, bivalent logic does not apply in this case。


Fuzzy Sets：Membership Functions

from Bezdek

Fuzzy membership function: the shape of the fuzzy set over the range of the numeric variable�

$> Can be any shape, including arbitrary or irregular �$> Is normalized to values between 0 and 1 �$> Often uses triangular approximations to save

computation time�


Crisp sets VS Fuzzy Sets

C={Lines longer than 4cm} C={Long lines}


For contiguous data:

C={MEN OLDER THAN 50 YEARS OLD} C={OLD MEN}


Example of Fuzzification Assume inside temperature is 67.5 F, change in temperature last five minutes is -1.6 F, and outdoor temperature is 52 F.

Now find fuzzy values needed for our four example rules:

For InTemp,

0.0)5.67( and ,75.0)5.67(,25.0)5.67( _ === warmtooecomfortablcool µµµ

. For DeltaInTemp,

0.0)6.1( and ,2.0)6.1( ,8.0)6.1( _arg__ =−=−=− positiveelzeronearnegativesmall µµµ

For OutTemp, 9.0)52( =chillyµ.


Fuzzy …… Crisp

• Fuzzy logic comprises fuzzy sets and approximate reasoning �

•  A fuzzy “fact” is any assertion or piece of information, and can have a “degree of truth”, usually a value between 0 and 1 �

�•  Fuzziness: “A type of imprecision which is associated

with ... Classes in which there is no sharp transition from membership to non-membership” - Zadeh (1970) �


Fuzziness ……probability

•  Probability is used, for example, in weather forecasting �•  Probability is a number between 0 and 1 that is the　certainty that an event will occur �•  The event occurrence is usually 0 or 1 in crisp logic, but fuzziness says that it happens to some degree�•  Fuzziness is more than probability; probability is a subset of fuzziness�• 　Probability is only valid for future/unknown events�•  Fuzzy set membership continues after the event �


Fuzzy relations and operations RealOons：Equality and Containment


Equality of Fuzzy Sets

* In traditional logic, sets containing the same members are equal: �${A,B,C} = {A,B,C}�

�* In fuzzy logic, however, two sets are equal if and only if all�

$elements have identical membership values: �$ ${.1/A,.6/B,.8C} = {.1/A,.6/B,.8/C}�


Fuzzy Containment

•  In traditional logic, � A B⊂

if and only if all elements in A are also in B. ��•  In fuzzy logic, containment means that the membership values�

$for each element in a subset is less than or equal to the�$membership value of the corresponding element in the�$superset. �

�•  Adding a hedge can create a subset or superset. �


Fuzzy Intersection

* In standard logic, the intersection of two sets contains those elements in both sets.

* In fuzzy logic, the weakest element determines the degree

of membership in the intersection

( ) ( ) ( )If m x and m x then m xA B A B= = ≡∩05 03 03. . .


Fuzzy Union

•  In traditional logic, all elements in either (or both) set(s)　are included �

�* 　In fuzzy logic, union is the maximum set membership value�

$�

( ) ( ) ( )If m x and m x then m xA B A B= = =∪07 09 09. . .


Fuzzy Complement

•  In tradiOonal logic, the complement of a set is all of the elements not in the set.

•  In fuzzy logic, the value of the complement of a

membership is (1 -‐ membership_value)


Examples: IntersecOon, union, complement


U={u1,u2,u3,u4,u5} �A=0.2/u1+0.7/u2+1/u3+0.5/u5 �B=0.5/u1+0.3/u2+0.1/u4+0.7/u5�

A …?... B =

B =


Rough Sets Rough Sets: Background •  vagueness

•  boundary region approach(Gottlob Frege )

•  existing of objects which cannot be uniquely classified to the set or its complement

•  another approach to vagueness

• imprecision in the approach is expressed by a boundary region of a set

•  defined quite generally by means of topological operations, interior and closure, called approximations

lower approximaOon upper approximaOon


•  Human knowledge about a domain is expressed by classification �•  Rough set theory treats knowledge as an ability to classify perceived objects into categories�•  Objects belonging to the same category are considered to be indistinguishable to each other. �•  The primary notions of rough set theory are the approximation space: lower and upper approximations of an object set�•  The lower approximation of an object set (S) is a set of objects surely belonging to S, while its upper approximation is a set of objects surely or possibly belonging to it �•  An object set defined through its lower and upper approximations is called a rough set��

Rough Sets: Introduction


Introduc*on

• Research on rough set theory and applications in China began in the middle 1990s.

• Chinese researchers achieved many significant results on rough set theory and applications.

• both the quality and quantity of Chinese research papers are growing very quickly

• many topics being investigated by Chinese researchers: fundamental of rough set, knowledge acquisition, granular computing based on rough set,extended rough set models, rough logic, applications of rough set, et al.


Basic Concepts

•  Knowledge�

•  Indiscernibility Relation �

•  lower and upper approximations�

1. preliminary�

2. secondary�

•  Reduct �

•  Indiscernibility Matrix �

•  Attributes Significance�

�


Basic Concepts

PART I: preliminary

knowledge

approximate space: K=(U,R)


Basic Concepts

PART I: preliminary�

patients�

�

Patient��

Headache��

Muscle-pain ��

Temperature��

Flu ��p1 �

�yes��

yes��

normal ��

no ��p2 �

�yes��

yes��

high ��

yes��p3 �

�yes��

yes��

very high��

yes��p4 �

�no ��

yes��

normal ��

no ��p5 �

�no ��

no ��

high ��

no ��p6 �

�no ��

yes��

very high��

yes��

IS（Information System/Tables）

Attributes� Decision Attribute�

Condition Attribute�


Basic concepts of rough set theory : �•  lower approximation of a set X with respect to R : � is the set of all objects, which can be for certain classified as X with respect to R (are certainly X with respect to R). �•  upper approximation of a set X with respect to R: � is the set of all objects which can be possibly classified as X with respect to R (are possibly X in view of R). �•  boundary region of a set X with respect to R : � is the set of all objects, which can be classified neither as X nor as not-X with respect to R. ��

Basic Concepts PART I: preliminary�


X1 = {u | Flu(u) = yes}�

= {u2, u3, u6, u7}� RX1 = {u2, u3} � = {u2, u3, u6, u7, u8, u5}�

X2 = {u | Flu(u) = no}�

= {u1, u4, u5, u8}�

RX2 = {u1, u4}� = {u1, u4, u5, u8, u7, u6}�X1R X2R

U Headache Temp. Flu U1 Yes Normal No U2 Yes High Yes U3 Yes Very-high Yes U4 No Normal No U5 NNNooo HHHiiiggghhh NNNooo U6 No Very-high Yes U7 NNNooo HHHiiiggghhh YYYeeesss U8 No Very-high No

The indiscernibility classes defined by R = {Headache, Temp.} are {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}. �



RX1 = {u2, u3} � = {u2, u3, u6, u7, u8, u5}�

Lower & Upper Approximations (4) � R = {Headache, Temp.}�U/R = { {u1}, {u2}, {u3}, {u4}, {u5, u7}, {u6, u8}}��X1 = {u | Flu(u) = yes} = {u2,u3,u6,u7}�X2 = {u | Flu(u) = no} = {u1,u4,u5,u8}�

RX2 = {u1, u4}�

= {u1, u4, u5, u8, u7, u6}�

X1R

X2R

u1 �

u4 �u3 �

X1 � X2 �

u5 �u7 �u2 �

u6 � u8 �


INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Dept. of Computer Science and Technology, Nanjing University

| ( ) |( )| ( ) |BB XXB X

α −−

=•  accuracy of approximation: �

Basic Concepts

PART I: preliminary�

where |X| denotes the cardinality of�

Obviously �

If X is crisp with respect to B. �

If X is rough with respect to B. �

.φ≠X.10 ≤≤ Bα

,1)( =XBα,1)( <XBα


Basic Concepts

PART II: secondary�

B AÃ( ) ( )IND B IND A=

is a reduct of information system if �

and no proper subset of B has this property �

Reduct �Patient�

�Headache�

�Muscle-pain �

�Temperature�

�Flu ��p1 �

�yes��

yes��

normal ��

no ��p2 �

�yes��

yes��

high ��

yes��p3 �

�yes��

yes��

very high��

yes��p4 �

�no ��

yes��

normal ��

no ��p5 �

�no ��

no ��

high ��

no ��p6 �

�no ��

yes��

very high��

yes��

Reducts: {Headache, Temperature}�

or {Muscle-pain, Temperature} �

Core: CORE(P)=∩RED(P) �


Attributes Reduct

，if ，then S is the Reduct of D。

其中， , X∈U/D S P⊂ S PPOS (D)=POS (D)

PPOS (D) P_(X)=U

Basic Concepts PART II: secondary

Patient

Headache

Muscle-pain

Temperature

Flu p1

yes

yes

normal

no p2

yes

yes

high

yes p3

yes

yes

very high

yes p4

no

yes

normal

no p5

no

no

high

no p6

no

yes

very high

yes

Positive region

POS{M ，T}={p1,p2,p3,p4,p5,p6}


Patient Headache Temperature Flu

p1 no high yes p2 yes high yes p3 yes very high yes p4 no normal no p5 yes high no p6 no very high yes

Patient��

Headache��

Muscle-pain ��

Temperature��

Flu ��p1 �

�yes��

yes��

normal ��

no ��p2 �

�yes��

yes��

high ��

yes��p3 �

�yes��

yes��

very high��

yes��p4 �

�no ��

yes��

normal ��

no ��p5 �

�no ��

no ��

high ��

no ��p6 �

�no ��

yes��

very high��

yes��

Patient

Muscle-pain

Temperature

Flu

p1 yes high yes p2 no high yes p3 yes very high yes p4 yes normal no p5 no high no p6 yes very high yes

Basic Concepts PART II: secondary�


A.Skowron: Indiscernibility Matrix


p1 p2 p3 p4 p5 p6 p1 % T T H T H,T

p2 % p3 % H,T H,M,T ％

p4 % ％ T

p5 % M,T

p6 %

M(S)=[cij]n×n， cij={a∈A：a(xi)≠a(xj)，i,j=1,2,…,n}

Patient

Headache

Muscle-pain

Temperature

Flu p1

yes

yes

normal

no p2

yes

yes

high

yes p3

yes

yes

very high

yes p4

no

yes

normal

no p5

no

no

high

no p6

no

yes

very high

yes



Patient

Headache

Muscle-pain

Temperature

Flu p1

yes

yes

normal

no p2

yes

yes

high

yes p3

yes

yes

very high

yes p4

no

yes

normal

no p5

no

no

high

no p6

no

yes

very high

yes

headache muscle-pain temperature� flu �

Which is more important? �

(C,D)

( γ (C,D)-γ (C- {a} ,D) ) γ (C- {a} ,D)σ (a) = =1-

γ (C,D) γ (C,D)

Definition: σ (Headache) = 0, �

σ (Muscle-pain) = 0, �

σ (Temperature) = 0.75 �


Theory and Applications

Theory� in the view of algebra�

in the view of information theory �

in the view of logic �

Applications�

medical data analysis�

finance�

voice recognition �

image processing �

…�

�


•  Good at…�

discrete values�

uncertainty �

Advantages and Disadvantages

Disadvantages: �

discrete values�sensitive�


•  Models�

•  Data�•  Algorithms�

•  Application �

Trends and Challenges


Some cases

•  Classifications�

Fuzzy Sets and Rough Sets - Nanjing University · 2016. 1. 9. · Fuzzy Sets and Rough Sets !...

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Transcript of Fuzzy Sets and Rough Sets - Nanjing University · 2016. 1. 9. · Fuzzy Sets and Rough Sets !...