fuzzy sets and rough sets - nanjing university · 2016. 1. 9. · fuzzy sets and rough sets !...
TRANSCRIPT
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
“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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Lotfi Zadeh
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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) �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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’�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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) ��
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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. �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 $almost all human reasoning is imprecise�
n fuzzy sets, approximate reasoning, and fuzzy logic issues and applications�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzziness is not probability
• Probability is used, for example, in weather forecasting �
• Probability is a number between 0 and 1 that is the
certainty that an event will occur �
• The event occurrence is usually 0 or 1 in crisp logic, but fuzziness says that it happens to some degree�
• Fuzziness is more than probability; probability is a subset
of fuzziness�
• Probability is only valid for future/unknown events�
• Fuzzy set membership continues after the event �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Probability
• Probability is based on a closed world model in which it is�$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��
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzzy Sets
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzzy Sets Are Membership Functions
from Bezdek
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Representations of Membership Functions
⎭⎬⎫
⎩⎨⎧ ++++=
⎭⎬⎫
⎩⎨⎧ ++=
900
805.
701
605.
500
15.21
95.150.
75.10
Warm
TAMPBP
( ) ( ) 50/80_
2−−= pPRICEFAIR epµ
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Two Types of Fuzzy Membership Function
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
$elements have identical membership values: �$ ${.1/A,.6/B,.8C} = {.1/A,.6/B,.8/C}�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzzy Union
• In traditional logic, all elements in either (or both) set(s) �$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. . .
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzzy Relations and Operators
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Summary: FUZZY SETS Membership function and Fuzzy set operations
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Tom is rather tall, but Judy is short. �
�
If you are tall, than you are quite likely heavy. �
Examples on fuzzy concepts
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
• 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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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=
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
If a fuzzy concept can be rigidly described by Cantor’s
notion of sets or the bivalent (true/false or two-valued)…
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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’。
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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。
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Crisp sets VS Fuzzy Sets
C={Lines longer than 4cm} C={Long lines}
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
For contiguous data:
C={MEN OLDER THAN 50 YEARS OLD} C={OLD MEN}
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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.
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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) �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Fuzzy relations and operations RealOons:Equality and Containment
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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}�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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. �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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. . .
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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. . .
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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)
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Examples: IntersecOon, union, complement
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 =
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
• 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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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.
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Basic Concepts
• Knowledge�
• Indiscernibility Relation �
• lower and upper approximations�
1. preliminary�
2. secondary�
• Reduct �
• Indiscernibility Matrix �
• Attributes Significance�
�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Basic Concepts
PART I: preliminary
knowledge
approximate space: K=(U,R)
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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}. �
Basic Concepts PART I: preliminary�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
Basic Concepts PART I: preliminary�
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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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) �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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}
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
A.Skowron: Indiscernibility Matrix
Basic Concepts PART II: secondary
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
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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
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 �
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
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 �
…�
�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
• Good at…�
discrete values�
uncertainty �
Advantages and Disadvantages
Disadvantages: �
discrete values�sensitive�
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
• Models�
• Data�• Algorithms�
• Application �
Trends and Challenges
INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015
Some cases
• Classifications�