2007-7-14 1

Granular Computing and Its Potential Application

Granular Computing and Its Potential Application

Zeng Yizengyi@emails.bjut.edu.cn

SupervisorYiyu Yao

yyao@cs.uregina.ca

International WIC InstituteBeijing University of Technology

Only for Internal Study at WIC Institute

2007-7-14 2

Agenda

GrC Overview• Granularity and Information Granulation• Basic Issues of Granular Computing• Construction of Granules• Semantic Issue: GrC Models • Computing Tasks of GrC• Granular Logic and GrC-based Reasoning (Next Time)

Potential Applications• Data Mining and KDD• Automatic Animation Generation• Retrieval Support System• Social Network• Cognitive Informatics (Next Time)

• Multi-Agent and Distributed Reasoning (Next Time)

2007-7-14 3

Granularity and Information GranulationGranularity• 1979 Zadeh, L.A., Fuzzy sets and information granularity, Advances

in Fuzzy Set Theory and Applications,North-Holland .• 1985 J. Hobbs, Granularity, in IJCAI 85, pp. 432--435.• 1990 B.Zhang, L.Zhang, Problem Solving Theory and Its Application.

Some description on Granularity.Before 1997, Partition based granularity[1].

Information Granulation• Toward a theory of fuzzy information granulation and its

centrality in human reasoning and fuzzy logic, 1997.Covering based Information Granulation[1].In Fuzzy, vague perspective.

2007-7-14 4

Review of Partition and Covering

Partition{ {a,b},{c,d},{e,f} }

“By 'partition' we mean that every element of A is in exactly one little box as being a single object, instead of thinking of it as a plurality of objects.”

Covering{ {a,b},{a,c,d},{a,b,e,f} }

By 'covering' we mean that every element of A is not necessarily in one little box as being a single object.

From Elements of Set Theory, Herbert B. Enderton, 1977

A = { a, b, c, d, e, f }

2007-7-14 5

The Birth of Granular Computing1996 UC-BerkeleyProf. Zadeh Granular MathematicsProf. T.Y.Lin Granular Computing

“GrC is a superset of the theory of fuzzy information granulation, rough set theory and interval computations, and is a subset of granular mathematics.”

------ Lotfi Zadeh's Announcement.

“the notion of information granulation has not been fully explored in its own right. We hope the GrC Special Interest Group can explore, organize and unify these divergent concepts, theories, and applications into a well formulated theory of granular computing.”

------ T.Y. Lin's Announcement.

From http://www2.cs.uregina.ca/~yyao/GrC/

2007-7-14 6

Basic issues of Granular Computing

Construction of Granules• Formation• Representation• Interpretation

(Closeness, Dependency,Association)Computation with Granules• Approximation• Reasoning• Inference

From Y.Y.Yao, Granular Computing: basic issues and possible solutions, 2000

2007-7-14 7

Construction of GranulesConstructions of Granules by• Equality relations• Equivalence relations• Reflexive binary relations

).,(})(|{),(

vamvxIUxvaG ae

==∈=

).,(),(),(),(

)).,(),((})(|{)),(),(( )(

beae

ba

ba

bbaabae

vbGvaGvbmvam

vbvamIvxIUxvbvaG vx

∩=∩=∧=

∧=∈=∧ =

From Y.Y.Yao, Granular computing using information tables,2002

Granules induced by equality of attribute values

Sample construction of Granules induced by equality of attribute values

Sample construction of Granules induced by equality of attribute values

[3]With respect to attribute “Sky”

With respect to attribute “Water”

}.|),({}{ ae VvvaGa ∈≠= φπ

})}.{),((}),,,{),({()( 3421 NoRainnySkyGNoNoNosunnySkyGSky ee ===π

No Sky AirTemp Humidity Wind Water Forecast EnjoySport1 Sunny Warm Normal Strong Warm Same Yes2 Sunny Warm High Strong Warm Same Yes3 Rainy Cold High Strong Warm Change No4 Sunny Warm High Strong Cool Change Yes

Table Sample from Tom M. Mitchell, Machine Learning, 1997

})}.{),((}),,,{),({()( 4321 NoCoolWaterGNoNoNoWarmWaterGWater ee ===π

}.,,{},,{),(),(),(),()),(),((

})(|{)),(),((

321421

)(

NoNoNoNoNoNoWarmWaterGSunnySkyGWarmWatermSunnySkymWarmWaterSunnySkym

WarmISunnyxIUxWarmWaterSunnySkyG

ee

Waterskye x

∩=∩=∩=∧=

=∧=∈=∧

Granules induced by equivalence of attribute values

• Sample construction of Granules induced by equivalence of attribute values

}.','|)',({}][','|)',({

}][)(|{})(|{),(

vEvVvvamvvVvvam

vxIUxvExIUxvaG

aaEa

EaaaE

a

a

∈∪=∈∈∪=

=∈=∈=

No Sky AirTemp EnjoySport1 Sunny Warm Yes2 Sunny Warm Yes3 Rainy Cold No4 Sunny Warm Yes

( , )EG Sky v

No Sky-and-AirTemp

EnjoySport

1 Good Yes2 Good Yes3 Bad No4 Good Yes

Original Information Table

Quotient Information Table

[3]

).][,'(}][)]([|{

)][,'(

a

aa

a

EEEa

EE

vamvxIUx

vaG

==∈=

[3]

∽ ( , )EG AirTemp v

{ ( , ), ( , ),( , ), ( , )}

( ){ ( , ),( , ).

e e

e e

e

e

G Sky sunny G Sky RainnyG AirTemp Warm G AirTemp Cold

Sky and AirTempG Sky and AirTemp Good

G Sky and AirTemp Bad

Π − −= − −

− −Table Sample from Tom M. Mitchell, Machine Learning, 1997

Granules induced by similarity of attribute values

( ) { ' | ' , ' }.pa aaR v v v V v R v= ∈

aR aVDefine is a binary relation on .

is -related to ; is a binary (reflexive) relation on . 'v vaR aR aV(Predecessor Neighborhood)

( , ) { | ( ) }{ | ( ) ( )}

{ ( , ') | ' , ' ( )}.

s a ap

a ap

a a

G a v x U I x R vx U I x R v

m a v v V v R v

= ∈= ∈ ∈= ∪ ∈ ∈

Equality and Equivalence Relations : Granules belongs to one equivalence class. [3]

Reflexive Relation: May belong to more than one granules. [3]

[3]

Article Number

Name SportType Article_Type

1 Yao.Ming Basket Ball Sports**2 Yao Ming Basket Ball Sports, Life3 Jordan Basket Ball Life

**4 Ronaldinho FootBall Sports, Life

If X is small then Y is smallIf X is medium then Y is largeIf X is large then Y is small

Fuzzy Sets’ PerspectiveRelationship between granules: fuzzy graphs or fuzzy if-then rules.

If X is A1 then Y is B1If X is A2 then Y is B2……If X is An then Y is Bn

Sample from L.A. Zadeh, From Computing with Numbers to Computing with Words – From Manipulation of Measurements to Manipulation of Perceptions, IEEE Trans. on Circuits and Systems 45,1, pp.105-119, Jan. 1999.

Semantic Issue: GrC Models

Y

XAi

Bi

Equality, possibility, fuzzy, veristic constraints

Fuzzy Set definition of a granule R}. X|{X G isr =

Y

X

Fuzzy Sets’ PerspectiveComputing with Words

Y

XAi

Bi

ααα

α αμ

∑

∑

×=

≥=

×=

×+×+×=

iii

f

iii

BAfvuvuf

BAff

}),(|),{(

smalllargelargemediumsmallsmall

“words serve as values of variables and play the role of fuzzy constraints.”

What is maximum value of ?Fuzzy Graph is

Using to compute the max value.∑ ×

iii BAf

f

cuts−α

[4]

R}. X|{X G isr =

Mike is a man.He does not like fish.John probably didn’t hurt by the rock.

Look at article [4] for more samples

Rough Sets’ Perspective

“The theory of Rough Sets deals mainly with the approximationaspect of information granulation.”[2]

.][)(][

EXx

xXaprE⊆∪=

[5]

.][)(][

EXx

xXaprE φ≠∩∪=

).()( XaprXXapr ⊆⊆ This picture is extract from J.W.Han, Data Mining: Concepts and Techniques,2001.

[5]

Data Analysis, attribute reduction, dependency analysis, decision rules learning, and Mining Information Table [3]

Set-theoretic ModelAny Algebra, Its power algebra is given by Power operation may carry some properties of .

).,...,,( 21 kfffU ).,...,,2( 21++ +

kU fff

+f fInterval number algebra

Define interval numbers:

Interval set algebra}.|{],[ axaxaa ≤≤= },|2{],[ 2121 AXAXAA U ⊆⊆∈==Α

].,[0],/1,/1[],[

},|/{/)].,,,max(),,,,[min(

},|{],,[

},|{],,[

},|{].,[B ],[

bbbbaa

ByAxyxBAbabababababababa

ByAxyxBAbaba

ByAxyxBAbaba

ByAxyxBAbbandaaA

∉•=

∈∈==

∈∈•=•−−=

∈∈−=−++=

∈∈+=+==

].,[},|{\

],,[},|{

],,[},|{

].,[B ],[

1221

2211

2211

2121

BABABYAXYXBA

BABABYAXYXBA

BABABYAXYXBA

BBandAAA

−−=∈∈−=

∪∪=∈∈∪=∪

∩∩=∈∈∩=∩

==

From Y.Y.Yao, Granular Computing: basic issues and possible solutions, 2000

Set-theoretic Model

Sample for Interval set algebraA: “International WIC Institute, Beijing University of Technology”A = [ A1, A2, A3, A4, A5, A6, A7, A8 ]B: “Beijing University of Technology ,International WIC Institute”B = [ B1, B2, B3, B4, B5, B6, B7, B8 ]Description of Full ordered relations.Question: What are the n-ary subsets’ computing tasks.

Another method to describe the ordered information: Fuzzy Sets’perspective and its Fuzzy Constraints.

A1 is a predecessor of A2.A3 is a successor of A2.

Computing Tasks of GrCRecommended reading on the Computing Tasks

• Yao, Y.Y. and Zhong, N. Granular computing using information tables, 2002.

• Yao, Y.Y., Information granulation and rough set approximation, 2001.

• Yao, J.T. and Yao, Y.Y., Induction of Classification Rules by Granular Computing , 2002.

• Yao, Y.Y., Granular computing using neighborhood systems, 1999.

Neighborhood systemsU: non empty universe; V: data set; Binary Relation

N(p) is a sub set of U.

NS: {N(p)} is a neighborhood system. [6]

B V U⊆ ×, ( ) { : }.p V N p u uBp∀ ∈ =

Sample of Neighborhood systems

Ball Color

1 Red

2 Red

3 Red

4 Saffron Yellow

5 Saffron Yellow

6 Yellow

7 Yellow

8 Yellow

9 Yellow

NS1={1,2,3,4,5}

NS2={1,2,3,4,5}

NS3={1,2,3,4,5}

NS4={1,2,3,4,5,6,7,8,9}

NS5={1,2,3,4,5,6,7,8,9}

NS6={4,5,6,7,8,9}

NS7={4,5,6,7,8,9}

NS8={4,5,6,7,8,9}

NS9={4,5,6,7,8,9}

Sample table from Qing Liu, Rough Sets and Rough Reasoning,2001.

Granular Computing on Binary Number[Red] = {u1,u3,u8,u9,u12}

[Yellow] = {u2,u7,u10}

[Blue] = {u4,u6}

[Black] = {u11}

[White] = {u5}

[B100] = {u1,u5,u6}

[B200] = {u2,u8,u12}

[B300] = {u3,u9,u11}

[B400] = {u4}

[B500] = {u7,u10}

U Color Type Priceu1 Red B100 Mediumu2 Yellow B200 Expensiveu3 Red B300 Expensiveu4 Blue B400 Mediumu5 White B100 Cheapu6 Blue B100 Expensiveu7 Yellow B500 Mediumu8 Red B200 Expensiveu9 Red B300 Expensive

u10 Yellow B500 Mediumu11 Black B300 Mediumu12 Red B200 Expensive

U/IND(Color) = {[Red], [Yellow], [Blue], [Black], [White]};U/IND(Type) = {[B100], [B200], [B300], [B400], [B500]};Sample table from Qing Liu, Rough Sets and Rough Reasoning,2001.

Association Rules using GrC

Combination Binary“AND”computing

Result Total number of 1

[Yellow]AND[Expensive] 010000100100 AND011001011001

010000000000 1

[Blue]AND[Expensive] 000101000000 AND011001011001

000001000000 1

[White]AND[Expensive] 000010000000 AND011001011001

000000000000 0

[Black]AND[Expensive] 000000000010 AND011001011001

000000000000 0

[Red]AND[Expensive] 101000011001 AND011001011001

001000011001 4

Rule [Red]AND[Expensive] = 4 (support = 12*10%)

Sample table from Qing Liu, Rough Sets and Rough Reasoning,2001.

2007-7-14 20

GrC-based Reasoning

For Next Time• In Fuzzy Sets Perspective• In Rough Sets Perspective• Decision Logic Language• Multi-Agent and Distributed

Reasoning

Decision Rule Granule:Decision Algorithm Granule:Program Granule:

);)(,( Granule ; ),( ϕϕ mva );))(,(),(,(( φφϕϕ mm

}g,...,{gG ;))}(,{())(,( k11 =∪= = iiki gmgGmG

}S,...,{SG (S);))(,( k1=∪= ∈ mGmG GS [6]

Potential Applications of GrC• Data Mining and KDD• Automatic Animation Generation• Retrieval Support System• Social Network• Cognitive Informatics (Next Time)• Multi-Agent and Distributed Reasoning (Next Time)

2007-7-14 22

GrC for Data Mining

Classification[1] Yao, J.T. and Yao, Y.Y., Induction of Classification Rules by

Granular Computing , 2002. [2] Yao, Y.Y., and Yao, J.T., Granular computing as a basis for consistent

classification problems, 2002. Clustering[3] Witold Pedrycz, Granular Modeling: The Synergy of Granular

Computing and Fuzzy Logic, 2004.Association Rule Mining[4] Qing Liu, Rough Sets and Rough Reasoning, Science Press, 2001.

GrC for Automatic Animation GenerationDeleted for confidential reason

GrC for IRSS

Term Space Granulation

Document Space Granulation

Retrieval Result Granulation

User Space Granulation

Retrieval Result

User

Information Retrieval System

Information Retrieval Support System

…

GrC for IRSS

2007-7-14 26

GrC for IRSS

Picture 1. Traditional Retrieval SystemExtract from B.Y.Ricardo, R.N.Berthier, et al. Modern Information Retrieval. the ACM press,1999

Picture 2. IRSS using GrC

Mathematical Foundation for IRSS Space Granulation

The measure of a single granule [7]

Confidence or absolute support of provided by ：[7]

The strength of the inference (The Conditional Entropy)：[7]

The strength of the inference ：[7]

Consistent classification problems (using logically implication)[7]

φ

UmG )()( φφ =

)()()()()()( φψφφψφψφ mmmmmAS ∩=∧=⇒

)|(log)|()|(1

φψφψφ iPiPHn

i∑=

−=Ψ

=Ψ=Ψ ∑=

)|()()|(1

jHjPHm

jφφφ )|(log)(

1 1jiPjiP

m

j

n

iφψφψ ∧−∑∑

= =

ciclass =⇒φ

ψ

ψφ ⇒

ψ⇒Φ

GrC for Social Network

Deleted for confidential reason

2007-7-14 29

For Next Time• GrC-based Reasoning• GrC for Cognitive Informatics• Multi-Agent and Distributed Environment Reasoning

2007-7-14 30

GrC related International ConferencesIEEE-GrC 2006 http://www.cs.sjsu.edu/~grc/International Conference on Granular Computing

RSCTC http://rsctc2006.med.shimane-u.ac.jp/2006 The Fifth International Conference on Rough Sets and Current

Trends in Computing

RSFDGrC 2005 http://rsfdgrc.cs.uregina.ca/The Tenth International Conference on Rough Sets, Fuzzy Sets, Data

Mining, and Granular Computing

RSKT 2006 http://cs.cqupt.edu.cn/crssc/rskt2006/International Conference on Rough Sets and Knowledge Technology

IFTGrC2006International Forum on Theory of Granular Computing from Rough Set

Perspective

2007-7-14 31

Recommended ReadingsFor Basic Concepts and Perspectives

Please attend Prof. Yao’s talk on GrC between July,2nd and July,5th, and ask for further reading…

Yao, Y.Y., Perspectives of Granular Computing , Proceedings of 2005 IEEE International Conference on Granular Computing, Vol. 1, pp. 85-90, 2005.

Yao, Y.Y., Granular Computing , Computer Science (Ji Suan Ji Ke Xue), Vol. 31, pp. 1-5, 2004, Proceedings of The 4th Chinese National Conference on Rough Sets and Soft Computing.

Granular Computing Information Centerhttp://www.cs.sjsu.edu/~grc/grcinfo_center/grcinfo_index.php

2007-7-14 32

ReferencesReferences[1] Q. Liu, Granular Computing and Recent Research, CRSSC2004.[2] Y.Y.Yao, Granular Computing: basic issues and possible solutions,

2000.[3] Y.Y.Yao, Granular computing using information tables,2002.[4] L.A. Zadeh, From Computing with Numbers to Computing with Words

– From Manipulation of Measurements to Manipulation of Perceptions, 1999.

[5] Klir, G.J. and Yuan,B., Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice Hall, New Jersey, 1995.

[6] Qing Liu, Rough Sets and Rough Reasoning, Science Press, 2001.[7] Yao, J.T. and Yao, Y.Y., Induction of Classification Rules by Granular

Computing , 2002. [8] J.W.Han, Data Mining: Concepts and Techniques,2001.[9] B.Y.Ricardo, R.N.Berthier, et al. Modern Information Retrieval. the

ACM press,1999 <http://www2.dcc.ufmg.br/livros/irbook/>

Thank You!

International WIC InstituteBeijing University of Technology