Transactions on Rough Sets XIII

284 Pages·2011·4 MB·English

by  Chien-Chung Chan

Lecture Notes in Computer Science 6499
CommencedPublicationin1973
FoundingandFormerSeriesEditors:
GerhardGoos,JurisHartmanis,andJanvanLeeuwen
EditorialBoard
DavidHutchison
LancasterUniversity,UK
TakeoKanade
CarnegieMellonUniversity,Pittsburgh,PA,USA
JosefKittler
UniversityofSurrey,Guildford,UK
JonM.Kleinberg
CornellUniversity,Ithaca,NY,USA
FriedemannMattern
ETHZurich,Switzerland
JohnC.Mitchell
StanfordUniversity,CA,USA
MoniNaor
WeizmannInstituteofScience,Rehovot,Israel
OscarNierstrasz
UniversityofBern,Switzerland
C.PanduRangan
IndianInstituteofTechnology,Madras,India
BernhardSteffen
TUDortmundUniversity,Germany
MadhuSudan
MicrosoftResearch,Cambridge,MA,USA
DemetriTerzopoulos
UniversityofCalifornia,LosAngeles,CA,USA
DougTygar
UniversityofCalifornia,Berkeley,CA,USA
MosheY.Vardi
RiceUniversity,Houston,TX,USA
GerhardWeikum
MaxPlanckInstituteforInformatics,Saarbruecken,Germany
James F. Peters Andrzej Skowron
Chien-Chung Chan Jerzy W. Grzymala-Busse
Wojciech P. Ziarko (Eds.)
Transactions on
Rough Sets XIII
1 3
Editors-in-Chief
JamesF.Peters
UniversityofManitoba,Winnipeg,Manitoba,Canada
E-mail:[email protected]
AndrzejSkowron
WarsawUniversity,Warsaw,Poland
E-mail:[email protected]
GuestEditors
Chien-ChungChan
TheUniversityofAkron,OH,USA
E-mail:[email protected]
JerzyW.Grzymala-Busse
TheUniversityofKansas,Lawrence,KS,USA
E-mail:[email protected]
WojciechP.Ziarko
UniversityofRegina,SK,Canada
E-mail:[email protected]
ISSN0302-9743(LNCS) e-ISSN1611-3349(LNCS)
ISSN1861-2059(TRS) e-ISSN1861-2067(TRS)
ISBN978-3-642-18301-0 e-ISBN978-3-642-18302-7
DOI10.1007/978-3-642-18302-7
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010942518
CRSubjectClassification(1998):I.2,H.3,H.2.8,I.4,F.2,G.2
©Springer-VerlagBerlinHeidelberg2011
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Preface
Volume XIII of the Transactions on Rough Sets (TRS) consists of extended
versions of selected papers presented at the Rough Sets and Current Trends
in Computing Conference (RSCTC 2008) organized in Akron, OH, USA, in
October 2008 that are part of Lecture Notes in Artificial Intelligence volume
5306 edited by Chien-Chung Chan, Jerzy W. Grzymala-Busse and Wojciech P.
Ziarko;italsoincludessomeregularpapers.Theselectionofpapersacceptedfor
RSCTC 2008 was made by the editors and the authors were invited to submit
extended versions to this issue. The 13 submissions received on this invitation
wentthroughtworoundsofreviews,and10paperswereacceptedforpublication.
The editors of the special issue are particularly grateful to all the authors
of submitted papers, and to the following reviewers: Salvatore Greco, Masahiro
Inuiguchi, Wojciech Jaworski, Richard Jensen, L(cid:2) ukasz Kobylin´ski, Tianrui Li,
Tsau Young Lin, Pawan Lingras, Dun Liu, Dariusz Maylyszko, Georg Peters,
Sheela Ramanna, Jerzy Stefanowski, Marcin Szczuka, Guoyin Wang, Wei-Zhi
Wu and JingTao Yao. Their laudable efforts made possible a careful selection
and revision of submitted manuscripts.
The articles of this special issue on Rough Sets and Current Trends in Com-
puting introduceanumber ofnew advancesinboththe foundationsandthe ap-
plications of roughsets. The advances in the foundations concernmathematical
structuresofgeneralizedroughsetsininfinite universes,approximationsofarbi-
trary binary relations, and attribute reduction in decision-theoretic rough sets.
Methodological advances introduce rough set-based and hybrid methodologies
forlearningtheory,attributionreduction,decisionanalysis,riskassessment,and
data-mining tasks such as classificationand clustering. In addition, this volume
contains regular articles on mining temporal software metrics data, C-GAME
discretizationmethod,perceptualtoleranceintersectionasanexample ofa near
set operation and compression of spatial data with quadtree structures.
The editors and authors of this volume extend their gratitude to Alfred
Hofmann, Anna Kramer, Ursula Barth, Christine Reiss and the LNCS staff at
Springer for their support in making this volume of the TRS possible.
TheEditors-in-ChiefhavebeensupportedbytheStateCommitteeforScien-
tificResearchoftheRepublicofPoland(KBN)researchgrantsNN516368334,N
N516077837,theNaturalSciencesandEngineeringResearchCouncilofCanada
(NSERC) research grant 185986, Canadian Network of Excellence (NCE), and
a Canadian Arthritis Network (CAN) grant SRI-BIO-05.
September 2010 Chien-Chung Chan
Jerzy W. Grzymala-Busse
Wojciech P. Ziarko
James F. Peters
Andrzej Skowron
LNCS Transactions on Rough Sets
The Transactions on Rough Sets series has as its principal aim the fostering of
professional exchanges between scientists and practitioners who are interested
in the foundations and applications of rough sets. Topics include foundations
and applications of rough sets as well as foundations and applications of hybrid
methodscombiningroughsetswithotherapproachesimportantforthedevelop-
ment of intelligent systems. The journal includes high-quality research articles
accepted for publication on the basis of thorough peer reviews. Dissertations
and monographs up to 250 pages that include new research results can also be
considered as regular papers. Extended and revised versions of selected papers
from conferences can also be included in regularor special issues of the journal.
Editors-in-Chief: James F. Peters, Andrzej Skowron
Managing Editor: Sheela Ramanna
Technical Editor: Marcin Szczuka
Editorial Board
Mohua Banerjee Ewa Orl(cid:2)owska
Jan Bazan Sankar K. Pal
Gianpiero Cattaneo Lech Polkowski
Mihir K. Chakraborty Henri Prade
Davide Ciucci Sheela Ramanna
Chris Cornelis Roman Sl(cid:2)owin´ski
Ivo Du¨ntsch Jerzy Stefanowski
Anna Gomolin´ska Jaros(cid:2)law Stepaniuk
Salvatore Greco Zbigniew Suraj
Jerzy W. Grzymal(cid:2)a-Busse Marcin Szczuka
Masahiro Inuiguchi Dominik S´le¸zak
Jouni Ja¨rvinen Roman S´winiarski
Richard Jensen Shusaku Tsumoto
Boz˙ena Kostek Guoyin Wang
Churn-Jung Liau Marcin Wolski
PawanLingras Wei-Zhi Wu
Victor Marek Yiyu Yao
Mikhail Moshkov Ning Zhong
Hung Son Nguyen Wojciech Ziarko
Table of Contents
Bit-Vector Representation of Dominance-Based Approximation Space... 1
Chien-Chung Chan and Gwo-Hshiung Tzeng
Approximations of Arbitrary Binary Relations by Partial Orders:
Classical and Rough Set Models ................................... 17
Ryszard Janicki
Hybridization of Rough Sets and Statistical Learning Theory .......... 39
Wojciech Jaworski
Fuzzy-Rough Nearest Neighbour Classification....................... 56
Richard Jensen and Chris Cornelis
Efficient Mining of Jumping Emerging Patterns with Occurrence
Counts for Classification .......................................... 73
L(cid:2) ukasz Kobylin´ski and Krzysztof Walczak
Rough Entropy Hierarchical Agglomerative Clustering in Image
Segmentation.................................................... 89
Dariusz Mal(cid:2)yszko and Jarosl(cid:2)aw Stepaniuk
Software Defect Prediction Based on Source Code Metrics Time
Series .......................................................... 104
L(cid:2) ukasz Pul(cid:2)awski
Risk Assessment in Granular Environments ......................... 121
Marcin Szczuka
Core-Generating Discretization for Rough Set Feature Selection........ 135
David Tian, Xiao-jun Zeng, and John Keane
Perceptual Tolerance Intersection .................................. 159
Piotr Wasilewski, James F. Peters, and Sheela Ramanna
Some Mathematical Structures of Generalized Rough Sets in Infinite
Universes of Discourse ............................................ 175
Wei-Zhi Wu and Ju-Sheng Mi
Quadtree Representation and Compression of Spatial Data............ 207
Xiang Yin, Ivo Du¨ntsch, and Gu¨nther Gediga
VIII Table of Contents
Solving the Attribute Reduction Problem with Ant Colony
Optimization .................................................... 240
Hong Yu, Guoyin Wang, and Fakuan Lan
A Note on Attribute Reduction in the Decision-Theoretic Rough Set
Model .......................................................... 260
Yan Zhao, S.K.M. Wong, and Yiyu Yao
Author Index.................................................. 277
Bit-Vector Representation of Dominance-Based
Approximation Space
Chien-Chung Chan1 and Gwo-Hshiung Tzeng2,3
1 Department of Computer Science, University of Akron
Akron,OH,44325-4003, USA
[email protected]
2 Department of Business and Entrepreneurial Administration, Kainan University
No. 1 Kainan Road, Luchu,Taoyuan County 338, Taiwan
[email protected]
3 Instituteof Management of Technology, National Chiao TungUniversity
1001 Ta-Hsueh Road, Hsinchu 300, Taiwan
[email protected]
Abstract. Dominance-based Rough Set Approach (DRSA) introduced
by Greco et al. is an extension of Pawlak’s classical rough set theory by
using dominance relations in place of equivalence relations for approxi-
matingsetsofpreferenceordereddecision classes. Theelementarygran-
ules in DRSA are P-dominating and P-dominated sets. Recently,Chan
and Tzeng introduced the concept of indexed blocks for representing
dominance-basedapproximationspacewithgeneralized dominancerela-
tions on evaluations ofobjects. This papershows howto deriveindexed
blocksfrom P-dominatingandP-dominatedsetsin DRSA.Approxima-
tionsaregeneralizedtoanyfamilyofdecisionclassesintermsofindexed
blocks formulated as binary neighborhood systems. We present algo-
rithms for generating indexed blocks from multi-criteria decision tables
andforencodingindexedblocksasbit-vectorstofacilitatethecomputa-
tion of approximations and rule generation. A new form of representing
decisionrulesbyusingintervalandset-differenceoperatorsisintroduced,
and we give a procedure of how to generate this type of rules that can
be implemented as SQL queries.
Keywords: Rough sets, Dominance-based rough sets, Multiple criteria
decisionanalysis(MCDA),Neighborhoodsystems,Granularcomputing.
1 Introduction
The Dominance-Based Rough Set Approach (DRSA) to multiple criteria deci-
sion analysis was introduced by Greco, Matarazzo and Slowinski [1, 2, 3] as an
extension of Pawlak’s classicalrough sets (CRS) [8, 9, 10]. In DRSA, attributes
withpreference-ordereddomainsarecalledcriteria,andtheindiscernibilityrela-
tioninCRSisgeneralizedtoadominancerelationthatisreflexiveandtransitive.
Itis alsoassumedthatdecisionclassesareorderedby somepreferenceordering.
The fundamental principle in DRSA is that the assignment of decision classes
J.F.Petersetal.(Eds.):TransactionsonRoughSetsXIII,LNCS6499,pp.1–16,2011.
(cid:2)c Springer-VerlagBerlinHeidelberg2011
2 C-C. Chan and G-H.Tzeng
to objects based on evaluation values of given criteria follows the monotonic
dominance principle. The dominance principle requires that if criteria values of
object xarebetterthanthoseofobjecty,thenx should be assigned to a class not
worsethany.InDRSA, anapproximationspaceofdecisionclassesisdefinedby
dominating and dominated sets of objects. Let x be an object and P be a set
of evaluation criteria, then the P-dominating set of x consists of those objects
with evaluation values at least as good as x’s with respect to P, and objects
with evaluation values at most as good as x’s belong to the P-dominated set
of x. In DRSA, decision classes considered for approximations are upward and
downwardunionsofdecisionclasses.Ithasbeenshowntobeaneffectivetoolfor
multiple criteria decision analysis problems [12] and has been applied to solve
multiple criteria sorting problems [4, 5].
The representation of dominance-based approximation space by a family of
indexedblockswasintroducedin[13].Indexedblocksaresetsofobjectsindexed
by pairsofdecisionvalues.The basicidea is to use abinary relationondecision
values as indices for grouping objects in light of dominance principle where
inconsistency is defined as a result of violating the dominance principle. A set
of ordered pairs is derived from objects violating dominance principle involving
a pair of decision values, which is used as the index for the set of ordered pairs.
Forexample,objects thatareconsistentlyassignedto decisionclassiforma set
of ordered pairs with decision index (i,i). A set of ordered pairs with decision
index (i,j), i(cid:2)=j, corresponds to objects that violate dominance principle with
respecttodecisionvaluesiandj.Eachindexedsetoforderedpairsinducesaset
of objects, called a block, which is indexed by the same pair of decision values.
Theseblocksarecalledindexedblocks,whichareelementarysets(granules)ofa
dominance-basedapproximationspace.In this approach,approximationsof any
union of decision classes are computed by neighborhoods of indexed blocks.
In this paper,indexedblocks areformulatedinterms ofbinaryneighborhood
systems [15]. We show how to derive indexed blocks from P-dominating and
P-dominated sets. Basically, an object y in P-dominating of x or P-dominated
set of x is in a inconsistent indexed block B(i,j), if the decision class assign-
ment to x and y violates the dominance principle. Strategies for generating
indexed blocks from multi-criteria decision tables were introduced in [13]. Here,
we introduce algorithms to compute indexed blocks from inconsistent ordered
pairs. To facilitate the computation of approximations and the generation of
decision rules, bit-vector encodings for indexed blocks is introduced. In our ap-
proach, conditional elements of decision rules are formulated as conjuncts of
intervals, which may be modified with EXCEPT clauses corresponding to set-
difference operators. This type of rules is suitable for SQL (Standard Query
Language)implementations. We introduce a procedure to generate these con-
juncts from approximationsofany unions of decisionclasses by using bit-vector
representation of indexed blocks.
Theremainderofthispaperisorganizedasfollows.Relatedconceptsofrough
sets,dominance-basedroughsets,andindexedblocksaregiveninSection2.The
relationship between indexed blocks and dominating and dominated sets and