Transactions on Rough Sets XVII

301 Pages·2014·16.697 MB·English

by  James F. Peters, Andrzej Skowron (eds.)

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Transactions on
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Rough Sets XVII
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James F. Peters · Andrzej Skowron
Editors-in-Chief
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Lecture Notes in Computer Science 8375
CommencedPublicationin1973
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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
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 (Eds.)
Transactions on
Rough Sets XVII
1 3
Editors-in-Chief
JamesF.Peters
UniversityofManitoba
Winnipeg,MB,Canada
E-mail:[email protected]
AndrzejSkowron
UniversityofWarsaw
Warsaw,Poland
E-mail:[email protected]
ISSN0302-9743(LNCS) e-ISSN1611-3349(LNCS)
ISSN1861-2059(TRS) e-ISSN1861-2067(TRS)
ISBN978-3-642-54755-3 e-ISBN978-3-642-54756-0
DOI10.1007/978-3-642-54756-0
SpringerHeidelbergNewYorkDordrechtLondon
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Preface
Volume XVII of the Transactions on Rough Sets (TRS) is a continuation of a
numberofresearchstreamsthathavegrownoutoftheseminalworkofZdzisl(cid:2)aw
Pawlak1duringthefirstdecadeofthetwenty-firstcentury.Theresearchstreams
representedinthepaperscoverboththetheoryandapplicationsofrough,fuzzy,
and near sets as well as their combinations.
DavideCiucciandDidierDuboispresentacomprehensivesurveyonthecon-
nections between three-valued logics and rough sets from the point of view of
incomplete informationmanagement.Ivo Du¨ntschandGu¨nther Gedigapropose
procedures to compute confidence intervals for standard errors of indices such
as γ and α to measure quality of approximation in rough set data analysis.
Christopher Henry and Garrett Smith present an application to demonstrate
descriptive-based approaches to nearness and proximity within the context of
digital image analysis.Victor MarekandAndrzej Skowronexplorepropertiesof
roughsetsrelatedtooneoftheclassicstructuresofcombinatoricsandcomputer
science,namely,matroid.MariuszPodsiadl(cid:2)oandHenrykRybin´skiprovideade-
tailed review of the currently available literature covering applications of rough
sets in economy and finance. The classic rough set model and its important
extensions are applied to areas of risk management, financial market predic-
tion, valuation, and portfolio management. Sai Prasad and Raghavendra Rao
present reduct computation algortihm(s) using a fuzzy rough set approach and
the effectiveness of their algorithm(s) is empirically demonstrated by compara-
tive analysis with existing reduct approaches. This volume also includes a long
paper by Andrzej Janusz based on his PhD thesis on algorithms for similarity
relation learning from high-dimensional data.
The editors would like to express gratitude to the authors of all submitted
papers.Specialthanksareduetothefollowingreviewers:JerzyGrzyma(cid:2)la-Busse,
ChrisCornellis,IvoDu¨ntsch,JouniJ¨arvinen,HenrykRybinski,SheelaRamanna,
Dominik S´l¸ezak, Marcin Wolski, JingTao Yao, and Yiyu Yao.
The editorsand authorsof this volume extendtheir gratitude to Alfred Hof-
mann and the LNCS staff at Springer for their support in making this volume
of the TRS possible.
The Editors-in-Chief were supported by the Polish National Science Centre
grants DEC-2011/01/B/ ST6/03867, DEC-2011/01/D/ST6/06981, and DEC-
2012/05/B/ST6/03215aswellasbythePolishNationalCentreforResearchand
1 See, e.g., Pawlak, Z., A Treatise on Rough Sets, Transactions on Rough Sets IV,
(2006), 1-17. See, also, Pawlak, Z., Skowron, A.:Rudimentsof rough sets, Informa-
tion Sciences 177 (2007) 3-27; Pawlak, Z., Skowron, A.: Rough sets: Some exten-
sions, Information Sciences 177 (2007) 28-40; Pawlak, Z., Skowron, A.: Rough sets
and Boolean reasoning, Information Sciences 177 (2007) 41-73.
VI Preface
Development(NCBiR)undergrantSYNAT No.SP/I/1/77065/10inthe frame-
workofthe strategicscientificresearchandexperimentaldevelopmentprogram:
“InterdisciplinarySystemforInteractiveScientificandScientific-TechnicalInfor-
mation”andbygrantNo.OROB/0010/03/001intheframeworkoftheDefence
and Security Programmes and Projects “Modern Engineering Tools for Deci-
sion Support for Commanders of the State Fire Service of Poland during Fire
and Rescue Operationsin the Buildings”as well as by the Natural Sciences and
Engineering Research Council of Canada (NSERC) discovery grant 185986.
January 2014 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 regular or 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 Sankar K. Pal
Jan Bazan Lech Polkowski
Gianpiero Cattaneo Henri Prade
Mihir K. Chakraborty Sheela Ramanna
Davide Ciucci Roman S(cid:2)lowin´ski
Chris Cornelis Jerzy Stefanowski
Ivo Du¨ntsch Jaros(cid:2)law Stepaniuk
Anna Gomolin´ska Zbigniew Suraj
Salvatore Greco Marcin Szczuka
Jerzy W. Grzymal(cid:2)a-Busse Dominik S´l¸ezak
Masahiro Inuiguchi Roman S´winiarski
Jouni J¨arvinen Shusaku Tsumoto
Richard Jensen Guoyin Wang
Boz˙ena Kostek Marcin Wolski
Churn-Jung Liau Wei-Zhi Wu
PawanLingras Yiyu Yao
Victor Marek Ning Zhong
Mikhail Moshkov Wojciech Ziarko
Hung Son Nguyen
Ewa Orl(cid:2)owska
Table of Contents
Three-Valued Logics, Uncertainty Management and Rough Sets........ 1
Davide Ciucci and Didier Dubois
Standard Errorsof Indices in Rough Set Data Analysis ............... 33
Gu¨nther Gediga and Ivo Du¨ntsch
Proximity System: A Description-Based System for Quantifying the
Nearness or Apartness of Visual Rough Sets......................... 48
Christopher J. Henry and Garrett Smith
Rough Sets and Matroids ......................................... 74
Victor W. Marek and Andrzej Skowron
An Efficient Approach for Fuzzy Decision Reduct Computation ........ 82
P.S.V.S. Sai Prasad and C. Raghavendra Rao
Rough Sets in Economy and Finance ............................... 109
Mariusz Podsiad(cid:2)lo and Henryk Rybin´ski
Algorithms for Similarity Relation Learning from High Dimensional
Data ........................................................... 174
Andrzej Janusz
Author Index.................................................. 293
Three-Valued Logics, Uncertainty Management
and Rough Sets
Davide Ciucci1 and Didier Dubois2
1 DISCo- Universit`a di Milano – Bicocca,
Viale Sarca 336 – U14, 20126 Milano Italia
2 IRIT,Universit´e Paul Sabatier,
118 routede Narbonne,31062 Toulouse cedex 4 France
Abstract. This paper is a survey of the connections between three-
valued logics and rough sets from the point of view of incomplete infor-
mation management. Based on the fact that many three-valued logics
canbeputunderauniquealgebraicumbrella,weshowhowtotranslate
three-valued conjunctions and implications into operations on ill-known
sets such as rough sets. We then show that while such translations may
providemathematically elegant algebraic settingsfor rough sets, thein-
terpretability of these connectives in terms of an original set approxi-
mated via an equivalence relation is very limited, thus casting doubts
onthepracticalrelevanceoftruth-functionallogical renderingsofrough
sets.
1 Introduction
Rough sets have often been studied under a three-valued logic framework and
differentauthorshavetriedtoconnectroughsetstodifferentlogics:L(cid:2) ukasiewicz
[9, 11], Nelson [58, 59], G¨odel, Gaines-Rescher three-valued logics [49, 41]. De-
spitetheformalcorrectnessoftheseapproaches,littleattentionhasbeendevoted
to the interpretation of these logics in the rough set context. Moreover, a com-
prehensive study on the three-valued connectives that can be defined on rough
sets is needed and, as we will see, it can be accomplished starting from known
results in three-valued logics.
Three-valued logics are apparently simple; they are straightforward general-
izations of Boolean logic based on the most simple bipolar scale {0,1,1} where
2
1 (resp. 0) has a positive (resp. negative) flavor, and 1 is neutral. Further, they
2
arewidely usedinseveralappliedcontextssuchaslogicprogramming[43], elec-
tronic circuits [67], databases [27], and, of course, rough sets. However, there
have been several different meanings attached to the third value, some having
an epistemic nature. There is not a clear result on the definition of its connec-
tives in connection with this meaning. Here is a list of these interpretations of
thethirdtruth-value,differentfromtrueandfalse:Possible (duetoL(cid:2)ukasiewicz
[17]), Unknown (Kleene [52]), Undefined (also Kleene), Half-true (in fuzzy logic
[48]), Borderline (in logics ofvagueness, like in Shapiro [66]), Inconsistent (that
isbothtrueandfalse,asinparaconsistentlogicsorthelogicofparadoxbyPriest
J.F.PetersandA.Skowron(Eds.):TransactionsonRoughSetsXVII,LNCS8375,pp.1–32,2014.
(cid:2)c Springer-VerlagBerlinHeidelberg2014
2 D. Ciucci and D.Dubois
[63]), or yet Irrelevant as in relevance logics [2] or the logic of conditionals [38].
Sometimes, two of these notions are simultaneously used as Inconsistent and
Unknown in Belnap four-valued logic [14].
Three-valued logics go along with three-valued sets having central elements
and peripheral ones [46]. However the meaning of such central and peripheral
elementsdependsonthemeaningofthethirdtruth-value.Itdependsonwhether
it has an epistemic flavoror not; a peripheral element can be understood in one
of the following ways:
1. either as an untypical element of a non-classical set,
2. or as anelementthat cannotbe definitely classifiedas belonging ornot to a
crisp set due to incomplete information,
3. or as anelementthat cannotbe definitely classifiedas belonging ornot to a
crisp set due to conflicting information,
4. or as an element for which membership or non membership makes no sense,
due to irrelevance or the dubious existence of such an element.
Case 2 is the one we are concerned with in this paper. Then the three truth-
values refer to the epistemic status of otherwise Booleanpropositions (provably
true, provably false or unknown [39]). This is typically the case of ill-known or
intervalsets[72],wherethecentralelementsareelementsthatcertainlybelongto
someill-knownset, the thirdtruth-valueisassimilatedto{0,1}andunderstood
as the hesitancy between membership and non-membership. They are special
cases of interval-valued fuzzy sets [77] or twofold fuzzy sets [37]. One of the
causes of a set being ill-known can be the lack of precisionon the value of some
oftheattributesthatdescribeit(forinstance,asetofsinglepersonsisill-known
if the marital status of some of the persons is ill-known).
A rough set, viewed as a pair of nested approximations is a typical example
of ill-known set, where the lack of knowledge comes from an equivalence rela-
tion between possibly indistinguishable elements, this indistinguishability being
due to the use of a language that is not expressive enough (incomplete set of
attributes or attributes that are too coarsely defined). This situation contrasts
withthe caseofsetsthatareill-knowndue tothe lackofknowledgeofattribute
values; see Couso and Dubois [28] when the two causes of partial ignorance
appear simultaneously.
Inrecentpapers[23–26],wehavestudiedvariousthree-valuedlogicsofpartial
knowledge,wherethe thirdtruth-value meansunknown. Ithas beenshownthat
alargeclassofthree-valuedlogics(includingL(cid:2) ukasiewiczL )iscompatiblewith
3
this understanding of the third truth-value, but their translations into a very
elementarymodallogicindicatethatsuchthree-valuedlogicscannotaccountfor
partialignorancejointlyaffectingseveralBooleanvariables:onlystatesofpartial
ignorancethatcanbedescribedindependentlyforeachvariablecanbeaccounted
for in a three-valued logic. This is the price paid for truth-functionality.
In this paper, we examine the situation of three-valued logics of rough sets.
While the aforementioned limitation is still valid (since rough sets do not be-
have truth-functionally in general), there is an additional constraint in this
case.Namely, the approximationpairs aregeneratedby anequivalence relation,