Table Of ContentStudies in Computational Intelligence 957
Jan W. Owsiński · Jarosław Stańczak ·
Karol Opara · Sławomir Zadrożny ·
Janusz Kacprzyk
Reverse
Clustering
Formulation, Interpretation and Case
Studies
Studies in Computational Intelligence
Volume 957
Series Editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
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Jan W. Owsi ski Jaros aw Sta czak
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Karol Opara S awomir Zadro ny
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Janusz Kacprzyk
Reverse Clustering
Formulation, Interpretation and Case Studies
123
Jan W.Owsiński Jarosław Stańczak
Polish Academy ofSciences Polish Academy ofSciences
Systems Research Institute Systems Research Institute
Warsaw,Poland Warsaw,Poland
KarolOpara Sławomir Zadrożny
Polish Academy ofSciences Polish Academy ofSciences
Systems Research Institute Systems Research Institute
Warsaw,Poland Warsaw,Poland
JanuszKacprzyk
Polish Academy ofSciences
Systems Research Institute
Warsaw,Poland
ISSN 1860-949X ISSN 1860-9503 (electronic)
Studies in Computational Intelligence
ISBN978-3-030-69358-9 ISBN978-3-030-69359-6 (eBook)
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Preface
We witness nowadays an explosive growth and development of methods and
techniques,relatedtodataanalysis,thisgrowthbeingconditioned,ontheonehand,
by the rapidly expanding availability of data in virtually all domains of human
activity, and, on the other hand, the very substantive progress in technical and
scientific capabilities of dealing with the increasing volumes of data. All this
amounts to a dramatic change, especially in quantitative terms.
Yet, as researchers and practitioners involved in the work on methodological
side of data analysis know very well, many of the fundamental substantive prob-
lemsinthisdomainstillrequiresolutions,oratleast—bettersolutions—thanthose
available now. This concerns, in particular, such fundamental areas as clustering,
classification, rule extraction, and so on. The primary issue is here constituted by
the opposition between precision or accuracy and speed or computational cost
(when the problem at hand is already truly well-defined). One cannot forget, nei-
ther, of the very strong data dependence of effectiveness and efficiency of many
of the methodologies being applied nowadays, making the situation even more
difficult.
Thepresentbookaddressesthisnexusofissues,aiming,inthiscase,apparently
attheinterfaceofclusteringandclassification,but,infact,beingrelevanttoamuch
broader domain, with much broader implications in terms of applicability and
interpretation.Namely,itdescribestheparadigmof“reverseclustering”,introduced
bythepresentauthors.Theparadigmconcernsthesituation,inwhichwearegiven
acertaindata set,composedofentities,observations,objects…,whichisusualfor
the data analysis situation, and, at the same time, we are given, or we consider, a
certainpartitionofthisdataset.Wedonotassumeapriorianythingaboutthedata
set,noraboutthepartition,and,essentiallyimportantly,abouttherelationbetween
thedatasetandthepartition.Thus,thepartitionmaybetheresultofadefinitekind
of analysis of the given data set, but may, as well, result from quite a different
mechanism (e.g. a division of the set of objects according to some variable or
criterion not contained in the data set at hand).
v
vi Preface
Under these circumstances—the data set and the partition being given—we try
toreconstructthepartitiononthebasisofthedataset,usingclusteranalysis.Wetry
to find the entire clustering procedure that will yield, for this given data set, a
partition that is as close to the given one as possible. Thus, the result of the pro-
cedure is both the clustering procedure, defined by a number of attributes (clus-
tering method, its parameters, variable selection, distance definition,…) and the
concrete partition found.
It is obvious that the paradigm borders upon classification (for a very specific
formulation/interpretation of the situation faced), but extends to a much broader
domain,inwhichtheperceptionoftheproblemitselfandthemeaningofsolutions
can vary very widely. This is, in particular, shown in the present book.
In the current stage of work, the results obtained and largely contained in this
book pertain mainly to the substantive aspect of the paradigm, while the technical
aspects of the respective algorithms are, as of now, left to future research.
The reverse clustering paradigm constitutes a new perspective on quite a broad
spectrumofproblemsindataanalysis,and,asthebookshows,itcanprovidevery
interesting, instructive and significant results, under a wide variety of interpreta-
tionalassumptions.Wesincerelyhope,therefore,thatthisbookdoesnotonlygive
the Readers a new material and fresh insight into some problems of data analysis,
but may also provoke them to deeper studies in the direction here indicated.
Warsaw, Poland Jan W. Owsiński
Jarosław Stańczak
Karol Opara
Sławomir Zadrożny
Janusz Kacprzyk
Introduction
This book is devoted to an approach or a paradigm, developed by the authors and
appliedtoaseriesofcases,ofdiversecharacter,mostlybasedonreal-lifedata;the
approach (or paradigm) belonging to the broadly understood domain of data
analysis—more precisely: classification and cluster analysis. We call the approach
“reverse clustering” because of its logic, which is formulated as follows:
Assumewedisposeofasetofdata,X,composedofnobjectsorobservations,
indexedi,i=1,…,n,eachofthesebeingdescribedbyavectorofmfeatures
or variables, indexed k, the respective vector being denoted x = {x ,…,x
i i1 ik,
…,xim}. At the same time, assume we dispose of a partition of the set X of
objectsintosubsets,thispartitionbeingdenotedP .Forthesedata,wetryto
A
obtainapartitionP thatisasclosetoP aspossible,byapplyingclustering
B A
algorithmstothesetX.Thereby,wefindboththepartitionP thatisasclose
B
as possible to P and the concrete clustering procedure, with all its param-
A
eters, which yields the partition P .
B
The above does not explicitly state the purpose of the exercise (to say nothing
ofthetechnicaldetails),butitcaneasilybededucedthatwhatisaimedatisclosely
related tothenotionofclassification.Whiletheclose relation with classificationis
not only obvious, but definitely true, the paradigm has a much wider spectrum of
applications and meanings, as this is explained in Chap. 2 of the book, following
the more precise presentation of this paradigm, given in Chap. 1.
Theparadigmisconstituted,first,bytheabovestatementoftheproblem,which
then has to be expressed in pragmatic technical terms, involving
(1) the space of clustering algorithms with its granularity (what algorithms are
accountedforandwhatparameters,definingtheentireclusteringprocedure,are
being subject of the search for P );
B
vii
viii Introduction
(2) themeasureofsimilaritybetweenthepartitionofthesetX,givenattheoutset,
i.e.P ,andthepartitions,obtainedfromtheclusteringalgorithms,thismeasure
A
beingmaximised(orthemeasureofdistancebetweenthem,beingminimised);
and
(3) the technique of search for the P given the data of the concrete problem.
B
Thisparadigmis,however,also,andperhapsevenmoreimportantly,constituted
by the interpretation of the entire setting, and the particular instances of this
interpretation—asmentioned,treatedatlengthinChap.2.Thisisimportantinsofar
asit places theparadigmagainstthebackground ofthe data analysisdomain, with
special emphasis on classification and related fields. These various interpretation
instances are associated primarily with the status of the partition P , namely its
A
source, the degree of credibility we assign to it, as well as its actual or presumed
connectionwiththedatasetX.Dependingonthese,andontheresultsobtained,the
status of the obtained partition P , including validity and applicability, will also
B
vary significantly.
Owing to this variety of interpretations, the paradigm may find application in a
broad spectrum of analytic, but also cognitive, situations. The subsequent chapters
of the book, starting with the third one, are exactly devoted to the presentation
of the cases treated, which definitely differ not only as to their substance matter
(domain, from which the data come), but, largely, as to the interpretation of the
actualproblemandtheresultsobtained.Theimplicationisthattheparadigmcanbe
used in many data analytic circumstances for diverse purposes, whenever the
structuration of the data set into groups is appropriate.
Theparadigmofreverseclusteringhasbeenpresentedalreadyinseveralpapers
bythesameteamofauthors,e.g.inOwsińskietal.(2017a,b),Owsiński,Stańczak
andZadrożny(2018).Thepresentbookaimsatamorecompletepresentationofthe
paradigm and its interpretations. The book does not go into the computational and
numerical issues and details, which are, of course, of very high importance.
Namely,themainpurposeofthebookistopresenttheapproachanditscapacities
in terms of various kinds of situations, problems and interpretations of respective
results. We do indeed hope it conveys the intended message in an effective and
interesting manner.
The book is structured in the following manner: first, Chap. 1 presents the
schemeoftheapproach,characterised,inparticular,asithasbeenusedinthecases
illustratedinthisbook,alongwithnotationused.Then,Chap.2outlinesthecontext
of the reverse clustering, starting with other approaches, which concern similar
kindsofproblems,relatedtodataanalysis,includingalsoanamplereferencetothe
very general idea of reverse engineering, as well as explainable artificial intelli-
gence or data analysis. Then, the context is shortly analysed in terms of more
detailed specific problems, arising in connection with both the reverse clustering
procedure and the data analytic methods in a more general perspective (like, e.g.
selection of variables, or definitions of distance). This chapter contains also a very
importantsectiononthepotentialinterpretationsofthereverseclusteringparadigm
and its results. Chapter 3 constitutes a very short introduction to the cases studied
Introduction ix
and illustrated in the book, which are then presented in the consecutive chapters:
Chap. 4 is devoted to the motorway traffic data, Chap. 5 to environmental con-
taminationdata,Chaps.6and7totwoseparatecasesoftypologiesorclassifications
of administrative units in Poland, and, finally, Chap. 8 to some more academic
exercises.ThebookcloseswithChap.9summarisingtheworkdoneandproposing
some new vistas.
This book is intended to offer the Readers truly interesting and novel perspec-
tives in data analysis, regarding the diverse ways offormulating and approaching
problems,andunderstandingtheresults,andweshallbeverysatisfiedifitdiditat
least in a perceptible degree.
Jan W. Owsiński
Jarosław Stańczak
Karol Opara
Sławomir Zadrożny
Janusz Kacprzyk
