Non-Additive Measures: Theory and Applications

207 Pages·2014·3.01 MB·English

by  Vicenç Torra (auth.)

Studies in Fuzziness and Soft Computing
Vicenç Torra
Yasuo Narukawa
Michio Sugeno Editors

Non-Additive
Measures
Theory and Applications
Studies in Fuzziness and Soft Computing
Volume 310
SeriesEditors
JanuszKacprzyk,PolishAcademyofSciences,Warsaw,Poland
e-mail:[email protected]
Forfurthervolumes:
http://www.springer.com/series/2941
AboutthisSeries
Theseries“StudiesinFuzzinessandSoftComputing”containspublicationsonvarious
topicsin the area ofsoftcomputing,whichincludefuzzysets, roughsets, neuralnet-
works, evolutionarycomputation,probabilisticand evidentialreasoning,multi-valued
logic,andrelatedfields.Thepublicationswithin“StudiesinFuzzinessandSoftCom-
puting” are primarily monographsand edited volumes. They cover significant recent
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tantfeatureoftheseriesisitsshortpublicationtimeandworld-widedistribution.This
permitsarapidandbroaddisseminationofresearchresults.
· ·
Vicenç Torra Yasuo Narukawa Michio Sugeno
Editors
Non-Additive Measures
Theory and Applications
ABC
Editors
VicençTorra MichioSugeno
IIIA-CSIC EuropeanCentreforSoftComputing
Bellaterra Mieres
Catalonia Spain
Spain
YasuoNarukawa
TohoGakuen
Kunitachi
Tokyo
Japan
ISSN1434-9922 ISSN1860-0808 (electronic)
ISBN978-3-319-03154-5 ISBN978-3-319-03155-2 (eBook)
DOI10.1007/978-3-319-03155-2
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013951835
(cid:2)c SpringerInternationalPublishingSwitzerland2014
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Preface
A measureassignsvaluesto sets andgeneralizesthe conceptoflength, areaand
volume. Probability measures are a well known example of measures. Lebesgue
measuresare another example. Most typically, measuresare additive, as length,
areas and volumes are. That is, the measure of the union of two disjoint sets is
the sum of these two measures
μ(A∪B)=μ(A)+μ(B)
for A∩B =∅. Again, probabilities are an example of additive measures.
Althoughnotsomuchknown,non-additivemeasureshavealsobeenstudiedin
theliteraturebothfortheirmathematicalpropertiesaswellasfortheirapplica-
tiontorealproblems.Non-additivemeasuresreplaceadditivitymymonotonicity.
That is,
μ(A)≤μ(C) if A⊆C.
As all additive measures are monotonic, non-additive measures generalize addi-
tive ones.
Non-additivemeasuresarealsoknownbythetermcapacities,andfuzzymea-
sures.
Non-additive measures permit us to represent interaction between the ele-
ments. For example, we might have μ(A∪B) < μ(A)+μ(B) (negative inter-
action between A and B), and μ(A∪B) > μ(A)+μ(B) (positive interaction
between A and B).
Then, in the same way that there are fundamental concepts in measure the-
orybasedonadditivemeasures,therearesomebasedonnon-additivemeasures.
Some of these concepts are generalizations of corresponding concepts for addi-
tive measures. For example, the Choquet integral [1] is a generalization of the
Lebesgue integral in the sense that the Choquet integral of a function f with
respecttoanadditivemeasurecorrespondstothe Lebesgueintegral.Othercon-
ceptswereintroducedasnew inthis non-additivesetting.This isthe caseofthe
integral introduced by Sugeno in 1972 [4,5] which is now known as the Sugeno
integral.
VI Preface
This book has its origin in the 9th International Conference on Modeling
Decisions for Artificial Intelligence (MDAI 2012) that took place in Girona1
and, more specifically, in the panel session Fuzzy measures, fuzzy integrals and
aggregation operatorsholdintheconference.Thepanelgatheredkeyresearchers
with the aim of discussing new and challenging lines for future research in the
area of non-additive measures and integrals. The chapters of this book, written
by most of the panelists and two additional invited authors, are state-of-the-art
descriptions of the field that cover the lines of researchdiscussed in the panel.
Thefirstchapterisareviewofusesandapplicationsofnon-additivemeasures
andintegrals.Thechapterpresentsmostrelevantdefinitionsandalsopointsout
totheotherchaptersinthebookforfurtherdetailsandreferences.Linksbetween
non-additive integrals and aggregationoperators [6] are also highlighted.
In the second chapter, Narukawapresents an overviewof integrationwith re-
spectto anon-additivemeasure.Thechaptergivesspecialemphasisto integrals
over continuous domains. The Sugeno, Choquet and generalized integrals are
presented and their properties reviewed. The case of multidimensional integrals
arediscussedandaFubini-liketheoremispresented.Thechapterconcludeswith
the M¨obius transform and generalizations of the M¨obius transform.
In the third chapter, Mesiar and Stupnˇanov´a focuses on different integrals
with respect to a non-additive measure. The authors discuss the approach to
integration introduced by Even and Lehrer [2] (decomposition integrals), the
Choquet and Sugeno integrals and also Shilkret and universal integrals.
Chapter four, by Honda, focuses on the definition ofentropy for non-additive
measures(orcapacities).First,Hondareviewsthedefinitionofentropyforprob-
abilitiesandthenintroducesdifferentgeneralizationsthatexistfornon-additive
measures.Theauthornotonlyconsidersthecaseofmeasuresdefinedon2X but
alsoonmeasuresdefinedonsetsystems(basedonasubsetof2X). The problem
of the axiomatization of entropies is also discussed.
Non-additivemeasuresandintegralshavebeenusedinapplications.Thefifth
chapter by Ozaki focuses on their application to economics. More especifically,
theauthorconsidersdecisiontheoryunderrisk,anddecisiontheoryunderuncer-
tainty. The chapter describes some of the problems and paradoxes that cannot
be solved using additive models (as e.g. Ellsberg’s paradox).
Fujimoto in Chapter six surveys cooperative game theory, an important ap-
plication area for non-additive measures. Non-additive measures permit to rep-
resent coalitions in game theory. The chapter discusses with detail the case in
which not all coalitions can be formed, and how we can deal with this situa-
tion.Thechapteralsodiscussesindicesthathavebeendefinedforgames(asthe
Shapley index [3]).
Flaminio, Godo, and Kroupa focus in Chapter seven on belief functions on
MV-algebras of fuzzy sets. Belief functions are totally monotone non-additive
measures.TheauthorsdiscusstwowaysofextendingbelieffunctionsonBoolean
algebras of events to MV-algebras of events.
1 http://www.mdai.cat/mdai2012
Preface VII
We hope that this book will provide a reference to students, researchers and
practitioners in the field.
The editors of this book would like to thank Prof. Kacprzyk for his encour-
agement to edit it and publish it in this series. Partial support by the Spanish
MINECO (TIN2011-15580-E)is acknowledged.
August 2013 Vicen¸c Torra
L’Alcalat´en Yasuo Narukawa
Michio Sugeno
References
1. Choquet, G. (1953/54) Theory of capacities, Ann.Inst.Fourier 5 131-295.
2. Even,Y.,Lehrer,E.(2013)Decomposition-Integral:UnifyingChoquetandtheCon-
cave Integrals, Submitted.
http://www.tau.ac.il/∼ lehrer/Papers/decomposition.pdf (access August 2013)
3. Shapley,L. (1953) A valuefor n-person games, Annalsof Mathematical Studies28
307-317.
4. Sugeno, M. (1972) Fuzzy measures and fuzzy integrals (in Japanese), Trans. of the
Soc. of Instrumentand Control Engineers 8:2
5. Sugeno,M.(1974) TheoryofFuzzyIntegralsanditsApplications,Ph.D.Disserta-
tion, Tokyo Instituteof Technology, Tokyo, Japan.
6. Torra, V., Narukawa, Y. (2007) Modeling decisions: information fusion and aggre-
gation operators, Springer.
Contents
Use and Applications of Non-Additive Measures and
Integrals........................................................ 1
Vicen¸c Torra
Integral with Respect to a Non Additive Measure:
An Overview ................................................... 35
Yasuo Narukawa
Integral Sums and Integrals .................................... 63
Radko Mesiar, Andrea Stupnˇanov´a
Entropy of Capacity ............................................ 79
Aoi Honda
Integral with Respect to Non-additive Measure in Economics... 97
Hiroyuki Ozaki
Cooperative Game as Non-Additive Measure ................... 131
Katsushige Fujimoto
Belief Functions on MV-Algebras of Fuzzy Sets:
An Overview ................................................... 173
Tommaso Flaminio, Llu´ıs Godo, Toma´ˇs Kroupa
Author Index................................................... 201
Use and Applications of Non-Additive Measures
and Integrals
Vicenc¸ Torra
IIIA,Institut d’Investigaci´o en Intel·lig`encia artificial,
CSIC, Consejo Superiorde Investigaciones Cient´ıficas,
Campus Universitat Auto`noma deBarcelona s/n,
08193 Bellaterra, Catalonia
[email protected], [email protected]
Abstract. Non-additive measures (also known as fuzzy measures and
capacities) andintegralshavebeenusedinseveraltypesofapplications.
Inthischapterwereviewthemaindefinitionsrelatedtothesemeasures,
motivatetheirusefromthepointofviewoftheapplications,anddescribe
their usein different contexts.
1 Introduction
Non-additivemeasuresandintegralshavebeen usedindifferentareas.They are
used to overcome the shortcommings of other (simpler) models. In particular,
they are used when models based on additive measures are not appropriate. In
this chapter we review the basic definitions of this topic and describe some of
the problems that can be solved using non-additive measures.
Forthesakeofcompleteness,Section2reviewsdefinitionsandresultsthatare
needed in the rest of the chapter. Then, Section 3 focuses on decision making.
Wedescribesomeparadoxesthatcanbesolvedusingnon-additivemeasuresand
integralsandthatcannotbesolvedwithsomeofthealternativemodels.Section4
focuses on the use of these integrals on subjective evaluation. Section 5 reviews
someotherapplicationsine.g. the fieldofcomputervision.The chapterfinishes
with some conclusions.
Inthis chapterwealsoestablishsomelinks betweenthe topics describedhere
and the other chapters of this book.
2 Preliminaries
This section reviews a few definitions on probability, measures, aggregationop-
eratorsandintegrals.We startreviewingthe definition ofσ-algebra.The notion
of σ-algebra is introduced because for some sets X it is, in general, not possi-
ble to consider all subsets of X and define the probability for these subsets. In
particular, generally it is not possible when X is not finite. A σ-algebra is a set
of sets with appropriate properties to define the measure on them. See [26] for
details.
V.Torra,Y.Narukawa,andM.Sugeno(eds.),Non-Additive Measures, 1
StudiesinFuzzinessandSoftComputing310,
DOI:10.1007/978-3-319-03155-2_1, (cid:2)c SpringerInternationalPublishingSwitzerland2014