Non-Additive Measures: Theory and Applications PDF
Preview Non-Additive Measures: Theory and Applications
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 developmentsin the field, both ofa foundationaland applicablecharacter.An impor- 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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown orhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnectionwithreviews orscholarly analysis ormaterial suppliedspecifically forthepurposeofbeingentered andexecuted ona computersystem,forexclusive usebythepurchaser ofthework.Duplication ofthis publication orparts thereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation,initscur- rentversion,andpermissionforusemustalways beobtained fromSpringer. Permissionsforusemaybe obtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliabletoprosecutionunder therespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication, neither the authors northe editors nor the publisher can accept any legal responsibility for any errors or omissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothematerial containedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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
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