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1、ComputationalStatistics&DataAnalysis50(2005)1452–1477www.elsevier.com/locate/csdaFuzzyunsupervisedclassificationofmultivariatetimetrajectorieswiththeShannonentropyregularizationRenatoCoppia,∗,PierpaoloD’UrsobaDipartimentodiStatistica,ProbabilitàeStatisticheApplicate,UniversitàdegliStudidiRoma“L
2、aSapienza”,P.leA.Moro,5-00185Rome,ItalybDipartimentodiScienzeEconomiche,GestionalieSociali,UniversitàdegliStudidelMolise,ViaDeSanctis,86100Campobasso,ItalyReceived14January2005;accepted14January2005AbstractFuzzyunsupervisedclusteringmodelsbasedonentropyregularizationaresuggestedinordertoclassi
3、fytime-varyingdata.Inparticular,intheproposedmodels,objectivefunctions,whicharethesumoftwoterms,areminimized.Thefirsttermisadynamicgeneralizationofintra-clusterdistance,inafuzzyframework,thattakesintoaccounttheinstantaneousand/orlongitudinalfeaturesofthetime-varyingobservations(theso-calledmult
4、ivariatetimetrajectories);inthisway,thewithinclusterdispersionisminimized(maximizetheinternalcohesion).ThesecondtermrepresentstheShannonentropymeasureasappliedtofuzzypartitions(entropyregularization);then,agivenmeasureofentropyismaximizedor,equivalently,theconverseoftheentropyisminimized.Overa
5、ll,thetotalfunctionaldependingonboththepreviousaspectsisoptimized.Thedynamicfuzzyentropyclus-teringmodelshavebeenappliedtoameteorologicaldatasetandanempiricalcomparisonwiththeinstantaneousand/orlongitudinalfuzzyC-meansclusteringmodelshasbeenmade.©2005ElsevierB.V.Allrightsreserved.Keywords:Mult
6、ivariatetime-varyingdata;Dynamicfuzzyclustering;Uncertainty;Shannonentropy;Entropyregularization;Maximumentropy∗Correspondingauthor.Tel.:+390649910731;fax:+39064959241.E-mailaddresses:renato.coppi@uniroma1.it(R.Coppi),durso@unimol.it,pierpaolo.durso@uniroma1.it(P.D’Urso).0167-9473/$-seefrontma
7、tter©2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.csda.2005.01.008R.Coppi,P.D’Urso/ComputationalStatistics&DataAnalysis50(2005)1452–147714531.IntroductionAbasicprincipleofclusteringtechniquesconsistsinsearchingforanappropriatepartiti