14.114.214.314.414.514.114.1“”“”“”14.1RuiXu200514.1“”“hardclustering”14.1nK“”KkPk,,1),(L=ωKkpkk,,1),,|(L=θωx∑==KkkkkPpp1)(),|()|(ωθωθxx14.1ndissimilarityKKn“”14.1jiiTiijjTijiijpdkpkjkijiijdkkjkijiijdTdiiinSCosddsMahalanobixxddMinkowskixxddRxxSxxxxMxMxxxxxxxxxx===⎟⎠⎞⎜⎝⎛−==⎟⎠⎞⎜⎝⎛−==∈==−==∑∑111211211),(),(),(),,(),,,(LL14.114.1HierarchicalClusteringHierarchicalClusteringPartitionalClusteringPartitionalClusteringNeuralNetworkbasedmethodsNeuralNetworkbasedmethodsSquarederrorbasedmethodsSquarederrorbasedmethodsCombinatorialsearchtechniquesCombinatorialsearchtechniquesTop-downDivisionTop-downDivisionBottom-upAgglomerativeBottom-upAgglomerativeOthermethodsOthermethodsFuzzy14.214.2——14.2∑∑===××=≠==≠=≤===∈==ninjjilkkKkkkkKnnjinnijdTdiiindLJlkCCSCCCnnKCCCddRxxS111111)),(,()(,)3)2)1),(},,{)),(()(D),,(),,,(xxxxxxxααφφIULLL14.2K-(K-meansC-means)14.2LDA∑∑∑∑∑∑∑=∈==∈====+=−−=−−=−−=KkkkniiCxikkbwtKkTkkkbTkiKkCkiwniTiitnnnnnkiki111111,1))(())(())((mxmxmSSSmmmmSmxmxSmxmxSx14.2)(max)(min)()()()()())(()(11bwbwtniiTiniTiittrtrtrtrtrtrtrSSSSSmxmxmxmxS+==−−=⎥⎦⎤⎢⎣⎡−−=∑∑==14.214.2∑∑∑∑∑∑∑∑∑∑=∈=∈=∈=∈=∈=−=−=−−=−−=KkCjikKkCjikKkCkiKkCkiTkiKkCTkikiwkjikjikikikidnntrtr1,21,21211),(2121)()(]))([()(xxxxxxxxxxxmxmxmxmxmxS14.2K-(K-means)∑∑∑∈=∈=−==kikiCikkKkCkiCwCCntrCJxxxmmxS1||||min)(min)(min1214.2“”Kkk,,1,L=m2,,12min,kiKkjijiifCmxmxx−=−∈=L14.2K-1.2.3.4.2-32,,12min,kiKkjijiifCmxmxx−=−∈=L14.2K-O(ndK)“”outlier14.2CkCl22||||1||||1lllkkknnnnJmxmx−++−−−=∆14.2K-(K-means)⎩⎨⎧∉∈=−=−==∑∑∑∑===∈kikikiKknikikiKkCkiCwCCCCtrCJkixxmxmxSx01||||min||||min)(min)(min11212θθθ14.2K-——kiuniuumuJkiKkkinKkiKKknikimkiMUMU0.,,1,1,)(],,[1,||||min),(min11112,,≥====−=∑∑∑=×==LLUmmMmxMU14.2Lagrange()∑∑∑∑∑∑∑=========−=∂∂−−−=nimkiniimkiknikimkikniKkkiiKknikimkinuuuJuuJ111111121)()(02)1(||||),,,,(xmmxmmxMUλλλL14.2()()∑∑∑∑∑∑∑=−−=−=−−====−−==⎟⎟⎠⎞⎜⎜⎝⎛−⇒===⎟⎟⎠⎞⎜⎜⎝⎛−==−−=∂∂−−−=KjmjimkikiKkmkiiKkkimkiikiikimkikiniKkkiiKknikimkinumuniKkmumuuJuuJ11121121112111221111121||||1||||11||||1,,1,,,1,||||0||||)1(||||),,,,(mxmxmxmxmxmxMUλλλλλλLLL14.2()())()()()(||||1||||11)1(1)1()1(1112)(112)()1(∑∑∑=+=++=−−+=−−=njmtkjnjjmtkjtkKjmtjimtkitkiuuuxmmxmx14.2())()()()(1,1,)(1)1()(1)()()1(1)1(1)1(1)1()1(1)()()()(1)()()()(tkjnjtkjtknjtktkjtkjjnjtkjnjmtkjnjjmtkjtknitkitkitktkinjtkjtkmtkitkiuuwwuwmxmmmxxxm−+=+−=======∑∑∑∑∑∑∑=+=+=+=+=++==∆∆ααααηαη14.2K-——K-∑∑∑∑∑∑======+−=−=−⎩⎨⎧∉∈==−=nljljklkjknjjikjkiinjjkjkikikikikiniikikkKknikikiCCKnKnKnCCnCJ1,212121112),(1),(2),(||)(1)(||||~)(||01,)(1~||~)(||min)(minxxxxxxxxmxxxxmmxθθθφθφφθφθφθ14.2K-K“”K14.2“”“”“”∑∑∈∈=−=kikiCiCikdxxxxxxm),(minarg||||minarg22∑∈∈=kikCiCkdxxxxm),(minarg214.2K-medoids1.medoids2.medoids3.medoids4.2-3∑∈∈≡kikCiCkdxxxxm),(minarg2()()kiKkjijiddifCmxmxx,min,,2,,12L==∈14.314.3——14.3x1,x2,x3,x4,x5x1,x2,x3x4,x5x1,x2x1x2x3x4x514.31.2.3.4.214.31.51.11.21.31.52.22.11.81.71.71.61.71.42.01.8x1x2x3x4x5x6x9x8x75.01.014.3——Single-Linkage),(min),(,minjiCClkdCCdljkixxxx∈∈=14.3Single-Linkagex1x2x3x4x5x6x7x8x91.01.11.21.31.41.61.75.014.3——Complete-Linkage),(max),(,maxjiCClkdCCdljkixxxx∈∈=14.3Complete-Linkagex1x2x3x4x5x6x7x8x91.01.11.41.51.62.02.214.3Average-Linkage——Average-Linkage∑∑∈∈=kiljCCjilklkaveragedCCCCdxxxx),(1),(14.3Average-Linkagex1x2x3x4x5x6x7x8x91.01.11.41.41.61.751.814.3Single-LinkageAverage-LinkageComplete-Linkage),(),(),(),(),(),(),(ljlijiljjliilkljiCCdCCdCCdCCdCCdCCdCCCd−+++==γβααU),(min),(,minjiCClkdCCdljkixxxx∈∈=∑∑∈∈=kiljCCjilklkaveragedCCCCdxxxx),(1),(),(max),(,maxjiCClkdCCdljkixxxx∈∈=14.3),(),()),(),,(min(),min(5.05.05.05.0,0,5.0minminminminlkljiljlijliljliljlilkljiCCdCCCdCCdCCdddddddd====−−+=−====Uγβαα14.3iCkCjCjikjilkCCCkjilCCdCCdU=≠∀≥,,),,(),(14.3),(),(),(),(),(),(),(ljlijiljjliilkljiCCdCCdCCdCCdCCdCCdCCCd−+++==γβααU1,0,0≥++≥≥βααααjiji{}0,max0≤≤−−≥cjiααγ14.3),(),(),()),(),(()),(),((),(),(0),(),(,0),(),()),(),(()),(),((),(),(),(),(),(),(),(1jiijlijiljjjiliijilkijilkljlijiljjjiliijiljlijiljjliilkjiCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCdCCd≥−+−+−+≥≤≤−≥≥−+−+−+≥−+++=−−≥γααγαγγααγβααααβ14.31.2.3.214.3O(n²)14.314.3——GGG(MinimumSpanningTree)14.31.11.21.31.61.71.4x1x2x3x4x5x6x9x8x75.01.014.31.2.14.3x1,x2,x3,x4,x5,x6,x7,x8,x9x1,x2,x3,x4,x5x6,x7,x8,x9x6,x7x8,x9x8x9x4,x5x1,x2,x3x1,x2x3x1x2x6x714.314.3{}{}TChdCChCdmedChCdChC=∈=∈=∑∈)(),(21)(,),,()(,),,(max)(,321yxyxyxyxyxyx14.31.51.11.21.31.52.22.11.81.71.71.61.71.42.01.8x1x2x3x4x5x6x9x8x75.01.054.14.15101)(5.1)(2.2)(131211=×===ChChCh275.12.1081)(7.1)(0.2)(232221=×===ChChCh14.3{}jijijiCCChChCCd,,)(),(max),(min∀14.3{}{}{}54.1)(),(max7.1)(),(max2.2)(),(max0.5),(23132212211121min====ChChChChChChCCd1.51.11.21.31.52.22.11.81.61.71.61.71.42.01.8x1x2x3x4x5x6x9x8x75.01.014.414.414.414.514.514.5[1]A.K.Jain,M.N.MurtyandP.J.Flynn,Dataclustering:areview.ACMComputingSurveys,Vol.31,Issue3,Sept.1999.[2]DanielFasulo,Ananalysisofrecentworkonclusteringalgorithms.TechnicalReport01-03-02,UniversityofWashington,April1999.[3]RuiXuandDonaldWunschII,Surveyofclusterin