第九章+無監(jiān)督學習與半監(jiān)督學習-Clustering

JiafengGuoUnsupervisedLearning——ClusteringOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringSupervisedvs.UnsupervisedLearning

WhydoUnsupervisedLearning?Rawdatacheap.Labeleddataexpensive.Savememory/computation.Reducenoiseinhigh-dimensionaldata.Usefulinexploratorydataanalysis.Oftenapre-processingstepforsupervisedlearning.Discovergroupssuchthatsampleswithinagrouparemoresimilartoeachotherthansamplesacrossgroups.ClusterAnalysisAvariablecanbeunobserved(latent).Itisanimaginaryquantitymeanttoprovidesomesimplifiedandabstractiveviewofthedatagenerationprocess.E.g.,speechrecognitionmodels,mixturemodelsItisareal-worldobjectand/orphenomena,butdifficultorimpossibletomeasure.E.g.,thetemperatureofastar,causesofadisease,evolutionaryancestorsItisareal-worldobjectand/orphenomena,butsometimeswasnotmeasured,becauseoffaultysensors;orwasmeasurewithanoisychannel,etc.E.g.,trafficradio,aircraftsignalonaradarscreenDiscretelatentvariablescanbeusedtopartition/clusterdataintosub-groups.Continuouslatentvariablescanbeusedfordimensionalityreduction.UnobservedVariablesOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringImageSegmentation/pff/segmentHumanPopulationEranElhaiketal.NatureClusteringGraphsNewman,2008VectorquantizationtocompressimagesBishop,PRMLAdissimilarity/distancefunctionbetweensamples.Alossfunctiontoevaluateclusters.Algorithmthatoptimizesthislossfunction.IngredientsofclusteranalysisOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringChoiceofdissimilarity/distancefunctionisapplicationdependent.Needtoconsiderthetypeoffeatures.Categorical,ordinalorquantitative.Possibletolearndissimilarityfromdata.Dissimilarity/DistanceFunction

DistanceFunction

StandardizationWithoutstandardizationWith

standardizationStandardizationnotalwayshelpfulWithoutstandardizationWith

standardizationOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringPerformanceEvaluationofClustering:ValidityindexEvaluationmetrics:referencemodel(externalindex)comparewithreferencenon-referencemodel(internalindex)measuredistanceofinner-classandinter-classEvaluationofClustering

ReferenceModelm(m-1)/2referencesamenotclusteringsameabnotcd

ExternalIndexOnlyhavingresultofclustering,howcanweevaluateit?Intra-clustersimilarity:largerisbetterInter-clustersimilarity:smallerisbetterNon-referencemodel

Non-referencemodel

InternalIndex

OutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringK-means:

Idea

HowdoweminimizeJw.r.t(rik,uk)?ChickenandeggproblemIfprototypesknown,canassignresponsibilitiesIfresponsibilitiesknown,cancomputeprototypesWeuseaniterativeprocedureK-means:minimizingthelossfunction

K-meansAlgorithmsSomeheuristicsRandomlypickKdatapointsasprototypesPickprototypei+1tobethefarthestfromprototypes{1,2….i}HowdoweinitializeK-means?Evolutionofk-Means(a)originaldataset;(b)randominitialization;(c-f)illustrationofrunningtwoiterationsofk-means.(ImagesfromMichaelJordan)LossfunctionJaftereachiterationk-meansisexactlycoordinatedescentonthereconstructionerrorE.Emonotonicallydecreases,andthevalueofEconverges,sodotheclusteringresults.Itispossiblefork-meanstooscillatebetweenafewdifferentclusterings,butthisalmostneverhappensinpractice.Eisnon-convex,socoordinatedescentonEcannotguaranteedtoconvergetoglobalminimum.Onecommonthingtodoisrunningk-meansmanytimesandpickthebestone.ConvergenceofK-meansLikechoosingKinkNN.ThelossfunctionJgenerallydecreaseswithK.HowtochooseK?HowtochooseK?GapstatisticCross-validation:Partitiondataintotwosets.Estimateprototypesononeandusethesetocomputethelossfunctionontheother.Stabilityofclusters:Measurethechangeintheclustersobtainedbyresamplingorsplittingthedata.Non-parametricapproach:PlaceaprioronK.MoredetailsintheBayesiannon-parametriclecture.Hardassignmentsofdatapointstoclusterscancauseasmallperturbationtoadatapointtoflipittoanothercluster.Solution:GMMAssumessphericalclustersandequalprobabilitiesforeachcluster.Solution:GMMClusterschangearbitrarilyfordifferentK.Solution:HierarchicalclusteringSensitivetooutliers.Solution:Usearobustlossfunction.Workspoorlyonnon-convexclusters.Solution:Spectralclustering.LimitationsofK-meansOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringMultivariateNormalDistribution

GaussianMixtureModel

TheLearningisHard

HowtoSolveit?

TheExpectation-Maximization(EM)AlgorithmAverygeneraltreatmentoftheEMalgorithm,andIntheprocessprovideaproofthattheEMalgorithmderivedheuristicallybeforeforGaussianmixturesdoesindeedmaximizethelikelihoodfunction,andThisdiscussionwillalsoformthebasisforthederivationofthevariationalinferenceframeworkTheEMAlgorithminGeneral

TheEMAlgorithminGeneralTheEMAlgorithminGeneral

Maximizingoverq(Z)wouldgivethetrueposteriorEM:VariationalViewpointEStepMStep

TheEMAlgorithm

InitialConfiguratinE-StepM-StepTheEMAlgorithmTheEMAlgorithm

Convergence

GMM:RelationtoK-meansIllustrationK-meansvsGMMLossfunction:minimizesumofsquareddistance.Hardassignmentofpointstoclusters.Assumessphericalclusterswithequalprobabilityofacluster.Minimizenegativeloglikelihood.Softassignmentofpointstoclusters.Canbeusedfornon-sphericalclusterswithdifferentprobabilities.K-meansGMMOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringSquaredEuclideandistancelossfunctionofK-meansnotrobust.Onlythedissimilaritymatrixmaybegiven.Attributesnotquantitative.K-medoids

K-medoidsOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClusteringOrganizetheclustersinahierarchicalway.Producesarootedbinarytree(dendrogram).HierarchicalClusteringHierarchicalClusteringBottom-up(agglomerative):Recursivelymergetwogroupswiththesmallestbetween-clustersimilarity.Top-down(divisive):Recursivelysplitaleast-coherent(e.g.largestdiameter)cluster.Userscanthenchooseacutthroughthehierarchytorepresentthemostnaturaldivisionintoclusters(e.g.whereintergroupsimilarityexceedssomethreshold).

HierarchicalClusteringOutlineIntroductionApplicationsofClusteringDistanceFunctionsEvaluationMetricsClusteringAlgorithmsK-meansGaussianMixtureModelsandEMAlgorithmK-medoidsHierarchicalClusteringDensity-basedClustering

DBSCAN1Esteretal.Adensity-basedalgorithmfordiscoveringclustersinlargespatialdatabaseswithnoise.ProceedingsoftheSecondInternationalConferenceonKnowledgeDiscoveryandDataMining(KDD).1996.Twopointspandqaredensity-connectedifthereisapointosuchthatbothpandqarereachablefromoAclustersatisfiestwoproperties:Allpointswithintheclusteraremutuallydensity-connected;Ifapointisdensity-reachablefromanypointofthecluster,itispartoftheclusteraswellDBSCAN

DBSCANAdvantagesNotneedtospecifythenumberofclustersArbitraryshapeclusterRobusttooutliersDisadvantagesDifficultparameterselectionNotproperfordatasetswithlargedifferencesindensitiesAnalysisofDBSCAN

Mean-ShiftClustering2Fukunaga,Keinosuke;LarryD.Hostetler.TheEstimationoftheGradientofaDensityFunction,withApplicationsinPatternRecognition.IEEETransactionsonInformationTheory21(1):32–40.Jan.1975.Cheng,Yizong.Me