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1、ATutorialonPrincipalComponentAnalysisJonathonShlens?CenterforNeuralScience,NewYorkUniversityNewYorkCity,NY10003-6603andSystemsNeurobiologyLaboratory,SalkInsituteforBiologicalStudiesLaJolla,CA92037(Dated:April22,2009;Version3.01)Principalcomponentanalysis(PCA)isamainstayofmoder
2、ndataanalysis-ablackboxthatiswidelyusedbut(sometimes)poorlyunderstood.Thegoalofthispaperistodispelthemagicbehindthisblackbox.Thismanuscriptfocusesonbuildingasolidintuitionforhowandwhyprincipalcomponentanalysisworks.Thismanuscriptcrystallizesthisknowledgebyderivingfromsimpleint
3、uitions,themathematicsbehindPCA.Thistutorialdoesnotshyawayfromexplainingtheideasinformally,nordoesitshyawayfromthemathematics.Thehopeisthatbyaddressingbothaspects,readersofalllevelswillbeabletogainabetterunderstandingofPCAaswellasthewhen,thehowandthewhyofapplyingthistechnique.
4、I.INTRODUCTIONII.MOTIVATION:ATOYEXAMPLEPrincipalcomponentanalysis(PCA)isastandardtoolinmod-Hereistheperspective:weareanexperimenter.Wearetryingerndataanalysis-indiverse?eldsfromneurosciencetocom-tounderstandsomephenomenonbymeasuringvariousquan-putergraphics-becauseitisasimple,
5、non-parametricmethodtities(e.g.spectra,voltages,velocities,etc.)inoursystem.forextractingrelevantinformationfromconfusingdatasets.Unfortunately,wecannot?gureoutwhatishappeningbe-WithminimaleffortPCAprovidesaroadmapforhowtore-causethedataappearsclouded,unclearandevenredundant.d
6、uceacomplexdatasettoalowerdimensiontorevealtheThisisnotatrivialproblem,butratherafundamentalobstaclesometimeshidden,simpli?edstructuresthatoftenunderlieit.inempiricalscience.Examplesaboundfromcomplexsys-temssuchasneuroscience,webindexing,meteorologyandThegoalofthistutorialisto
7、providebothanintuitivefeelforoceanography-thenumberofvariablestomeasurecanbePCA,andathoroughdiscussionofthistopic.Wewillbeginunwieldyandattimesevendeceptive,becausetheunderlyingwithasimpleexampleandprovideanintuitiveexplanationrelationshipscanoftenbequitesimple.ofthegoalofPCA.
8、Wewillcontinuebyaddingmathemati-calrigortoplaceitwithinthefra
