Transcription of Principal Component Analysis - Columbia University
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Principal Component Analysis - Columbia University
Principal Component AnalysisFrank WoodDecember 8, 2009 This lecture borrows andquotesfrom Joliffe s Principle Component Analysis book. Go buy it! Principal Component AnalysisThe central idea of Principal Component Analysis (PCA) isto reduce the dimensionality of a data set consisting of alarge number of interrelated variables, while retaining asmuch as possible of the variation present in the data is achieved by transforming to a new set of variables,the Principal components (PCs), which are uncorrelated,and which are ordered so that the firstfewretain most ofthe variation present inallof the original variables.[Jolliffe, Pricipal Component Analysis ,2ndedition]Datadistribution (inputs in regression Analysis )Figure: Gaussian PDFU ncorrelated projections of Principal variationFigure: Gaussian PDF with PC eigenvectorsPCA rotationFigure: PCA Projected Gaussian PDFPCA in a nutshellNotationIxis a vector ofprandom variablesI kis a vector ofpconstantsI kx= pj=1 kjxjProcedural descriptionIFind linear function ofx, 1xwith maximum find another linear function ofx, 2x, uncorrelated with 1xmaximum is hoped, in general, that most of the variation inxwil
Constrained maximization - method of Lagrange multipliers I If we recognize that the quantity to be maximized 0 k = 0 k = 0 k = then we should choose k to be as big as possible. So, calling 1 the largest eigenvector of and 1 the corresponding eigenvector then the solution to 1 = 1 1 is the 1st principal component of x. I In general
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