Principal Component Analysis - Columbia University
Principal Component AnalysisFrank WoodDecember 8, 2009This 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 PDFUncorrelated 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.
PCA in a nutshell Notation I x is a vector of p random variables I k is a vector of p constants I 0 k x = P p j=1 kjx j Procedural description I Find linear function of x, 0 1x with maximum variance. I Next nd another linear function of x, 0 2x, uncorrelated with 0 1x maximum variance. I Iterate. Goal It is hoped, in general, that most of the variation in x will be
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