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 descriptionIF
y = B0x where y is a q-element vector and B0is a q p matrix, and let y = B0B be the variance-covariance matrix for y. Then the trace of y, denoted tr(y), is maximized by taking B = A q; where A q consists of the rst q columns of A. What this means is that if …
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