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Principal Components Analysis - CMU Statistics

Chapter 18 Principal Components AnalysisPrincipal Components Analysis (PCA) is one of a family of techniques for takinghigh-dimensional data, and using the dependencies between the variables to representit in a more tractable, lower-dimensional form, without losing too much is one of the simplest and most robust ways of doing suchdimensionalityreduction. It is also one of the oldest, and has been rediscovered many times inmany fields, so it is also known as the Karhunen-Lo ve transformation, the Hotellingtransformation, the method of empirical orthogonal functions, and singular valuedecomposition1. We will call it Mathematics of Principal ComponentsWe start withp-dimensional vectors, and want to summarize them by projectingdown into aq-dimensional subspace. Our summary will be the projection of theoriginal vectors on toqdirections, theprincipal Components , which span the are several equivalent ways of deriving the Principal Components mathe-matically.

know that v is a covariance matrix, so it is symmetric, and then linear algebra tells us that the eigenvectors must be orthogonal to one another. Again because v is a covariance matrix, it is a positive matrix, in the sense that￿x ·v￿x ≥0 for any￿x. This …

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