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Unformatted text preview: A TUTORIAL ON PRINCIPAL COMPONENT ANALYSIS Derivation, Discussion and Singular Value Decomposition Jon Shlens | jonshlens@ucsd.edu 25 March 2003 | Version 1 Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but poorly understood. The goal of this paper is to dispel the magic behind this black box. This tutorial focuses on building a solid intuition for how and why principal component analysis works; furthermore, it crystallizes this knowledge by deriving from first prin- cipals, the mathematics behind PCA . This tutorial does not shy away from explaining the ideas infor- mally, nor does it shy away from the mathematics. The hope is that by addressing both aspects, readers of all levels will be able to gain a better understand- ing of the power of PCA as well as the when, the how and the why of applying this technique. 1 Overview Principal component analysis ( PCA ) has been called one of the most valuable results from applied lin- ear algebra. PCA is used abundantly in all forms of analysis - from neuroscience to computer graphics- because it is a simple, non-parametric method of extracting relevant information from confusing data sets. With minimal additional effort PCA provides a roadmap for how to reduce a complex data set to a lower dimension to reveal the sometimes hidden, simplified dynamics that often underlie it. The goal of this tutorial is to provide both an intu- itive feel for PCA , and a thorough discussion of this topic. We will begin with a simple example and pro- vide an intuitive explanation of the goal of PCA . We will continue by adding mathematical rigor to place it within the framework of linear algebra and explic- itly solve this problem. We will see how and why PCA is intimately related to the mathematical tech- nique of singular value decomposition ( SVD ). This understanding will lead us to a prescription for how to apply PCA in the real world. We will discuss both the assumptions behind this technique as well as pos- sible extensions to overcome these limitations. The discussion and explanations in this paper are informal in the spirit of a tutorial. The goal of this paper is to educate . Occasionally, rigorous mathe- matical proofs are necessary although relegated to the Appendix. Although not as vital to the tutorial, the proofs are presented for the adventurous reader who desires a more complete understanding of the math. The only assumption is that the reader has a working knowledge of linear algebra. Nothing more. Please feel free to contact me with any suggestions, corrections or comments. 2 Motivation: A Toy Example Here is the perspective: we are an experimenter. We are trying to understand some phenomenon by mea- suring various quantities (e.g. spectra, voltages, ve- locities, etc.) in our system. Unfortunately, we can not figure out what is happening because the data appears clouded, unclear and even redundant. This is not a trivial problem, but rather a fundamental...
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This note was uploaded on 05/08/2010 for the course CS 6.345 taught by Professor Glass during the Spring '10 term at MIT.

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