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prc - STAT 760 October 3 2011 Lecture on principal...

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STAT 760, October 3, 2011 Lecture on principal components Setup: See text about perceived utility of dimension reduction schemes. They are widely used, though sometimes difficult to interpret, and not so easily motivated from any first-principles on the dynamics or statistical fluctuations, and Prof Newton rarely uses them! There are different ways to tell the story about principal components. Roughly, they extend to multiple dimensions the following observation (show a 2-d scatterplot of points, centered): that onto any line (through the origin) points in the sample can be projected (perpendicularly, as opposed to in linear regression), and that one can consider the difference vector d i = x i - ˆ x i between each point and its projection, and the total error i || d i || 2 , as a function of the line. It turns out that the line which minimizes the error in this approximation also maximizes the variability of the projected points, when these points are viewed in the topology of the line. Population perpective: We start the story from the perspective of population principal compo- nents , as many accounts do (e.g. our text, Section 7.2). Let X be a random vector in R p which, wlog, has mean 0. Our focus is on its covariance matrix Σ and with constructing linear combina- tions that themselves are uncorrelated and explain ever increasing amounts of the overall variation. Specifically, if we take a vector v R p and define the new random variable ξ = v t X , then ξ has mean 0 and variance v t Σ v . We consider only unit vectors v t v
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