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August, 1972 Regression Optimality of Principal Components
R. L. Obenchain
Ann. Math. Statist. 43(4): 1317-1319 (August, 1972). DOI: 10.1214/aoms/1177692482


Consider $p \geqq 2$ random variables, and let $A_1, \cdots, A_p$ denote the hyperplanes corresponding to the linear regression of each variable onto the other $(p - 1)$ variables. Let $A_0$ denote the hyperplane which passes through the centroid of the distribution and is spanned by the direction vectors defining the first $(p - 1)$ principal components. A new optimality property of $A_0$ is established; $A_0$ is the best single approximation to $A_1, \cdots, A_p$ when each regression hyperplane is given a certain weighting inversely proportional to the variability associated with its orientation and its prediction rescaling. When $p > 2$ and $k = 1, \cdots, p - 2$, certain $k$-dimensional linear subspaces of $A_0$ are also shown to have regression optimality properties.


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R. L. Obenchain. "Regression Optimality of Principal Components." Ann. Math. Statist. 43 (4) 1317 - 1319, August, 1972.


Published: August, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0242.62030
MathSciNet: MR312646
Digital Object Identifier: 10.1214/aoms/1177692482

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 4 • August, 1972
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