Abstract
Principal component analysis is a classical multivariate technique dating back to publications by Pearson (1901) and Hotelling (1933). Pearson focused on the aspect of approximation: Given a p-variate random vector (or a “system of points in space,” in Pearson’s terminology), find an optimal approximation in a linear subspace of lower dimension. More specifically, Pearson studied the problem of fitting a line to multivariate data so as to minimize the sum of squared deviations of the points from the line, deviation being measured orthogonally to the line. We will discuss Pearson’s approach in Section 8.3; however, it will be treated in a somewhat more abstract way by studying approximations of multivariate random vectors using the criterion of mean-squared error.
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Suggested Further Reading
Airoldi, J.-P., and Flury, B. 1988. An application of common principal component analysis to cranial morphometry of Microtus californicus and M. ochrogaster (Mammalia, Rodentia). Journal of Zoology (London) 216, 21–36. With discussion and rejoinder pp. 41–43.
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Hotelling, H. 1931. The generalization of Student’s ratio. Annals of Mathematical Statistics 2, 360–378.
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Anderson, T.W. 1963. Asymptotic theory for principal component analysis. Annals of Mathematical Statistics 34, 122–148.
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Flury, B. (1977). Linear Principal Component Analysis. In: A First Course in Multivariate Statistics. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2765-4_8
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DOI: https://doi.org/10.1007/978-1-4757-2765-4_8
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