Linear Principal Component Analysis
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.
KeywordsCovariance Matrix Random Vector Orthogonal Projection Sample Covariance Matrix Standard Distance
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Suggested Further Reading
- Jolicoeur, P., and Mosimann, J.E. 1960. Size and shape variation in the painted turtle: A principal component analysis. Growth 24, 339–354.Google Scholar
- Hills, M. 1982. Allometry, in Encyclopedia of Statistical Sciences, S. Kotz and N.L. Johnson, eds. New York: Wiley, pp. 48–54.Google Scholar
- Klingenberg, C.P. 1996. Multivariate allometry. In Advances in Morphometrics, L.F. Marcus, M. Corti, A. Loy, G.J.P. Naylor, and D.E. Slice, eds. New York: Plenum Press, pp. 23–49.Google Scholar