Linear Principal Component Analysis

  • Bernard Flury
Part of the Springer Texts in Statistics book series (STS)


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.


Covariance Matrix Random Vector Orthogonal Projection Sample Covariance Matrix Standard Distance 
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Suggested Further Reading

  1. 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.CrossRefGoogle Scholar
  2. Jolliffe, I.T. 1986. Principal Component Analysis. New York: Springer.CrossRefGoogle Scholar
  3. 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
  4. Rao, C.R. 1964. The use and interpretation of principal components in applied research. Sankhya A 26, 329–358.zbMATHGoogle Scholar
  5. Hastie, T., and Stuetzle, W. 1989. Principal curves. Journal of the American Statistical Association 84, 502–516.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Tarpey, T., and Flury, B. 1996. Self—consistency: A fundamental concept in statistics. Statistical Science 11, 229–243.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Hotelling, H. 1931. The generalization of Student’s ratio. Annals of Mathematical Statistics 2, 360–378.CrossRefGoogle Scholar
  8. Pearson, K. 1901. On lines and planes of closest fit to systems of points in space. Philosophical Magazine Ser. B 2, 559–572.CrossRefGoogle Scholar
  9. Hills, M. 1982. Allometry, in Encyclopedia of Statistical Sciences, S. Kotz and N.L. Johnson, eds. New York: Wiley, pp. 48–54.Google Scholar
  10. 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
  11. Anderson, T.W. 1963. Asymptotic theory for principal component analysis. Annals of Mathematical Statistics 34, 122–148.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Bernard Flury
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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