Advertisement

Discrimination and Classification, Round 2

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

Abstract

In this chapter we continue the theory of classification developed in Chapter 5 on a somewhat more general level. We start out with some basic consideration of optimality. In the notation introduced in Section 5.4, Y will denote a p-variate random vector measured in k groups (or populations). Let X denote a discrete random variable that indicates group membership, i.e., takes values 1, … , k. The probabilities
$$ {\pi _j} = \Pr \left[ {X = j} \right]\quad j = 1, \ldots ,k, $$
(1)
will be referred to as prior probabilities, as usual. Suppose that the distribution of Y in the jth group is given by a pdf f j (y), which may be regarded as the conditional pdf of Y, given X = j. Assume for simplicity that Y is continuous with sample space ℝ p in each group. Then the joint pdf of X and Y, as seen from Sec tion 2.8, is
$$ {f_{XY}}\left( {j,y} \right) = \left\{ \begin{gathered} {\pi _j}{f_j}\left( y \right)\;for{\kern 1pt} j = 1, \ldots ,k,y \in {\mathbb{R}^p} \hfill \\ 0\quad otherwise. \hfill \\ \end{gathered} \right. $$
(2)

Keywords

Covariance Matrice Canonical Variate Classification Rule Classification Region Standard Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Suggested Further Reading

  1. Everitt, B.S., and Hand, D.J. 1981. Finite Mixture Distributions. London: Chapman and Hall.zbMATHCrossRefGoogle Scholar
  2. McLachlan, G.J. 1992. Discriminant Analysis and Statistical Pattern Recognition. New York: Wiley.CrossRefGoogle Scholar
  3. Anderson, T.W. 1984. An Introduction to Multivariate Statistical Analysis, 2nd ed. New York: Wiley.zbMATHGoogle Scholar
  4. Krzanowski, W.J. 1988. Principles of Multivariate Analysis: A User’s Perspective. Oxford: Clarendon Press.zbMATHGoogle Scholar
  5. Bartlett, M.S., and Please, N.W. 1963. Discrimination in the case of zero mean differences. Biometrika 50, 17–21.MathSciNetzbMATHGoogle Scholar
  6. Bensmail, H., and Celeux, G. 1996. Regularized discriminant analysis through eigenvalue decomposition. Journal of the American Statistical Association 91, 1743–1748.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Flury, L., Boukai, B., and Flury, B. 1997. The discrimination subspace model. Journal of the American Statistical Association 92, in press.Google Scholar
  8. Friedman, J.H. 1989. Regularized discriminant analysis. Journal of the American Statistical Association 84, 165–175.MathSciNetCrossRefGoogle Scholar
  9. Fisher, R.A. 1936. The use of multiple measurements in taxonomic problems. Annals of Eugenics 7, 179–188.CrossRefGoogle Scholar
  10. Johnson, R.A., and Wichern, D.W. 1988. Applied Multivariate Statistical Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice—Hall.zbMATHGoogle Scholar
  11. Hand, D.J., and Taylor, C.C. 1987. Multivariate Analysis of Variance and Repeated Measures. London: Chapman and Hall.CrossRefGoogle Scholar
  12. Rencher, A.C. 1995. Methods of Multivariate Analysis. New York: Wiley.zbMATHGoogle Scholar
  13. Dobson, A.J. 1990. An Introduction to Generalized Linear Models. London: Chapman and Hall.zbMATHGoogle Scholar
  14. Efron, B. 1975. The efficiency of logistic regression compared to normal discriminant analysis. Journal of the American Statistical Association 70, 892–898.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hosmer, D.W., and Lemeshow, S. 1989. Applied Logistic Regression. New York: Wiley.Google Scholar
  16. Lavine, M. 1991. Problems in extrapolation illustrated with space shuttle O-ring data. Journal of the American Statistical Association 86, 919–922.CrossRefGoogle Scholar
  17. McCullagh, P., and Nelder, J.A. 1989. Generalized Linear Models, 2nd ed. London: ChapmanandHall.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

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

Personalised recommendations