A First Course in Multivariate Statistics pp 453-562 | Cite as

# Discrimination and Classification, Round 2

Chapter

## 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, will be referred to as *pdf*of*pdf*of

**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)

*prior probabilities*, as usual. Suppose that the distribution of**Y**in the*j*th group is given by a*pdf f*_{ j }(**y**), which may be regarded as the conditional**Y**, given*X = j*. Assume for simplicity that*Y*is continuous with sample space ℝ^{ p }in each group. Then the joint*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.

## Suggested Further Reading

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