## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 23, Number 1 (1952), 88-102.

### An Application of Information Theory to Multivariate Analysis

#### Abstract

The problem considered is that of finding the "best" linear function for discriminating between two multivariate normal populations, $\pi_1$ and $\pi_2$, without limitation to the case of equal covariance matrices. The "best" linear function is found by maximizing the divergence, $J'(1, 2)$, between the distributions of the linear function. Comparison with the divergence, $J(1, 2)$, between $\pi_1$ and $\pi_2$ offers a measure of the discriminating efficiency of the linear function, since $J(1, 2) \geq J'(1, 2)$. The divergence, a special case of which is Mahalanobis's Generalized Distance, is defined in terms of a measure of information which is essentially that of Shannon and Wiener. Appropriate assumptions about $\pi_1$ and $\pi_2$ lead to discriminant analysis (Sections 4, 7), principal components (Section 5), and canonical correlations (Section 6).

#### Article information

**Source**

Ann. Math. Statist., Volume 23, Number 1 (1952), 88-102.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177729487

**Digital Object Identifier**

doi:10.1214/aoms/1177729487

**Mathematical Reviews number (MathSciNet)**

MR47297

**Zentralblatt MATH identifier**

0047.13503

**JSTOR**

links.jstor.org

#### Citation

Kullback, S. An Application of Information Theory to Multivariate Analysis. Ann. Math. Statist. 23 (1952), no. 1, 88--102. doi:10.1214/aoms/1177729487. https://projecteuclid.org/euclid.aoms/1177729487

#### See also

- Part II: S. Kullback. An Application of Information Theory to Multivariate Analysis, II. Ann. Math. Statist., Volume 27, Number 1 (1956), 122--146.Project Euclid: euclid.aoms/1177728353