Uses Fisher’s linear discriminant analysis method to reduce the number of variables.

Routine DMSCR is a natural generalization of R.A. Fisher’s linear discrimination procedure for two groups. This method of discrimination obtains those linear combinations of the observed random variables that maximize the between‑groups variation relative to the within groups variation. Denote the first of these linear combinations by

where β1 is a column vector of coefficients of length NVAR and x is an observation to be classified. On the basis of one linear combination, the discriminant rule assigns the observation, z, to a group (characterized by the group mean) by minimizing the Euclidean distance between z and the group mean.

To obtain β1 (see, e.g., Tatsuoka 1971, page 158), let Sp denote the pooled within‑groups covariance matrix (Sp is defined and can be computed via routine DSCRM) and let Sb denote the between‑groups covariance matrix defined by

where g is the number of groups,

and N is the total number of observations used in computing COV. Then, β1, such that

can be computed as the maximum of

This yields β1 as the eigenvector associated with the largest eigenvalue from

Generally,

has rank m, where m = min(g ‑ 1, p) and p = NVAR.

has m such eigenvectors, and the matrix COEF is obtained as (β1, β2, …., βm), where each βi is an eigenvector.

The matrix CMEAN is taken as the within‑group means vector of the linear combinations zi defined by the β’s. For each observation x, scores

can be computed, because of the restriction on βi, the sample variance of the zi is 1.0. The observation is classified into the group (as specified by the group mean of the zi’s) to which, on the basis of the zi, the Euclidean distance is the least.

Note that the linear combinations zi have meaning even when discrimination is not desired. The linear combination of the observed variables that most separates the g groups is z1; z2, giving the second highest such separation orthogonal to the first, and so on. Thus, a plot of the mean vectors of the first two variables gives a good two‑dimensional summarization of the relationships between the groups.

The following example illustrates a typical sequence. Fisher’s iris data is used. (See routine GDATA, Chapter 19, “Utilities”). Routine DSCRM is first used to perform a discriminant analysis based on all the variables. COV, XMEAN, and NI are obtained from DSCRM. Function DMSCR, which uses these arrays, is then called.