Multiple Discriminant Analysis
DESCRIPTION:
Finds the linear discriminant function in order to distinguish between a number of groups.
USAGE: 
discr(x, k)
REQUIRED ARGUMENTS:
x:
matrix of data. The rows of x must be ordered by groups. The rows represent observations, and the columns represent variables. Missing values are not accepted.
k:
either the number of groups (if groups are equal in size), or the vector of group sizes; i.e., first k[1] rows form group 1, next k[2] group 2, etc. Missing values are not accepted.
VALUE:
a list describing the discriminant analysis, with the following components:
cor:
vector of discriminant correlations (correlations between linear combinations of the variables and of the groups).
vars:
square matrix (of size ncol(x)) of linear combinations of variables. The columns are the linear combinations of the input variables, i.e., x %*% vars produces the matrix of discriminant variables.
groups:
square matrix (of size the number of groups) of linear combinations of groups predicted. The columns (except the last) are contrasts such that the correlation between the ith contrast of groups (represented for the data by a vector of length nrow(x) with values from group[,i]) and the ith discriminant variable is the ith element of cor.
DETAILS:
The method that discr implements is linear discriminant analysis. This was devised by R. A. Fisher as a "sensible" way of distinguishing between groups. The first discriminant function (represented by vars[,1]) is the linear function of the variables that maximizes the ratio of the between-group sum of squares to the within-group sum of squares. The second discriminant function is the linear combination that is uncorrelated with (but not necessarily orthogonal to) the first which has the same optimality criterion. The third and later discriminant functions are defined analogously. The greatest possible number of linear discriminant functions is min(ncol(x), k-1) where k is the number of groups. This type of analysis is optimal if the groups are all distributed multivariate normal with the same variance matrix.

An observation can be classified by computing its Euclidean distance from the group centroids, projected onto a subspace defined by a subset of the canonical variates. The observation is assigned to the closest group.

The cor component of the output is a measure of the ability to discriminate between the groups.

BACKGROUND:
Discriminant analysis, in general, is a process of dividing up ncol(x) dimensional space into k pieces such that the groups are as distinct as possible. While cluster analysis seeks to divide the observations into groups, discriminant analysis presumes the groups are known and seeks to understand what makes them different.

Multivariate planing can be used as an exploratory method to see if (non-linear) discrimination is possible, see mstree.

REFERENCES:
Discriminant analysis is discussed in many multivariate statistics books, among which are:

Dillon, W. R. and Goldstein, M. (1984). Multivariate Analysis, Methods and Applications. Wiley, New York.

Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations. Wiley, New York.

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.

SEE ALSO:
cbind , hclust , mstree , prcomp , rbind .
EXAMPLES: 
# discrimination using a grouping variable
discr.group <- function(x, group) {
      size <- table(category(sort(group))
      discr(x[order(group),], size)
}


# a discrimination analysis of the iris data
iris.var <- rbind(iris[,,1], iris[,,2], iris[,,3])
iris.dis <- discr(iris.var, 3)
iris.dv <- iris.var %*% iris.dis$vars
brush(cbind(iris.dv, rep(1:3, c(50, 50, 50))))


iris.x <-   iris.dv[,1] ; iris.y <- iris.dv[,2]
iris.lab <- c(rep("S", 50), rep("C", 50), rep("V", 50))
plot(iris.x, iris.y, type = "n", xlab = "first discriminant variable",
     ylab = "second discriminant variable")
     text(iris.x, iris.y, iris.lab)