Discriminant analysis's method of classification works by forming a discriminant function. This discriminant function creates borders between groups of data points. When a new data point is input, the point falls between a set of borders and is "classified" as being in that group. For actual classification purposes, discriminant coefficients (the coefficients of the discriminant function) are enough to apply discriminant analysis.
Classification in discriminant analysis draws similarities to regression analysis. Regression analysis usually takes in variables in terms of real numbers. Its output is also a real number. The objective of regression analysis is to use the variables to "predict" the real number as output. Discriminant analysis performs by a similar means: It takes in variables in the same way as that of regression analysis, but the output is no longer a real number. The objective of discriminant analysis is to use the variables to "classify" the data point into a single group.
There are two main methods of classification in discriminant analysis--Fisher's and Mahalanobis's. While the overall process of discriminant analysis remains the same, the choice of using Fisher's or Mahalanobis's method in the process of classifying data points can affect the classifications to some degree. Fisher's approach attempts to maximize the ratio of the between-group variation in discriminant scores as given by the discriminant function to the within-group variation. Mahalanobis's method, on the other hand, is to calculate the covariance-adjusted distance from a data point to the centroid of each group; the group with the centroid that yields the smallest distance is then defined as the parent of the data point.
The output of any application of any form of discriminant analysis will yield a classification for a data point. The number of groups can be chosen by the user. The classification in the two-group form of discriminant analysis is more simple to apply and yields easy-to-interpret graphs. In the cases of three or more classification possibilities, the method of discriminant analysis has a greater effect. In fact, in the two-group case, Fisher's and Mahalanobis's approaches will be equivalent.