Team-H

Discriminant Function Analysis

Discriminant Analysis may be used for two objectives:

·         Either   we want to assess the adequacy of classification, given the group memberships of the objects under study;

·         We wish to assign objects to one of a number of (known) groups of objects. Discriminant Analysis may thus have a descriptive or a predictive objective.

In both cases, some group assignments must be known before carrying out the Discriminant Analysis. Such group assignments, or labeling, may be arrived at in any way. Hence Discriminant Analysis can be employed as a useful complement to Cluster Analysis (in order to judge the results of the latter) or Principal Components Analysis. Alternatively, in star-galaxy separation, for instance, using digitised images, the analyst may define group (stars, galaxies) membership visually for a conveniently small training set or design set.  

Multiple Discriminant Analysis

(MDA) is also termed Discriminant  Factor Analysis and Canonical Discriminant Analysis.The rows of the data matrix to be examined constitute points in a multidimensional space, as also do the group mean vectors. Discriminating axes are determined in this space, in such a way that optimal separation of the predefined groups is attained.Tthe problem becomes mathematically the eigenreduction of a real, symmetric matrix. The eigenvalues represent the discriminating power of the associated eigenvectors. The n_Ygroups lie in a space of dimension at most n_Y - 1. This will be the number of discriminant axes or factors obtainable in the most common practical case when n > m > n_Y (where n is the number of rows, and m the number of columns of the input data matrix).

 

Things to consider

• Multivariate   normal distribution assumptions hold for the response variables.  This means that each of the dependent variables is normally distributed within groups, that any linear combination of the dependent variables is normally distributed, and that all subsets of the variables must be multivariate normal. 
• Each group must have a sufficiently large number of cases.
• Different classification methods may be used depending on whether the variance-covariance matrices are equal (or very similar) across groups.
• Non-parametric discriminant function analysis, called k^th nearest neighbor, can also be performed.