12 - Multivariate data exploration and discrimination

Summary

Earlier chapters have made extensive use of exploratory graphs that have examined variables and their pair wise relationships, as a preliminary to regression modeling. Scatter plot matrices have been an important tool, and will be used in this chapter also. The focus will move from regression to methods that seek insight into the pattern presented by multiple variables. While the methodology has applications in a regression context, this is not a primary focus.

There are a number of methods that project the data on to a low-dimensional space, commonly two dimensions, suggesting “views” of the data that may be insightful. In the absence of other sources of guidance, it is reasonable to begin with views that have been thus suggested. One of the most widely used methods for projecting onto a low-dimensional space is principal components analysis (PCA).

The PCA form of mathematical representation has applications in many contexts beyond those discussed here. As used here, PCA is a special case of a much wider class of multidimensional scaling (MDS) methods. Subsection 12.1.3 is a brief introduction to this wider class of methods.

Principal components analysis replaces the input variables by new derived variables, called principal components. The analysis orders the principal components according to the amounts that they contribute to the total of the variances of the original variables.