Decomposition of Data Matrices by Factors

It is of the highest importance in the art of detection to be able to recognize, out of a number of facts, which are incidental and which vital. Otherwise your energy and attention must be dissipated instead of being concentrated. Sherlock Holmes in “The Reigate Puzzle” In this chapter, we take a descriptive perspective and show how using a geometrical approach can be a good way to reduce the dimension of a data matrix. We derive the interesting projections with respect to a least-squares criterion. The results will be low-dimensional graphical pictures of the data matrix. This involves the decomposition of the data matrix into factors. These factors will be sorted in decreasing order of importance. The approach is very general and is the core idea of many multivariate techniques. We deliberately use the word “factor” here as a tool or transformation for structural interpretation in an exploratory analysis.