Principal Components Analysis

Abstract

In moving from multidimensional scaling to principal components analysis, we shift from the simplest and most commonsense approach conceptually to the most abstract and mathematical. Mastering the mathematical fundamentals of principal components analysis is a lot of work – work that does not finally bring much payback in making it easier to perform more reliable or successful analyses. In keeping with the overall approach of this book, we will give short shrift to the abstract mathematical fundamentals of principal components analysis and concentrate our attention on understanding its principles and concepts in ways that provide surer guides to effective use of the technique. This approach is very different from the one that is usually taken to the subject. Nevertheless, more archaeologists seem able to understand the principles of principal components analysis more readily, more deeply, and to better effect through such a commonsense approach than through an abstract mathematical explanation. Understanding and effective use of multidimensional scaling does not require much knowledge of how the iterative trial-and-error procedure that produces the configuration is programmed. In similar fashion, what principal components are and how they tell us about patterning in a multivariate dataset can be understood effectively without much knowledge of the particular mathematics that produce them. Principal components analysis is often confused with factor analysis. Opinion is divided about how much this confusion matters. There certainly are distinctions between the underlying logic of the two analytical techniques. On the other hand, their results are presented and interpreted in precisely the same way. At the practical level, it is extremely unusual to carry out the two analyses on the same data and get very different results. Not surprisingly, statpacks tend to have a focus on the practical, and principal components analysis and factor analysis are often combined into one set of routines where the choice between the two is simply one of the options to set. The difference between them certainly matters little to the commonsense approach of this chapter. The vocabulary we will use will be that of principal components analysis, but in actual fact, this chapter could just as easily be a chapter on factor analysis. Virtually the only difference would be to replace the words “principal component” or “component” with “factor.”