Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

270 11. Rotation and Interpretation of Principal Componentssome threshold. There are links here to variable selection (see Section 6.3).This strategy and some of the problems associated with it are discussedin Section 11.3, and the chapter concludes with a short section on the desirein some disciplines to attach physical interpretations to the principalcomponents.11.1 Rotation of Principal ComponentsIn Chapter 7 it was seen that rotation is an integral part of factor analysis,with the objective of making the rotated factors as simple as possible tointerpret. The same ideas can be used to simplify principal components.A principal component is a linear function of all the p original variables.If the coefficients or loadings for a PC are all of a similar size, or if a feware large and the remainder small, the component looks easy to interpret,although, as will be seen in Section 11.3, looks can sometimes be deceiving.Several examples in Chapter 4 are like this, for instance, components 1 and2 in Section 4.1. If there are intermediate loadings, as well as large andsmall ones, the component can be more difficult to interpret, for example,component 4 in Table 7.1.Suppose that it has been decided that the first m components accountfor most of the variation in a p-dimensional data set. It can then be arguedthat it is more important to interpret simply the m-dimensional spacedefined by these m components than it is to interpret each individual component.One way to tackle this objective is to rotate the axes within thism-dimensional space in a way that simplifies the interpretation of the axesas much as possible. More formally, suppose that A m is the (p × m) matrixwhose kth column is the vector of loadings for the kth PC. Following similarsteps to those in factor analysis (see Section 7.2), orthogonally rotatedPCs have loadings given by the columns of B m , where B m = A m T,andT is a (m × m) orthogonal matrix. The matrix T is chosen so as to optimizeone of many simplicity criteria available for factor analysis. Rotationof PCs is commonplace in some disciplines, such as atmospheric science,where there has been extensive discussion of its advantages and disadvantages(see, for example Richman (1986, 1987); Jolliffe (1987b); Rencher(1995, Section 12.8.2)). Oblique, instead of orthogonal, rotation is possible,and this gives extra flexibility (Cohen, 1983; Richman, 1986). As noted inChapter 7, the choice of simplicity criterion is usually less important thanthe choice of m, and in the examples of the present chapter only the wellknownvarimax criterion is used (see equation (7.2.2)). Note, however, thatJackson (1991, Section 8.5.1) gives an example in which the results fromtwo orthogonal rotation criteria (varimax and quartimax) have non-trivialdifferences. He states that neither is helpful in solving the problem that hewishes to address for his example.