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

408 Appendix A. Computation of Principal Componentsprogramming environment within which new statistical techniques can beimplemented. PCA is also included in some neural network software.All the main statistical software packages incorporate procedures for findingthe basic results of a PCA. There are some variations in output, such asthe choice of normalization constraints used for the vectors of loadings orcoefficients. This can cause confusion for the unwary user (see Section 11.1),who may also be confused by the way in which some software erroneouslytreats PCA as a special case of factor analysis (see Chapter 7). However,even with this misleading approach, numerically correct answers are producedby all the major statistical software packages, and provided that theuser is careful to ensure that he or she understands the details of how theoutput is presented, it is usually adequate to use whichever software is mostreadily available.Most statistical packages produce the basic results of a PCA satisfactorily,but few provide much in the way of extensions or add-ons as standardfeatures. Some allow (or even encourage) rotation, though not necessarilyin a sufficiently flexible manner to be useful, and some will display biplots.With most it is fairly straightforward to use the output from PCA in anotherpart of the package so that PC regression, or discriminant or clusteranalysis using PCs instead of the measured variables (see Chapters 8 and9) are easily done. Beyond that, there are two possibilities for many of theextensions to PCA. Either software is available from the originator of thetechnique, or extra functions or code can be added to the more flexiblesoftware, such as S-PLUS or R.A.1 Numerical Calculation of PrincipalComponentsMost users of PCA, whether statisticians or non-statisticians, have littledesire to know about efficient algorithms for computing PCs. Typically, astatistical program or package can be accessed that performs the analysisautomatically. Thus, the user does not need to write his or her own programs;often the user has little or no interest in whether or not the softwareavailable performs its analyses efficiently. As long as the results emerge, theuser is satisfied.However, the type of algorithm used can be important, in particular ifsome of the last few PCs are of interest or if the data set is very large.Many programs for PCA are geared to looking mainly at the first few PCs,especially if PCA is included only as part of a factor analysis routine. Inthis case, several algorithms can be used successfully, although some willencounter problems if any pairs of the eigenvalues are very close together.When the last few or all of the PCs are to be calculated, difficulties are morelikely to arise for some algorithms, particularly if some of the eigenvaluesare very small.