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

128 6. Choosing a Subset of Principal Components or Variablesrally, and a number of rules for selecting a subset of PCs have been putforward with this context very much in mind. The LEV diagram, discussedin Section 6.1.3, is one example, as is Beltrando’s (1990) method in Section6.1.6, but there are many others. In the fairly common situation wheredifferent observations correspond to different time points, Preisendorfer andMobley (1988) suggest that important PCs will be those for which there isa clear pattern, rather than pure randomness, present in their behaviourthrough time. The important PCs can then be discovered by forming atime series of each PC, and testing which time series are distinguishablefrom white noise. Many tests are available for this purpose in the timeseries literature, and Preisendorfer and Mobley (1988, Sections 5g–5j) discussthe use of a number of them. This type of test is perhaps relevantin cases where the set of multivariate observations form a time series (seeChapter 12), as in many atmospheric science applications, but in the moreusual (non-meteorological) situation where the observations are independent,such techniques are irrelevant, as the values of the PCs for differentobservations will also be independent. There is therefore no natural orderingof the observations, and if they are placed in a sequence, they shouldnecessarily look like a white noise series.Chapter 5 of Preisendorfer and Mobley (1988) gives a thorough review ofselection rules used in atmospheric science. In Sections 5c–5e they discussa number of rules similar in spirit to the rules of Sections 6.1.3 and 6.1.4above. They are, however, derived from consideration of a physical model,based on spring-coupled masses (Section 5b), where it is required to distinguishsignal (the important PCs) from noise (the unimportant PCs). Thedetails of the rules are, as a consequence, somewhat different from thoseof Sections 6.1.3 and 6.1.4. Two main ideas are described. The first, calledRule A 4 by Preisendorfer and Mobley (1988), has a passing resemblance toBartlett’s test of equality of eigenvalues, which was defined and discussedin Sections 3.7.3 and 6.1.4. Rule A 4 assumes that the last (p−q) populationeigenvalues are equal, and uses the asymptotic distribution of the averageof the last (p − q) sample eigenvalues to test whether the common populationvalue is equal to λ 0 . Choosing an appropriate value for λ 0 introducesa second step into the procedure and is a weakness of the rule.Rule N, described in Section 5d of Preisendorfer and Mobley (1988) ispopular in atmospheric science. It is similar to the techniques of parallelanalysis, discussed in Sections 6.1.3 and 6.1.5, and involves simulating alarge number of uncorrelated sets of data of the same size as the real dataset which is to be analysed, and computing the eigenvalues of each simulateddata set. To assess the significance of the eigenvalues for the realdata set, the eigenvalues are compared to percentiles derived empiricallyfrom the simulated data. The suggested rule keeps any components whoseeigenvalues lie above the 95% level in the cumulative distribution of thesimulated data. A disadvantage is that if the first eigenvalue for the datais very large, it makes it difficult for later eigenvalues to exceed their own