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

290 11. Rotation and Interpretation of Principal Componentscorrespond to measurements of the same quantity at p different spatial locationsand the n observations are taken at n different points in time (seeSection 12.2). Suppose that r jk is the correlation between the jth and kthvariables (spatial locations), and s 2 kis the sample variance at location k.Then the location j ∗ is found for which the criterion ∑ pk=1 r2 jk s2 kis maximized.The first EOT is then the vector whose elements are the coefficientsin separate regressions of each variable on the chosen variable j ∗ . The firstderived variable is not the linear combination of variables defined by thisEOT, but simply the time series at the chosen location j ∗ . To find a secondEOT, the residuals are calculated from the separate regressions that definethe first EOT, and the original analysis is repeated on the resulting matrixof residuals. Third, fourth, ... EOTs can be found in a similar way. At thisstage it should be noted that ‘correlation’ appears to be defined by van denDool et al. (2000) in a non-standard way. Both ‘variances’ and ‘covariances’are obtained from uncentred products, as in the uncentred version of PCA(see Section 14.2.3). This terminology is reasonable if means can be assumedto be zero. However, it does not appear that this assumption can bemade in van den Dool and co-workers’ examples, so that the ‘correlations’and ‘regressions’ employed in the EOT technique cannot be interpreted inthe usual way.Leaving aside for the moment the non-standard definition of correlations,it is of interest to investigate whether there are links between the procedurefor finding the optimal variable j ∗ that determines the first EOT and ‘principalvariables’ as defined by McCabe (1984) (see Section 6.3). Recall thatMcCabe’s (1984) idea is to find subsets of variables that optimize the samecriteria as are optimized by the linear combinations of variables that arethe PCs. Now from the discussion following Property A6 in Section 2.3 thefirst correlation matrix PC is the linear combination of variables that maximizesthe sum of squared correlations between the linear combination andeach of the variables, while the first covariance matrix PC similarly maximizesthe corresponding sum of squared covariances. If ‘linear combinationof variables’ is replaced by ‘variable,’ these criteria become ∑ p∑k=1 r2 jk andpk=1 r2 jk s2 k s2 j , respectively, compared to ∑ pk=1 r2 jk s2 kfor the first EOT. In asense, then, the first EOT is a compromise between the first principal variablesfor covariance and correlation matrices. However, the non-standarddefinitions of variances and correlations make it difficult to understandexactly what the results of the analysis represent.11.2.4 Some ComparisonsIn this subsection some comparisons are made between the two main techniquesdescribed in the section, and to other methods discussed earlierin the chapter. Further discussion of the properties of SCoT and SCoT-LASS can be found in Uddin (1999). Another approach to simplification isdescribed in Section 14.6.3.