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

10.1. Detection of Outliers Using Principal Components 243Table 10.1. Anatomical measurements: values of d 2 1i, d 2 2i, d 4i for the most extremeobservations.Number of PCs used, qq =1 q =2d 2 1i Obs. No. d 2 1i Obs. No. d 2 2i Obs. No. d 4i Obs. No.0.81 15 1.00 7 7.71 15 2.64 150.47 1 0.96 11 7.69 7 2.59 110.44 7 0.91 15 6.70 11 2.01 10.16 16 0.48 1 4.11 1 1.97 70.15 4 0.48 23 3.52 23 1.58 230.14 2 0.36 12 2.62 12 1.49 27q =3d 2 1i Obs. No. d 2 2i Obs. No. d 4i Obs. No.1.55 20 9.03 20 2.64 151.37 5 7.82 15 2.59 51.06 11 7.70 5 2.59 111.00 7 7.69 7 2.53 200.96 1 7.23 11 2.01 10.93 15 6.71 1 1.97 7observations on each statistic, where the number of PCs included, q, is1,2or 3. The observations that correspond to the most extreme values of d 2 1i ,d 2 2i and d 4i are identified in Table 10.1, and also on Figure 10.3.Note that when q = 1 the observations have the same ordering for allthree statistics, so only the values of d 2 1i are given in Table 10.1. When qis increased to 2 or 3, the six most extreme observations are the same (ina slightly different order) for both d 2 1i and d2 2i . With the exception of thesixth most extreme observation for q = 2, the same observations are alsoidentified by d 4i . Although the sets of the six most extreme observationsare virtually the same for d 2 1i , d2 2i and d 4i, there are some differences inordering. The most notable example is observation 15 which, for q =3,ismost extreme for d 4i but only sixth most extreme for d 2 1i .Observations 1, 7 and 15 are extreme on all seven statistics given in Table10.1, due to large contributions from the final PC alone for observation15, the last two PCs for observation 7, and the fifth and seventh PCs forobservation 1. Observations 11 and 20, which are not extreme for the finalPC, appear in the columns for q = 2 and 3 because of extreme behaviouron the sixth PC for observation 11, and on both the fifth and sixth PCsfor observation 20. Observation 16, which was discussed earlier as a clearoutlier on the second PC, appears in the list for q = 1, but is not notablyextreme for any of the last three PCs.