Page 276 - Jolliffe I. Principal Component Analysis

10.1. Detection of Outliers Using Principal Components
2
Table 10.1. Anatomical measurements: values of d 1i , d 2i , d 4i for the most extreme
observations.
Number of PCs used, q 2 243
q =1 q =2
d 2 Obs. No. d 2 Obs. No. d 2 Obs. No. Obs. No.
1i 1i 2i d 4i
0.81 15 1.00 7 7.71 15 2.64 15
0.47 1 0.96 11 7.69 7 2.59 11
0.44 7 0.91 15 6.70 11 2.01 1
0.16 16 0.48 1 4.11 1 1.97 7
0.15 4 0.48 23 3.52 23 1.58 23
0.14 2 0.36 12 2.62 12 1.49 27
q =3
d 2 Obs. No. d 2 Obs. No. Obs. No.
1i 2i d 4i
1.55 20 9.03 20 2.64 15
1.37 5 7.82 15 2.59 5
1.06 11 7.70 5 2.59 11
1.00 7 7.69 7 2.53 20
0.96 1 7.23 11 2.01 1
0.93 15 6.71 1 1.97 7

observations on each statistic, where the number of PCs included, q,is1,2
2
or 3. The observations that correspond to the most extreme values of d ,
1i
d 2 and d 4i are identified in Table 10.1, and also on Figure 10.3.
2i
Note that when q = 1 the observations have the same ordering for all
three statistics, so only the values of d 2 are given in Table 10.1. When q
1i
is increased to 2 or 3, the six most extreme observations are the same (in
2
a slightly different order) for both d 2 and d . With the exception of the
1i 2i
sixth most extreme observation for q = 2, the same observations are also
identified by d 4i . Although the sets of the six most extreme observations
2
are virtually the same for d , d 2 and d 4i , there are some differences in
1i 2i
ordering. The most notable example is observation 15 which, for q =3, is
2
most extreme for d 4i but only sixth most extreme for d .
1i
Observations 1, 7 and 15 are extreme on all seven statistics given in Ta-
ble 10.1, due to large contributions from the final PC alone for observation
15, the last two PCs for observation 7, and the fifth and seventh PCs for
observation 1. Observations 11 and 20, which are not extreme for the final
PC, appear in the columns for q = 2 and 3 because of extreme behaviour
on the sixth PC for observation 11, and on both the fifth and sixth PCs
for observation 20. Observation 16, which was discussed earlier as a clear
outlier on the second PC, appears in the list for q = 1, but is not notably
extreme for any of the last three PCs.