Page 177 - Jolliffe I. Principal Component Analysis

6. Choosing a Subset of Principal Components or Variables
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Table 6.4. Subsets of selected variables, Alate adelges.
(Each row corresponds to a selected subset with × denoting a selected variable.)
Variables
5 8 9 11 13 14 17 18 19
McCabe, using criterion (a)
best × × ×
"
Three variables
second best × × ×
best × × × ×
"
Four variables
second best × × × ×
Jolliffe, using criteria B2, B4
B2 × × ×
"
Three variables
B4 × × ×
B2 ×× × ×
"
Four variables
B4 × × × ×
Criterion (6.3.4)
Three variables × × ×
Four variables × × × ×
Criterion (6.3.5)
Three variables × × ×
Four variables × × × ×

largest coefficients on five of the seven discrete variables, and the third PC
(3.9%) is almost completely dominated by one variable, number of antennal
spines. This variable, which is one of the two variables negatively correlated
with size, has a coefficient in the third PC that is five times as large as any
other variable.
Table 6.4 gives various subsets of variables selected by Jolliffe (1973)
and by McCabe (1982) in an earlier version of his 1984 paper that included
additional examples. The subsets given by McCabe (1982) are the best two
according to his criterion (a), whereas those from Jolliffe (1973) are selected
by the criteria B2 and B4 discussed above. Only the results for m =3 are
given in Jolliffe (1973), but Table 6.4 also gives results for m = 4 using his
methods. In addition, the table includes the ‘best’ 3- and 4-variable subsets
according to the criteria (6.3.4) and (6.3.5).
There is considerable overlap between the various subsets selected. In
particular, variable 11 is an almost universal choice and variables 5, 13 and
17 also appear in subsets selected by at least three of the four methods.
Conversely, variables {1–4, 6, 7, 10, 12, 15, 16} appear in none of subsets of
Table 6.4. It should be noted the variable 11 is ‘number of antennal spines,’
which, as discussed above, dominates the third PC. Variables 5 and 17, mea-
suring number of spiracles and number of ovipositor spines, respectively, are