Page 439 - Jolliffe I. Principal Component Analysis

14. Generalizations and Adaptations of Principal Component Analysis
404
x j ) into parts due to each component. In this respect

compose var(
p
j=1
there are similarities with a method proposed by Vermeiren et al. (2001),
which they call extended principal component analysis. This also decom-

poses var( p x j ), but does so with rescaled PCs. Denote the kth such
j=1

component by z E = a x, where a E = c k a k , a k is the usual vector of
E
k k k

coefficients for the kth PC with a a k =1, and c k is a rescaling constant.
k

Vermeiren et al. (2001) stipulate that p z E = p j=1 x j and then show
k=1 k
that this condition is satisfied by c = A 1 p , where the kth column of A

is a k and the kth element of c is c k .Thus c k is the sum of the coefficients
in a k and will be large when all coefficients in a k are of the same sign,
or when a PC is dominated by a single variable. The importance of such
PCs is enhanced by the rescaling. Conversely, c k is small for PCs that are
contrasts between groups of variables, and rescaling makes these compo-
nents less important. The rescaled or ‘extended’ components are, like the
unscaled PCs z k , uncorrelated, so that
& p ' & p ' p p p
2 2
var x j =var z E = var(z )= c var(z k )= c l k .
E
k k k k
j=1 k=1 k=1 k=1 k=1
2
Hence var[ p x j ] may be decomposed into contributions c l k ,k =
j=1 k
1, 2,... ,p from each rescaled component. Vermeiren et al. (2001) suggest
that such a decomposition is relevant when the variables are constituents
of a financial portfolio.
14.6.4 Subjective Principal Components
Korhonen (1984) proposes a technique in which a user has input into the
form of the ‘components.’ The slightly tenuous link with PCA is that it is
assumed that the user wishes to maximize correlation between the chosen
component and one or more of the original variables. The remarks following
the spectral decomposition (Property A3) in Section 2.1, Property A6 in
Section 2.3, and the discussion of different normalization constraints at the
end of that section, together imply that the first few PCs tend to have
large correlations with the variables, especially in a correlation matrix-
based PCA. Korhonen’s (1984) procedure starts by presenting the user
with the correlations between the elements of x and the ‘component’ a x,

0
1
where a 0 is the isometric vector √ (1, 1,... , 1) (see Section 13.2). The user
p
is then invited to choose a variable for which the correlation is desired
to be larger. The implications for other correlations of modifying a 0 so
as to increase the selected correlation are displayed graphically. On the
basis of this information, the user then chooses by how much to increase
the correlation and hence change a 0 , giving the first subjective principal
component.
If second, third, ..., subjective components are desired, emphasizing
correlations with different variables, a similar procedure is repeated in the