Information Technology Reference
In-Depth Information
Myfanwy
Fair
Clerical
Alisdair
Dark
Scotland
Wales
F
University
School
Jane
Brown
Harriet
M
Manual
Ivor
George
Professional
England
Grey
Jeremy
Postgrad
Figure 8.5
The row chi-squared MCA biplot of Figure 8.1. The row points (the green
filled circles) are plotted using the first two columns (after discarding the column asso-
ciated with the singular value of unity) of
Z
0
=
U
and the column points (the filled
squares) are plotted as the category centroids
L
−
1
G
Z
0
.
8.3 The Burt matrix
Just as PCA may find the SVD of
X
by evaluating the spectral decomposition of
X
X
,
so may the SVD of the adjusted indicator matrix be found by evaluating the spectral
decomposition of
p
−
1
L
−
1
/
2
G
GL
−
1
/
2
.Now,
G
G
, known as the Burt matrix, is a block
matrix each of whose blocks
{
G
j
G
j
}
is the two-way contingency table for the
j
th and
j
th
categorical variables. Table 8.3 shows the Burt matrix derived from Tables 8.1 and 8.2.
This is a trivial example, but it suffices to show that a Burt matrix is symmetric, has
diagonal blocks giving the frequencies of the different categorical variables and has
off-diagonal blocks giving the pairwise contingency tables.
Because of the contingency table structure of the Burt matrix, we may wish to approx-
imate
G
G
itself, rather than just use it as a stepping-stone to calculating an SVD. In this
way we can have a multivariate extension of CA in which all the
2
p
(
p
−
1
)
contingency
tables are simultaneously approximated. Even better, the spectral decomposition of the
normalized Burt matrix
L
−
1
/
2
G
GL
−
1
/
2
,givenby
L
−
1
/
2
G
GL
−
1
/
2
2
V
,
=
p
V
(8.9)
