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right-hand side an area. This implies that if we rotate one set of the scaffolding axes
through 90 degrees we may present the biplot as two sets of points as in (i) but using
area (including the origin) rather than inner products for interpretation. This introduces
some new geometrical concepts that need to be assimilated (see Gower
et al.
, 2010).
As well as the partitioning of the inner product and the choices of points and axes
for display, all the devices such as axis translation and
λ
-scaling of axes discussed in
Chapter 2 remain available.
We conclude with a few words on terminology. The extensive CA literature has devel-
oped its own terminology, based on English translations from the seminal French text of
Benzecri (1973), the fount of much subsequent research. Benzecri used terms drawn from
basic concepts in mechanics, such as mass and inertia. We have used standard statistical
terminology, referring to row (or column) sums rather than mass, and sums of squares
rather than inertia. Greenacre (1984, 2007) also introduced some terminology, referring to
standard coordinates, which do not include weights, and principal coordinates, which do.
In the above, we ourselves have referred to symmetric and asymmetric representations
in the Greenacre sense. This conflicts with the distinction made in Chapter 1 between
symmetric and asymmetric classes of biplot, according to which all correspondence anal-
yses of two-way contingency tables would be symmetric. The problem is that asymmetry
(
sensu
Greenacre) arises when treating the two-way table in an asymmetric way, as if it
were a data matrix, by focusing on inter-row or inter-column chi-squared distance. As
there is some ambiguity in many of these terms, for an understanding of the biplots we
prefer to state algebraically, as in the final column of Table 7.1 and in the examples,
precisely what it is we plot in the different circumstances.
7.3 Interpolation of new (supplementary) points in CA biplots
We have seen that the model
Y
=
W
−
1
1
(
X
−
E
)
C
−
1
/
2
W
−
1
2
R
−
1
/
2
is approximated by
Y
=
W
−
1
1
U
JV
W
−
1
2
which, in turn, is written as an inner product
Y
=
AB
,
giving a biplot where the rows of
A
give the coordinates in
r
dimensions for plotting row
points and the rows of
B
give the coordinates for plotting column points. Immediately,
we have
YB
(
B
B
)
−
1
,
A
=
(7.40)
=
(
A
A
)
−
1
A
Y
,
B
which are known as
transition formulae
(see also Section 2.7) that relate the row and
column coordinates. From the form of (7.40) it is clear that the transition formulae are
a manifestation of the regression method discussed in Section 2.7.
To add a new row
x
:1
×
q
to
X
we require a new row
y
:1
×
q
given by
y
=
w
−
1
r
−
1
/
2
(
x
−
r
1
C
/
n
)
C
−
1
/
2
W
−
1
,
(7.41)
1
2
