Information Technology Reference
In-Depth Information
Figure 5.6
Euclidean space for samples
a
,
b
,
c
with
d
at the intersection of three
spheres.
The above shows how to do embedding in a simple case but fails when there are
more than four samples. Then we have to resort to approximations given by PCO, which
amounts to the same thing as doing PCA on the points embedded by the simple method.
Performing a PCO on a Euclidean embeddable distance matrix
D
, the representation of
Y
in
R
is given in Figure 5.7. This representation is exact and approximation is not
relevant. The best-fitting
r
-dimensional subspace
L
is obtained from the first
r
principal
components given by the first
r
columns of
Y
denoted by
Z
=
Y
r
, as shown in the form
of a biplot in Figure 5.8.
5.4 Providing nonlinear biplot axes for variables
A major difference between MDS methods and the methods of previous chapters is that,
as a consequence of working with distances, there is no provision for representing the
original variables in
R
, only the sample points. This is a serious drawback because
biplots are concerned with approximating the variables. The way forward is to note that
