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In-Depth Information
Figure
9.3
Pseudo-sample
interpolation
for
variable
height
where
τ
=
170
and
τ
=
172.
and a set of
L
k
CLPs for a categorical variable. Gower (1992) shows that for a PCO
for continuous variables, with a wide class of distance measures in common use, this
choice of pseudo-samples effectively leads to the nonlinear biplot. The only difference
is that the trajectory is parallel to the nonlinear biplot trajectory (see orthogonal parallel
translation: Section 2.4). We note that, unlike nonlinear biplots,
τ
=
0 is not common
to all the continuous trajectories. In Figure 9.3 the pseudo-samples for variable
k
=
1,
height
, are interpolated where
τ
=
170 and
τ
=
172. As an example, the pseudo-samples
for
τ
=
170 are given by: (170, blue); (170, brown); (170, green); (170, brown).
Since the second and fourth pseudo-samples are identical, their interpolated represen-
tations are at exactly the same point. The centroid of the four interpolated pseudo-samples
is shown as a green sphere.
As explained above, when the assumption of additive distance is valid the biplot
trajectory can be calculated by using the distance between the centroid of the pseudo-
samples and the original samples. Define the matrix
D
11
D
12
D
=
(9.9)
D
12
D
22
with
D
11
containing
ddistances
between the samples,
D
22
containing
ddistances
between
the pseudo-samples and
D
12
containing
ddistances
between the
i
th sample point and the
j
th pseudo-sample.