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interpolates that sample into the map. Also,
d
may also be chosen as a pseudo-sample
(see Section 5.4) and used to plot predictive trajectories for the variables. In this way
the AoD of the individual samples may be enhanced to include information on the
variables to give a true biplot. The regression method (Chapter 4) may be used to give
approximate linear biplot axes. Krzanowski (2004) suggests a half-way house where a
limited number of pseudo-samples (say 10) are fitted with linear axes. In principle, the
formulae for adding a point may also be used to construct the convex prediction regions
of generalized biplots, as described in Chapter 9.
The cloud of points surrounding each centroid may be enclosed in any region that
expresses spread, analogously to the confidence circles of CVA. Thus we may use minimal
covering circles or ellipses enclosing, say, all or 95% of the points, or we may use
-
bags, or we may use convex hulls (Section 2.9.3). Furthermore, a nonparametric testing
procedure can be used for testing as illustrated in the example below.
Finally, we note that the above uses unweighted centroids. Again, analogously to
CVA, we may use centroids weighted by sample sizes. The starting point is the weighted
PCO of
, where now (5.26) is replaced by
α
I
−
I
−
1n
n
n1
n
YY
=
.
(5.30)
Because
n
Y
0
, G, the centroid of the samples, is now at the weighted centroid of
the group centroids. As with CVA, the use of a weighted centroid does not affect the
distances between the individual centroids; but in approximations, groups with smaller
sample size will be less well represented than those with the larger sample sizes.
=
5.8.1 Proof of centroid property for interpolated
points in AoD
We now show that the centroids of the interpolated samples from the
k
th group coincide
with the
k
th column of
Y
, the coordinates of the group means. Of course, this is a trivial
result in the linear case, but it is not obvious that it carries over to the nonlinear case
and therefore requires proof.
We begin with the basic interpolation formula (5.27):
=
−
1
Y
(
δ
−
y
1
/
K
)
,
n
i
1
d
i
for
i
=
1, 2,
...
,
K
. To interpolate all the sam-
ples from the
k
th group requires the
n
k
columns of
D
pertaining to the
k
th group. This
gives
n
k
columns of
1
2
2
i
2
i
2
where
δ
={−
δ
}
with
δ
=
D
ii
−
(
δ
−
1
/
K
)
as follows:
&
)
&
)
n
1
1
D
1
k
1
1
D
11
1
n
k
D
22
1
n
k
.
D
KK
1
n
k
(
+
(
+
−
n
2
1
D
2
k
.
1
2
−
1
K
1
n
k
.
−
2
n
K
1
D
Kk
1
