Information Technology Reference
In-Depth Information
Figure 5.10
Interpolation of embedded nonlinear axes as nonlinear interpolation axes
in the biplot space.
In order to derive an explicit expression for interpolating new samples we make the
assumption that the variables contribute independently to the squared distance between a
new sample and one of the original samples - that is, that the distance measure is additive.
It is clear that Pythagorean distance as well as the distances (5.5) and (5.6) are additive,
but Mahalanobis distance is not. Consider the pseudo-sample (0,
...
,0,
x
k
,0,
...
,0
)
=
x
k
e
k
with associated vector
d
n
+
1
denoted by
d
n
+
1
(
x
k
)
. The squared distance from this
pseudo-sample to each of the original samples is of the form
p
d
n
+
1,
i
(
d
2
d
2
d
2
x
k
)
=
(
x
ih
,0
)
+
(
x
ik
,
x
k
)
−
(
x
ik
,0
).
h
=
1
Therefore,
p
p
p
p
d
2
d
n
+
1,
i
(
x
k
)
+
d
2
d
2
(
x
ik
,
x
k
)
=
(
x
ik
,0
)
−
p
(
x
ih
,0
).
(5.11)
k
=
1
k
=
1
k
=
1
h
=
1
