Information Technology Reference
In-Depth Information
A1
A1
Predicted values
A1: 2
A2: 2
Interpolate
A1: 2
A2: 2
0
0
1
1
2
2
3
3
4
4
5
5
A2
6
A2
6
7
7
(a)
(b)
A1
A1
Predicted values
A1: 3.6
A2: 3.6
Interpolate
A1: 3.6
A2: 3.6
0
0
1
1
2
2
3
3
4
4
5
A2
5
6
A2
6
7
7
(c)
(d)
Figure 2.21
Why interpolation and prediction with a single set of nonorthogonal biplot
axes are inconsistent.
is the centroid of the markers
µ
1
,
µ
2
,
µ
3
,
...
,
µ
p
. The interpretation is that every point
on the
k
th axis is translated obliquely by an amount
µ
k
(
e
k
V
r
)
, resulting in new axes,
concurrent at
µ
. Contrast this process with the orthogonal translation method that is
appropriate for the axes of predictive biplots. With interpolation, the position is better,
because we may choose 'nice' values of
µ
k
on all the axes rather than just
r
of them.
Figure 2.24 illustrates the process where
µ
k
has been chosen to be zero for each axis.
To interpolate a point with the new axes, we proceed, as before, by finding the centroid
of the markers but now we have to remember to add in the vector
µ
. All this means is
that we continue to extend
p
times from the old origin O rather than from the new point
of concurrency of the axes. This is why the point O is retained in Figure 2.24.
Oblique transformation is implemented in our
PCAbipl
function. It is only used
with argument
ax.type = "interpolative"
and is invoked by assigning to argument
oblique.trans
avectorof
p
elements specifying the values of each variable at the
point of concurrency of the biplot axes. Figure 2.25 shows the results of this process for
the aircraft data, where we have chosen
µ
1
=
0,
µ
2
=
0,
µ
3
=
0 and,
µ
4
=
0.
In Figure 2.26 we show how to effect interpolating the new sample with
SPR
=
8,
RGF
=
4,
PLF
=
0
.
3and
SLF
=
3 using the obliquely translated interpolative axes.
Note the role of the original origin marked with the black cross. Note also that the inter-
polated position is exactly as in Figure 2.23. Interpolation performed using the graphical
vector-sum method is implemented in the function
vectorsum.interp
described earlier
in this section.
