Information Technology Reference
In-Depth Information
PLF
0.7
3
0
0.6
3.5
1
0.5
4
−
4
−
2
0
2
4
6
8
10
12
SPR
2
0.4
a
e
3
4.5
b
0.3
d
f
4
k
t
5
s
i
0.2
w
p
m
n
h
c
5.5
5
0.1
g
j
6
q
6
0
u
v
r
SLF
6.5
−
0.1
7
−
0.2
RGF
−
0.3
Figure 2.19
The two-dimensional biplot of Figure 2.18 with orthogonal parallel trans-
lation rotated such that the
SPR
-axis is horizontal with calibrations increasing from left
to right, similar to the first Cartesian axis of a scatterplot.
Thus, to interpolate, we need the vector-sum of the points corresponding to the markers
x
1
,
x
2
,
x
3
,
...
,
x
p
. Geometrically, the sum of two vectors is usually thought of as being
formed as the diagonal of the parallelogram that has the two vectors for its sides. Sev-
eral vectors may be summed by constructing a series of such parallelograms. We may
get the same result more simply by finding the centroid of the points given by the
markers
x
1
,
x
2
,
x
3
,
...
,
x
p
and extending the result
p
times from the origin. How this is
done is indicated in Figure 2.23 for the point with
SPR
=
8,
RGF
=
4,
PLF
=
0
.
3and
SLF
=
3. Note that, if we interpolate one of the original samples, say
p
(F3H-2), its
interpolated value will occupy the same position as it does in the analysis shown in
Figure 2.17.
We have implemented the vector-sum method for interpolating a new sample point
in our R function
vectorsum.interp
. First, the biplot in Figure 2.23 is constructed
using
PCAbipl
with
scaled.mat=TRUE, ax.type="interpolative"
as described
earlier. Then
vectorsum.interp
is called with
vertex.points = 4
, allowing the
user to select with the mouse the four values on the respective axes of the point to be
interpolated. A polygon is then drawn connecting these vertex points and the coordinates
