Information Technology Reference
In-Depth Information
FibL (0.8)
RayH (0.63)
550
1600
5
27
500
24
25
22
26
23
21
1400
450
10
14
20
19
160
VesD (0.6)
20
18
12
140
17
13
VesL (0.29)
37
500
120
400
30
16
9
8
11
400
10
36
1200
32
15
100
300
40
15
7
80
6
350
34
50
28
5
29
RayW (0.84)
33
31
30
1000
35
300
1
20
4
3
800
Obul; n = 20
Oken; n = 7
Opor; n = 10
25
2
NumVes (0.7)
Figure 5.1
PCA biplot of the normalized
Ocotea
data. Axis predictivity is shown in
brackets after the axis label.
Although the above has been concerned with predicting the values of
X
, the regression
method may be used to add axes for completely new variables to an existing biplot, as we
already have illustrated for PCA and CVA biplots in Figures 3.24 and 4.13, respectively.
In nonlinear situations it is an open question how well the regression will work. In
the next section we shall see why it could be unsatisfactory.
As an example of an MDS biplot we again consider the
Ocotea
data. Its PCA biplot,
given in Figure 4.1, is reproduced in rotated form in Figure 5.1 to aid visual comparison
with the MDS biplot given in Figure 5.2.
The MDS biplot given in Figure 5.2 is obtained by calling our function
MDSbipl
.
This function accepts a data matrix
X
:
n
p
as input. Various distance metrics are
available for calculating the distances between the rows of
X
to form an
n
×
n
distance
matrix
D
∗
. For Figure 5.2 we used the default Pythagorean distance. The coordinates of
the sample points in two dimensions are needed for producing the
×
δ
ij
such that the stress
criterion (5.1) is minimized. The default method of
MDSbipl
is to use an implementation
of the SMACOF algorithm as explained in Borg and Groenen (2005). For further details
on SMACOF, see De Leeuw (1977) and Heiser and De Leeuw (1977).
In the construction of Figure 5.2 we initialize the SMACOF algorithm with an
n
×
2
matrix of randomly selected normal (0, 1) values. The function
MDSbipl
allows several
random starts of the SMACOF algorithm. The coordinates resulting in the smallest value
