Information Technology Reference
In-Depth Information
Table 10.3
The Rothkopf Morse data for the vowels together with the
M
(symmetric)
and
N
(skew-symmetric) matrices.
Rothkopf vowels
M
matrix
N
matrix
aei
o u a
e
i
o
u
a
e
i
o
u
a
92
3
46
6
37
92.0
4.5
55.0
6.5
25.5
0.0
−
1.5
−
9.0
−
0.5
11.5
e
6
97
17
5
3
4.5
97.0
13.5
5.0
3.5
1.5
0.0
3.5
0.0
−
0.5
i
64
10
93
7
10
55.0
13.5
93.0
4.5
15.5
9.0
−
3.5
0.0
2.5
−
5.5
o752 6 1 .5 .0 .5 .0 .5
.5 .0
−
2.5
0.0
2.5
u
14
4
21
6
93
25.5
3.5
15.5
8.5
93.0
−
11.5
0.5
5.5
−
2.5
0.0
effect is to scale the hedron uniformly and so can it be absorbed into the singular value;
there is no visual impact.
As an illustration of monoplots for skew-symmetric data we consider the data (in
percentage form) for the vowels in the Rothkopf Morse code data (taken from the full
asymmetric form of the data as available in Borg and Groenen, 2005). In Table 10.3
we show in addition to the data on the vowels the
M
and
N
matrices defined above.
Figure 10.11 shows a hedron plot of the
N
matrix given in Table 10.3. The matrix
of
the SVD (10.5) is diag(16.2237, 16.2337, 2.7913, 2.7913, 0). Figure 10.11 is obtained
with the following code:
> par (mfrow = c(2,2), mar = rep(0.25,4))
> MonoPlot.skew (X = Rothkopf.vowels)
> MonoPlot.skew (X = Rothkopf.vowels, form.triangle1 = c(1,5),
form.triangle2 = c(1,3))
> MonoPlot.skew (X = Rothkopf.vowels, form.triangle1 = c(2,3),
form.triangle2 = c(3,4))
> MonoPlot.skew (X = Rothkopf.vowels, form.triangle1 = c(1,4),
form.triangle2 = c(4,5))
In Figure 10.11, note the approximate collinearity with the origin of
e
,
o
and
a
,
indicating that there is little difference in the confusion, depending on the order in which
these vowels are presented. Also,
i
,
a
and
u
have approximate linear skew-symmetry.
We see that some triangles include the origin (e.g.
a
,
u
,
i
) and some exclude the origin
(e.g.
a
,
u
,
o
).
We started with the decomposition
X
=
M
+
N
. It is sometimes possible to com-
bine plots derived from
M
with those derived from
N
, especially when skew-symmetry
has approximate linear form. Supposing
M
has been derived from an MDS, then linear
skew-symmetry may be added either as a line or as an extra dimension, possibly allowing
contour plots. For example, Gower and Dijksterhuis (2004) derive a map from the sym-
metric part of flight times between US cities given in
X
and superimpose a line, derived
from the linear skew-symmetry, giving the direction of a jet stream. Gower (2008) dis-
cusses this type of modelling that combines symmetry with departures from symmetry,
so giving what may be considered a further type of biplot, not further discussed in
this topic.
