Information Technology Reference
In-Depth Information
10
Monoplots
At the outset we emphasized that the
bi-
of biplots referred not to the usual use of
two dimensions but to the two types of entities exhibited - typically units and vari-
ables. Many displays exhibit only one type of entity, but represented in two or more
dimensions. We term these
monoplots
. Often monoplots are concerned with symmet-
ric matrices - for example, (dis)similarity, proximity or correlation matrices. Some are
intrinsically monoplots, but others are monoplots because a deliberate choice is made to
display only one type of entity; if the other were included the monoplot would become
a biplot. Already, we have encountered some ambiguity in a sharp distinction between
monoplots and biplots. For example, different symbols are plotted in CA (Chapter 7) to
show the levels of the categorical variables labelling the rows and columns of a two-way
contingency table. There, both rows and columns refer to the same type of entity. The
situation is heightened in MCA (Chapter 8), where
p
categorical variables are presented,
each with its own symbol. The same kind of entity is presented, albeit with
p
instances,
each with its own plotting symbol - is this still a biplot? We would call it a monoplot,
but it would become a biplot were the
n
units (samples) displayed as a set of
n
points or
a density plot together with the information on the variables (Section 8.8). The position is
aggravated when we have categorical variables because the
L
k
levels of the
k
th variable
give rise to
L
k
points, corresponding to a single calibrated axis for a quantitative variable
(Chapter 8).
10.1 Multidimensional scaling
We saw in Chapter 5, on nonlinear biplots, how a matrix may be calculated giving the
distances
d
ii
between every pair of rows of a data matrix
X
and how nonlinear biplot tra-
jectories may be found and used to predict approximations to the entries of
X
. A special
case is PCA, where the distance is Pythagorean - that is,
d
ii
=
(
x
i
)
(
-
and the trajectories are linear. In general, multidimensional scaling is concerned with
finding an
r
-dimensional matrix
Z
whose rows generate Pythagorean distances
δ
ii
x
i
−
x
i
−
x
i
)
that
