There are in fact two ways of looking at , row by row or
column by column:
From this point of view our data matrix is representable as a
cloud of
points in
as shown in Figure 8.1.
From this point of view the data matrix is
a cloud of
points in
as shown in Figure 8.2.
When and/or
are large (larger than
or
),
we cannot produce interpretable graphs of
these clouds of points. Therefore, the aim of the factorial methods to be
developed here is two-fold.
We shall try to simultaneously approximate the column
space
and the row space
with smaller
subspaces. The hope is of course that this can be done without
loosing too much information
about the variation and structure of the point clouds in both spaces.
Ideally, this will provide insights into the structure of
through graphs in
,
or
.
The main focus then is to find the dimension reducing factors.