Suppose that is represented by a cloud
of
points (variables) in
(considering each column).
How can this cloud be projected into a lower dimensional space?
We start as before with one dimension. In other words,
we have to find a straight line
, which is defined by
the unit vector
, and which gives the best fit of the
initial cloud of
points.
Algebraically, this is the same problem as above (replace
by
and follow Section 8.2):
the representation of the
-th variable
is obtained by the projection of the corresponding point onto the
straight line
or the direction
.
Hence we have to find
such that
is maximized, or
equivalently, we have to find the unit vector
which maximizes
.
The solution is given by Theorem 2.5.
The coordinates of the variables on
are given by
, the first factorial axis.
The
variables are now represented by a linear combination of the
original individuals
, whose coefficients are
given by the vector
, i.e., for
![]() |
(8.5) |
The representation of the variables in a subspace of dimension
is done in the same manner as for the
individuals above.
The best subspace is generated by the
orthonormal eigenvectors
of
associated with the
eigenvalues
.
The coordinates of the
variables on the
-th factorial axis are given
by the factorial variables
.
Each factorial variable
is a linear combination of the original individuals
whose coefficients are given by the elements of the
-th vector
.
The representation in a subspace of dimension
is depicted in
Figure 8.5.