Information Technology Reference
In-Depth Information
Figure 4.6
Step I illustrating the transformation to the canonical space.
shows that the distances in the top right panel have become additive weighted distances.
The next action (see the bottom right panel) scales the dimensions such that we now
have ordinary Pythagorean distances derived from the inner product
(
XPD
)(
DP
X
)
.
These distances are also obtained for any rotation by an orthogonal matrix multiplication
(
XPD
)
R
R
(
DP
X
)
.Wehavechosenthescalingof
L
such that
L
WL
=
I
, which now
uniquely defines the final rotation to be given by the matrix
Q
. After this final stage in
the transformation we obtain the configuration in the bottom left panel.
Step II now requires a PCA of
XL
. In Figure 4.7(a) the three points (canonical means)
are shown in three dimensions and the best-fitting two-dimensional PCA plane is added
in Figure 4.7(b). Since we need to fit the plane to only three points, the fit is exact,
rather than a two-dimensional approximation. In Chapter 3 we proceeded to construct
