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Fig. 3.4.
CCA Projection of a hemisphere
Fig. 3.5.
CCA projection of a sphere
The rank of the differential of the application determines the local dimensions
of the variety.
In relation to PCA, that method is therefore used to represent data struc-
tures that are distributed in a nonlinear manner. It is similar to methods
based on Kohonen's self-organizing maps, but its principle is different. There
are no constraints on the points in the projection space. In theory, no neigh-
borhood is defined between the points in the projection space. That gives
great flexibility to the method.
3.5.1 Formal Presentation of Curvilinear Component Analysis
The co-ordinates of the
p
points are defined
by
x
i
∈
R
n
,
i
=
•
{
1
,...,p
}
in the original space;
by
y
i
∈
R
n
<n
,
i
=
•
{
1
,...,p
}
in the reduced space.
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