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Table representation
Feature space
visualization
Feature space
visualization
Violine database
Fig. 2.10.
Exploitation of human perceptive capabilities by appropriate presenta-
tion of multivariate data employing dimensionality reduction and interactive visu-
alization.
While in selection, according to a chosen criterion
J
and the applied selec-
tion matrix
A
S
(see Eq. 2.13), the best features are retained and the remain-
ing ones are discarded; in extraction all features are retained and subject
to transformation
A
. In both cases a mapping
Φ
:
R
M
→ R
m
optimizing
a criterion
J
with
m ≤ M
and
y
=[
y
1
,y
2
,...,y
m
]
T
is determined. Here
y
=
(
v
) can be a linear or nonlinear mapping and employ unsupervised as
well as supervised information. The optimization criterion or cost function
J
can represent various objectives, e.g., signal preservation, distance preserva-
tion, topology preservation, or discrimination gain for the underlying
L
-class
problem (see Fig. 2.12). For the latter case, selected instances of
J
will be
given in the following. Figure 2.11 gives a taxonomy of state-of-the-art dimen-
sionality reduction methods for multivariate data classification, analysis, and
visualization in a unified presentation. This taxonomy has been elaborated
on in the last few years and is continuously enhanced, including new meth-
ods. Most of the methods have been implemented in the QuickCog system
[2.28] [2.29] and compared in previous survey publications [2.24] and tutori-
als [2.29] [2.22]. The taxonomy given in Fig. 2.11 covers methods as, e.g., the
principal-component analysis (PCA) [2.12], scatter matrices (SCM) [2.12],
Sammon's nonlinear mapping (NLM) [2.48], and accelerated heuristic vari-
ants, the nonlinear discrimination analysis method of Koontz and Fukunaga
[2.32], or Kohonen's self-organizing map [2.19] (see also [2.24]). For visual-
ization purposes, in this work distance-preserving nonlinear mappings, e.g.,
the one introduced by Sammon [2.48] have been applied. Interpoint distances
A
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