Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 427

∆ρ ( λ k )= ρ ( λ k +1 )

− ρ ( λ k )

.

λ k +1 − λ k

Seven slopes were thus computed for k from 1 to 7. In order to keep infor-
mation about the spectrum intensity, a component of the observation vector
was the norm of the spectrum. Thus, each observation was encoded in an

8-dimensional vector

ρ

] T .

[ ∆ρ ( λ 1 ) ,...,∆ρ ( λ 7 ) , ρ

The sampled LAC image that was used for training was encoded according
to that second scheme; the result will be denoted App cod2 . Normalization
between -1 and 1 was performed as previously. Since the slopes and the norm
do not have the same order of magnitude, the normalization was implemented
separately for each component as

2 x

− min

min − 1 , max −

where x is a derivative (namely ( ∆ρ ( λ k ) k =1 ,..., 7) , min (resp . max) are the
minimum (resp. the maximum) over the set of all the derivatives in App cod2 .
For all test data, the same encodings were used. The following numerical
experiments are an illustration of the methodology, which has been described
in the section “Classification and PRSOM.” They use quantizations, which
are followed by classifications. The quantizations are obtained from the prob-
abilistic maps, and clustering is performed by hierarchical classification. All
self-organizing maps have the same architecture:
• The input layer has 8 units

• The map is 2-dimensional, with 10 10 neurons. The neighborhoods are

defined from the exponential kernel family K ( δ )=exp(

×

δ 2 ).

−

7.5.1.4 Quantization Using PRSOM

In the first part of the study, PRSOM was used for determining the patterns
that are representative summaries of the set of all observed spectra. In that
case, a fine quantization of the training set is sought. Actually, the result is
a summary of the training set; if it is statistically representative, it is also
a summary of all observations. Otherwise, the generalization may be poor
since a subset of the set of observations was overlooked. The two encoding
schemes that were described above (normalized radiance vales for the first
one, slopes + norm for the second one), resulted in different maps. Those
maps will illustrate the importance of the encoding process for quantization
and topological order. Each map quantizes the observation set intro 100 sub-
sets. Figure 7.23 shows the map that was obtained with App cod1 ;onthat
figure, the number that is located above the neuron indicates how many pixels
from the training set are allocated to that neuron. Figure 7.24 shows the