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is not homogeneous from a statistical point of view. Then it is inferred that
the expert has assigned to the same class instances of the observation space
that are quite different. The analysis of the most consistent partition, ob-
tained by hierarchical classification, allows analyzing the homogeneity of the
classification performed by the expert. Therefore, a refinement of the expert
classification into S classes, with S >S , may be designed.
7.4.3 Labeling and Classification
After labeling of the map, the probabilistic version of self-organizing maps
(PRSOM) can perform a probabilistic classification. As mentioned above, a
normal law is associated to each neuron. An observation z is assigned to a
neuron according to the probability p ( c
z ), which is defined by Bayes relation
as shown below. A probabilistic assignment is thus obtained. Since the map
is labeled according to one of the above procedures, the posterior probability
that the considered observation z belongs to class l can be estimated. The
PRSOM stem from a probabilistic modeling where it is assumed that the
observations are generated according to the mixture distribution:
p ( z )=
c
|
p ( c ) p c ( z )
where p c ( z ) is also a normal law local mixture
p c ( z )= 1
T c
K T ( δ ( c,r )) f r ( z , w r )
c
where T c = c K T ( δ ( c,r )) and f is a normal law with mean w r
and scalar
covariance matrix σ 2
r
I . The quantities p c ( z ) are computed from the neurons
of the map and the quantities p ( c ) are computed from the partition that
has been proposed by PRSOM. If N stands for the observation number of
the training set A and n c is the number of observations that are assigned to
neuron c by the allocation rule χ ( z ) = arg max c p ( z
c ), the prior probability
p ( c )ofneuron c can be estimated as n c /N . Then Bayes rule allows computing
the posterior probability of neuron c given observation z :
|
z )= p ( c ) p c ( z )
p ( z )
n c p c ( z )
p ( c
|
=
.
n r p r ( z )
r∈C
After training, the topological map that is proposed by PRSOM determines
the parameters of the normal laws that characterize the various neurons. For
any observation z , it is then possible, applying the above relation , to compute
the posterior probability of an obervation being assigned to a given neuron.
Since a class is represented by a subset of neurons, the posterior probability
that the observation z belongs to the class l i is derived from the neurons that
are labeled by l i . If the subset of those neurons is denoted C i ,onegets
 
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