Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 382

Fig. 7.1. General diagram of the modeling process: one observation z is associated
to an index c that is selected among p indices using a function χ ; that index allows
defining the associated reference w c

• by defining an allocation function χ from D into the index set { 1 ,...,p } ;

that function performs a partition P = { P 1 ,...,P c ,...,P p } of D into p

subsets, P c = { z ∈ D/χ ( z )= c } .

Figure 7.1 describes graphically the modeling process: one observation z
is associated to an index c that is selected among p indices using a function
χ ; that index allows defining the associated reference w c . Thus, the reference
vector w c is the representative example of the set P c . It summarizes all the
information contained in P c . In the following, reference w c or its associated
index c will be used, depending on the context, for representing the observation
subset P c . We will estimate the model parameters from the observations of
the training set A . Therefore, we denote by n c the number of elements of P c .
The knowledge of the reference vector set W and of the allocation func-
tion χ generates what is called a vector quantization . All known methods to
determine W and χ can be derived from a variational principle and amount
to a cost function minimization. Each method has a specific cost function.
The latter incorporates the specific properties of the associated quantization.
The vector quantization permits the allocation of a reference w χ ( z ) to any
observation z D . That reference index is χ ( z ). Furthermore, the knowledge

of the allocation function χ completely determines the partition of the set D
into p subsets.
Although the cost functions are different for different methods, all methods
that will be described share common features. In the following, the formalism
of dynamic clustering will be used. That approach is iterative. Each iteration
consists in two steps: a minimization step computes the reference vectors, and
an allocation step changes the allocation function χ . Under some assumptions,
the cost function decreases at each step and eventually converges towards a
local minimum. That minimum depends strongly on the choice of the reference
vector set that was selected to initialize the algorithm.

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