Neural computing in pharmaceutical products and process development - Computer-Aided Applications in Pharmaceutical Technology - page 151

norm in algebra - it is a function that assigns strictly positive length or
size to all vectors in vector space, other than the zero vector. It is
clear from the equation that neurons closer to the winning neuron
will undergo greater modifi cations in comparison to neurons further
away from the winning neuron. α (t) is the learning factor infl uencing the
extent to which a neuron should be modifi ed. It gradually decreases from
1 to 0 during SOM training - once it reaches 0, the training stops.
Modifi cation and/or control of α (t) directly infl uences the number of
training iterations.
Relationship between the α (t) , total number of iterations T , and initial
value of the learning factor α 0 (optionally defi ned by the user) can be
expressed as:

[5.36]

Parameter α (t) is analogous to the learning rate used in the BP algorithm
and determines how much the winning neuron and its neighborhood are
moved in the direction of the data vector x(t) . To reach a good statistical
accuracy, the number of iterations should be at least 500 times the
number of neurons in the SOM (Rantanen et al., 2001).
Initially the radius of the neighborhood is large σ (t) (more than half the
diameter of the network) and it decreases during the training iterations.
Therefore, in the last stages of SOM training, only a few neurons close to
the winning neurons are modifi ed, which means that at the beginning
of the training process, the map is globally modifi ed, whereas at the end
there is only fi ne-tuning of the map's features (Guha et al., 2004). A
group of neurons that have similar Euclidean distances from each other
can be considered as a cluster.
Both the learning rate and the neighborhood function decrease over
time for better convergence of the SOM algorithm. Convergence has
occurred when the weights for the neurons no longer change with
each iteration. In some instances, it is also possible to fi ne-tune the map's
cells using supervised learning principles (Kohonen, 1990). Many
computational aspects of the SOM algorithm are covered in Kohonen
reference (1997) and a software package (1996) available on the Internet
( http://www.cis.hut.fi /nnrc/nnrc-programs.html ).
Even though SOM creation is an unsupervised process, the user still
has to decide on the size of the map (number of neurons) infl uencing the
way the number of clusters are formed in the trained SOM. Fewer clusters
are more easily visualized, but they contain more variation in each cluster.
Increase in the number of clusters leads to a decrease in variation and

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