Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 389

Fig. 7.2. Examples of operation of the k -means algorithm: sensitivity to initial
conditions and to the number of reference vectors. Observations and reference vectors
are shown on the same picture. ( a ) Representation of the learning set A: the data
are generated from four Gaussian modes. ( b ) Evolution of the two reference vectors
that were initialized at bottom right of the picture: each reference vector is assigned
the observations that are generated from two Gaussian modes. Pictures ( c ) and ( d )
show the evolution of four reference vectors that were generated in two different
ways. ( c ) The reference vectors are initialized at the center of the picture; each of
them is assigned observations coming from one a Gaussian mode ( d ) The reference
vectors are initialized at the bottom right of the picture: three reference vectors
share the observations generated by two Gaussian modes; the last reference vector
collects the observations generated by the other two modes
selected among p Gaussian modes with the prior distribution α c . Equivalently,
to generate the data, one must first choose randomly the mode according
to the discrete probability α c , and then to generate the observation from
the probability law of the selected mode. Thus, that model generates a data
set, which is partitioned by construction into p subsets. The subset that is
labeled by index c contains about Nα c observations. Those observations are
split around the reference vector w c . The subset has an ellipsoidal shape