Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 392

Fig. 7.3. Example of application of the k -means algorithm: a Gaussian mixture
model of two modes with different anisotropic covariance matrixes generates the
data. Crosses denote the positions of the reference vectors. ( a ) Training set A.
( b ) The reference vectors and the associated partition after stabilization of the algo-
rithm; an oblique line separates the two sets. The algorithm has not reconstructed
the true partition

In order to reconstruct the original distributions, the assumption of
isotropic covariance must be relaxed. That is possible if the covariance ma-
trices Σ c of the Gaussians are not supposed to be identical (yet semi-positive
definite). Then, the n ( n

1) / 2 elements of the matrices must be estimated,
in addition to the reference vectors w c . The model is more complex since
it has a larger number of parameters. A maximum likelihood methodology
may perform the estimation, using the EM (Expectation-Maximization) algo-
rithm [Dempster et al. 1977].

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7.3 Self-Organizing Topological Maps

7.3.1 Self-Organizing Maps

In the early 1980's, Kohonen described a self-organization algorithm that de-
fines a projection of the data space D onto a discrete, low-dimensionality
space. That space has a non-oriented graph structure that is a generally a 1-,
2- or 3-dimensional mesh; that graph will be hereinafter termed the map. Ac-
tually, the set C is made of interconnected neurons: the connections between