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Implementation

We have used the package SOM_PAK from Kohonen's laboratory to implement the self-organising map algorithm. Except for the modifications described below, the default settings in the package are used. The rectangular map topology is preferred over the default hexagonal setting for practical reasons concerning map display and manipulation. We also use the two-phase ordering and convergence training procedure as recommended in the SOM_PAK documentation and described below.

Since the algorithm is randomly primed and guarantees neither identical nor perfect mappings, a procedure similar to the vfind program in the SOM_PAK package was implemented to select the best mapping from a number of trials (usually 10 or 20) with different random seeds. Good maps are those with low quantisation errors meaned over all input vectors, $v_i$. The simple quantisation error, $q_i$ is defined as the Euclidean distance between the input vector and the winning map vector:

$$q_i = \vert\vert v_i - r_w \vert\vert$$ (11)

A better quality measure is the weighted quantisation error, $q^k_i$:

$$q^k_i = \sum k(m,n,r_w,d)(\vert\vert v_i - r_{m,n} \vert\vert) \;\forall m,n$$ (12)

This can be described as the goodness of fit of $v_i$ to $r_w$ and its neighbouring map vectors described by the kernel function, $k$ (radius $d=2$). The values of the other parameters used in map generation are as follows: training rate $a = 0.05$; learning kernel radius $d = \min(M,N)/2$; training cycles $T=10000$.