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(the regularization term). For inadequate choices, the GTM may
reflect the structure in the data very poorly. The SOM has the same
problem, but needs to estimate somewhat less parameters.
An advantage of the GTM over the SOM is that the parameters that
need to be set by the user have a clear interpretation, unlike in the SOM
where unintuitive parameters as learning parameters and time para-
meters have to be defined. Furthermore, the end result is a probabilistic
model, which can easily be compared to other models (e.g. in terms of
likelihood) or combined with them.
Figure 7.12 shows some examples of GTMs (D ¼ 1) trained on uni-
formly distributed data. Figure 7.12(a) clearly shows how the GTM can
be overtrained if too many basis functions – here, 10 – are used. In
Figure 7.12(b), less basis functions are used and the manifold found is
much smoother. Another option is to use regularization, which also
gives a more smooth result as shown in Figure 7.12(c) but cannot
completely prevent extreme nonlinearities.
Listing 7.9 shows how a GTM can be trained and displayed in
PRTools.

Listing 7.9
PRTools code for training and plotting generative topographic mapping.

z ¼ rand(100,2); % Generate the data set z
w ¼ gtm(z,15); % Train a 1D GTM and show it
figure; clf; scatterd(z); plotgtm(w);

(a) (b) (c)

Figure 7.12 Trained generative topographic mappings. (a) K ¼ 14, M ¼ 10, ’ ¼ 0:2
and ¼ 0. (b) K ¼ 14, M ¼ 5, ’ ¼ 0:2and ¼ 0. (c) K ¼ 14, M ¼ 10, ’ ¼ 0:2and
¼ 0:01