In some cases it would be beneficial to have a single quality determinant rather than two. Of course the energy function of the SOM (equation 2.5) is such, but there have also been efforts to somehow combine the quality measures discussed above. Simple linear combination, as in [30], is not acceptable since no heuristic reason can be given for such a procedure.

Kaski and Lagus [18] proposed a quality measure, the topological quantization error , which combined the good aspects of the methods proposed earlier to form a measure of discontinuities in the representation of local data relations. In this measure the shortest path in input space from the sample vector through the best matching weight vector to the second-best matching weight vector is calculated as shown in figure 3.1. The path from the best to the second best unit is required to go by adjacent units. Finally the expectation over all paths is taken:

where *k* is replaced with *k*' after each step of the
summation.

**Figure 3.1:** *Calculation of the topological quantization error .
The two BMUs of the input vector marked with X are searched
and the minimum path from the input vector through
the best-matching weight vector to the
second best-matching weight vector is calculated.
*

Tue May 27 12:40:37 EET DST 1997