Study of the magnification effect on self-organizing maps - Bastos, Edson Giovanni Gonçalves da Cunha Moreira (2021)
Self-Organizing Maps (SOM), are a type of neuronal network (Kohonen, 1982b) that has been used mainly in data clustering problems, using unsupervised learning. Among the multiple areas of application, SOM has been used in various problems of direct interest to the Navy (V. J. Lobo, 2009), including route planning and the location of critical infrastructures. The SOM has also been used to sample large databases. In this sort of application, they have a behaviour called the magnification effect (Bauer & Der, 1996), which causes areas of the attribute space of data with less density to be overrepresented or magnified. This dissertation uses an experimental approach to mitigate the lack of theoretical explanation for this effect except for one-dimensional and quite simple cases. From experimental evidence obtained for carefully designed problems we infer a relationship between input data densities and output neuron densities that can be applied universally, or at least in a broad set of situations. A large number of experiments were conducted using one-dimensional to one-dimensional mappings followed by 2D to 2D, 3D to 1, 2 and 3D. We derived an empirical relationship whereby the density in the output space is equal to a constant times the density of the input space raised to the power of (alpha) which although depending on a number of factors can be approximated by the root index n of 2/3 where n is the input space dimension. The correlation that we found in our experiments, for both the well-known 1- dimensional case and for more general 2 to 3-dimensional cases is a useful guide to predict the magnification effect in practical situations.Therefore, in chapter 4 we produce a populational cartogram of Angola and we prove that our relation can be used to correct the magnification effect on 2-dimensional cases.