Self-organizing maps and boundary effects: quantifying the benefits of torus wrapping for mapping SOM trajectories

In this study the impact of a planar and toroidal self-organizing map (SOM) configuration are investigated with respect to their impact on SOM trajectories. Such trajectories are an encoding of processes within an n-dimensional input data set and offer an important means of visualizing and analyzing process complexity in large n-dimensional problem domains. However, discontinuity associated with boundaries in the standard, planar SOM results in error that limits their analytical use. Previous studies have recommended the use of a toroidal SOM to reduce these errors, but fall short of a fully quantified analysis of the benefits that result. In this study, the comparative analysis of fifteen pairs of identically initiated and trained SOMs, of planar and toroidal configuration, allows the error in trajectory magnitude to be quantified and visualized; both within the SOM and data space. This offers an important insight into the impact of planar SOM boundaries that goes beyond the general, statistical measures of clustering efficacy associated with previous work. The adoption of a toroidal SOM can be seen to improve the distribution of error in the trajectory sets, with the specific spatial configuration of SOM neurons associated with the largest errors changing from those at the corners of the planar SOM to a more complex and less predictable pattern in the toroidal SOM. However, this improvement is limited to the smallest 60% of errors, with torus and planar SOMs performing similarly for the largest 40%.