# One-Layer vs. Two-Layer SOM in the Context of Outlier Identification: A Simulation Study
## 6 Interpretation of the Results and Concluding Discussion  
The classic single-layer model obviates low-frequency components located close to similar higher frequency ones.  
That is why we developed the multilayer alternative with data segmentation in layers, the two-layer SOM. If it adequately represents the total set of points in the sample in addition to maintaining a better topological structure, it is a valid solution for its application  
Figure 18: Final 3D result.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Example** & **Image Distance** & **Graph Distance** \\ \hline
1-layer SOM & 1 & 32,456 & 5032 \\
2-layer SOM & 1 & 22,456 & 4625 \\
1-layer SOM & 2 & 22,456 & 4048 \\
2-layer SOM & 2 & 17,456 & 3740 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Table of the results of the different distance metrics of the algorithms used.  
in various areas where we can find these disparate low-frequency components, even more so if it can only be fuzzily delimited.  
We encountered several challenges. First was to define a stochastic model through which one can illustrate the contribution of our methodology, for example the low-frequency group of the population needed to be represented because, in the one-layer strategy, it was absorbed by higher frequency neighboring groups. Second was that an adequate stratified sampling procedure had to be developed. Some, but not excessive, prior information was required for adequate outlier group detection. Third, the computational complexity had to be controlled; parallelization was essential to this end. Fourth was the definition of one or more adequate distance measures to quantify conservation of topology: traditional distances measures were not satisfactory for our specific problem because some low-frequency regions of the population were not accounted for.  
From the results obtained in this article, it can be concluded that:  
1. The classic single-layer model ignored low-frequency components located between similar higher frequency ones. This was reflected in the distance matrix, as can be seen in the developed metrics. One was not adequately representing the original space or translating its topology. Not only was one ignoring one set of the sample, but this affected the representation of other points in the original space;
2. The stratification of the data (which may be given by prior knowledge) allowed, in two-layer SOMs, generating a map in which these components were also well represented;
3. For the two-layer map, stratification would result in the generation of intermediate-layer nodes that would represent points lying between components far apart, but included in the same stratum, and as inputs to the last layer, they would end up appearing to generate a more representative map, also in a topological sense.  
Therefore, the methodology developed for the bilayer SOM was proposed to solve the problem of the representativeness of low-frequency elements that can be assigned to other components of the system, as well as a better strategy to maintain the topological structure understood as distances.  
Conceptualization, G.A.V.C., J.M.M.M., and B.G.-P.; methodology, G.A.V.C., J.M.M.M., and B.G.-P.; software, G.A.V.C.; validation, G.A.V.C.; formal analysis, G.A.V.C. and J.M.M.M.; investigation, G.A.V.C., J.M.M.M., and B.G.-P.; writing--original draft preparation, G.A.V.C., J.M.M.M., and B.G.-P.; writing--review and editing, G.A.V.C., J.M.M.M., and B.G.-P.; visualization, G.A.V.C.; supervision, G.A.V.C., J.M.M.M., and B.G.-P.; project administration, G.A.V.C. All authors read and agreed to the published version of the manuscript.  
This research received no external funding  
Not applicable.  
The authors declare no conflict of interest.