Lengths of principal components to determinate SOM grid

"Most applications of the SOM are based on regular arrays of nodes. Sometimes one uses rectangular arrays of nodes for simplicity. However, the hexagonal arrays are visually much more illustrative and accurate, and are recommended. Whatever regular architectures are used, it is advisable to select the lengths of the horizontal and vertical dimensions of the array to correspond to the lengths of the two largest principal components (i.e., those with the highest eigenvalues of the input correlation matrix), because then the SOM complies better with the low-order signal statistics.The oblong regular array shave the advantage over the square ones of guaranteeing faster and safer convergence in learning"

This is taken from the article "Essentials of the self-organizing map" written by Teuvo Kohonen itself. What does he mean by lenghts of the two largest principal components and what is the rationale of choosing the dimensions of the grid in this way?

It means that the 2D SOM map can be sized using the proportion of the largest principal component to the second largest. The rationale is that the principle components corresponds with the global structure, so provide some information about the global shape of the data to get a map that is the same shape as the data. Linear initialisation using the two largest principal components leads to significantly faster convergence.

Also touched on in Kohonen's recent book:

"Because the SOM is trying to represent the distribution of high-dimensional data items by a two-dimensional projection image, it may be understandable that the scales of the horizontal and vertical directions of the SOM array should approximately comply with the extensions of the input-data distribution in the two principal dimensions, namely, those two orthogonal directions in which the variances of the data are largest." Kohonen, T. K. (2014). MATLAB Implementations and Applications of the Self-Organizing Map.

Kohonen wrote this in a different way in his earlier book.

"If one wants to obtain an approximately uniform lattice spacing in the SOM, the relative numbers of cells in the horizontal and vertical directions of the lattice, respectively, should be proportional to the two largest eigenvalues considered above." Kohonen, T. K. (2001). Self-Organizing Maps. Springer Series in Information Sciences (Vol. 30). Springer-Verlag Berlin. Page 142. http://doi.org/10.1007/978-3-642-56927-2

Thus, the PCA is an approximation to the final solution by SOM. If your data is well suited to this variance analysis, then it will significantly speed up the algorithm without affecting the results.

• Applying SVD to centred data matrix $X=UDV^{T}$, principal component are defined as column of $DV$? For example, if the ratio of the first two eigenvalues is $4/1$ I should use the same proportion for the size of the map? – momomi May 8 '17 at 8:14
• Yes, that's what Kohonen suggested, to set the height and width to the ratio of the first two eigenvectors. – Tom Anderson Sep 6 '17 at 5:14