The SOM has several beneficial features which make
it a useful method for data mining. One of the most important
features is the ability to preserve the topology in the projection.
There are several measures that can be used to quantify the goodness
of the map in order to obtain the optimal projection, including the
average quantization error and many topological errors. Many
researches have studied how the topology preservation should be
measured. One option consists of using the topographic error which
considers the ratio of data vectors for which the first and second best
BMUs are not adjacent. In this work we present a study of the
behaviour of the topographic error in different kinds of maps. We
have found that this error devaluates the rectangular maps and we
have studied the reasons why this happens. Finally, we suggest a new
topological error to improve the deficiency of the topographic error.
[1] J. Vesanto. "Using SOM in Data Mining," Thesis for the degree of
Licentiate of Science in Technology. Espoo, Finland, 2000.
[2] G. Deboeck, T. Kohonen. "Visual Explorations in Finance with Self-
Organizing Maps," Springer Finance. Springer-Verlag, London, 1998.
[3] K. Lagus. "Text Mining with the WEBSOM," Acta Polytechnica
Scandinaviva, Mathematics and Computing Series No.110, Espoo.
Finnish Academies of Technology, 2000.
[4] Dolnicar, S. The Use of Neural Networks in Marketing: Market
Segmentation with Self-Organizing Feature Maps. Proceeding of the
Workshop on Self-Organizing Maps (WSOM'97), 38-43 (1997).
[5] Kohonen, T., Oja, E., Simula, O., Visa, A., Kangas, J. Engineering
application of the self-organizing map. Proceedings of the IEEE, Vol.84,
No.10, 1358-1384 (1996)
[6] J Kohonen, T.: Self-Organizing Maps. Vol. 30. of Springer Series in
Information Sciences, 3rd ed., Springer-Verlag, Berlin Heidelberg
(2001)
[7] T. Kohonen, "Self-organized formation of topologically correct feature
maps," Biological Cybernetics Vol. 43, 59-69 (1982).
[8] H. Ritter, T. Martinetz, K. Schultzen. "Neural Computation and Self-
Organizing Maps: An Introduction," Reading, MA: Addison-Weley
(1992)
[9] S. Kaski, "Data Exploration Using Self-Organizing Maps." PhD thesis,
Helsinky University of Technology, Acta Polytechnica Scandinavica:
Mathematics, Computing and Management in Engineering, 82. (1997).
[10] B. D. Ripley, "Pattern Recognition and Neural Networks." Cambridge
University Press, Cambridge, Great Britain (1996).
[11] .J. Mao, A. K. Jain, "Artificial neural networks for feature extraction and
multivariate data projection." IEEE Transaction on Neural Networks,
Vol.6, 296-317 (1995)
[12] J. Vesanto, E. Alhoniemi, "Clustering of the Self-Organizing Map."
IEEE Transactions on Neural Networks (2000).
[13] K. Obermayer, H. Ritter, K. Schultzen, "Development and spatial
structure of cortical feature maps: A model study". Advances in Neural
Information Proceedings Systems, Vol. 3, 11-17, San Mateo, CA (1991).
[14] M. Herrmann, "Self-Organizing Feature Map with Self-Organizing
Neighborhood Widths." IEEE International Conference on Neural
Networks (1995).
[15] A. Flexer, "Limitations of self-organizing maps for vector quantization
and multidimensional scaling." Technical Report oefai-tr-96-23. The
Australian Research Institute for Artificial Intelligence. (1997).
[16] G. J. Goodhill, T. J. Sejnowski. "Quantifying neighbourhood
preservation in topographic mappings." Proceedings of the 3rd Joint
Sympostium on Neural Computation. La Jolla. CA Vol. 6, 61-82 (1996).
[17] H.U. Bauer, K. R. Pawelzik, "Quantifying the Neighborhood
Preservation of Self-Organizing Feature Maps." IEEE Transactions on
Neural Networks, Vol. 3, No. 4, 570-579 (1992).
[18] J. C. Bezdek, N.R. Pal, "An index of topological preservation for feature
extraction." Pattern Recognition, Vol. 28, No. 3, 381-391 (1995).
[19] K. Kiviluoto, "Topology Preservation in Self-Organizing Maps."
Proceedings of International Conference on Neural Networks (ICNN),
294-299 (1996).
[20] SomToolbox, Helsinky University of Technology Available:
www.cis.hut.fi/projects/somtoolbox
[1] J. Vesanto. "Using SOM in Data Mining," Thesis for the degree of
Licentiate of Science in Technology. Espoo, Finland, 2000.
[2] G. Deboeck, T. Kohonen. "Visual Explorations in Finance with Self-
Organizing Maps," Springer Finance. Springer-Verlag, London, 1998.
[3] K. Lagus. "Text Mining with the WEBSOM," Acta Polytechnica
Scandinaviva, Mathematics and Computing Series No.110, Espoo.
Finnish Academies of Technology, 2000.
[4] Dolnicar, S. The Use of Neural Networks in Marketing: Market
Segmentation with Self-Organizing Feature Maps. Proceeding of the
Workshop on Self-Organizing Maps (WSOM'97), 38-43 (1997).
[5] Kohonen, T., Oja, E., Simula, O., Visa, A., Kangas, J. Engineering
application of the self-organizing map. Proceedings of the IEEE, Vol.84,
No.10, 1358-1384 (1996)
[6] J Kohonen, T.: Self-Organizing Maps. Vol. 30. of Springer Series in
Information Sciences, 3rd ed., Springer-Verlag, Berlin Heidelberg
(2001)
[7] T. Kohonen, "Self-organized formation of topologically correct feature
maps," Biological Cybernetics Vol. 43, 59-69 (1982).
[8] H. Ritter, T. Martinetz, K. Schultzen. "Neural Computation and Self-
Organizing Maps: An Introduction," Reading, MA: Addison-Weley
(1992)
[9] S. Kaski, "Data Exploration Using Self-Organizing Maps." PhD thesis,
Helsinky University of Technology, Acta Polytechnica Scandinavica:
Mathematics, Computing and Management in Engineering, 82. (1997).
[10] B. D. Ripley, "Pattern Recognition and Neural Networks." Cambridge
University Press, Cambridge, Great Britain (1996).
[11] .J. Mao, A. K. Jain, "Artificial neural networks for feature extraction and
multivariate data projection." IEEE Transaction on Neural Networks,
Vol.6, 296-317 (1995)
[12] J. Vesanto, E. Alhoniemi, "Clustering of the Self-Organizing Map."
IEEE Transactions on Neural Networks (2000).
[13] K. Obermayer, H. Ritter, K. Schultzen, "Development and spatial
structure of cortical feature maps: A model study". Advances in Neural
Information Proceedings Systems, Vol. 3, 11-17, San Mateo, CA (1991).
[14] M. Herrmann, "Self-Organizing Feature Map with Self-Organizing
Neighborhood Widths." IEEE International Conference on Neural
Networks (1995).
[15] A. Flexer, "Limitations of self-organizing maps for vector quantization
and multidimensional scaling." Technical Report oefai-tr-96-23. The
Australian Research Institute for Artificial Intelligence. (1997).
[16] G. J. Goodhill, T. J. Sejnowski. "Quantifying neighbourhood
preservation in topographic mappings." Proceedings of the 3rd Joint
Sympostium on Neural Computation. La Jolla. CA Vol. 6, 61-82 (1996).
[17] H.U. Bauer, K. R. Pawelzik, "Quantifying the Neighborhood
Preservation of Self-Organizing Feature Maps." IEEE Transactions on
Neural Networks, Vol. 3, No. 4, 570-579 (1992).
[18] J. C. Bezdek, N.R. Pal, "An index of topological preservation for feature
extraction." Pattern Recognition, Vol. 28, No. 3, 381-391 (1995).
[19] K. Kiviluoto, "Topology Preservation in Self-Organizing Maps."
Proceedings of International Conference on Neural Networks (ICNN),
294-299 (1996).
[20] SomToolbox, Helsinky University of Technology Available:
www.cis.hut.fi/projects/somtoolbox
@article{"International Journal of Information, Control and Computer Sciences:55469", author = "E. Arsuaga Uriarte and F. Díaz Martín", title = "Topology Preservation in SOM", abstract = "The SOM has several beneficial features which make
it a useful method for data mining. One of the most important
features is the ability to preserve the topology in the projection.
There are several measures that can be used to quantify the goodness
of the map in order to obtain the optimal projection, including the
average quantization error and many topological errors. Many
researches have studied how the topology preservation should be
measured. One option consists of using the topographic error which
considers the ratio of data vectors for which the first and second best
BMUs are not adjacent. In this work we present a study of the
behaviour of the topographic error in different kinds of maps. We
have found that this error devaluates the rectangular maps and we
have studied the reasons why this happens. Finally, we suggest a new
topological error to improve the deficiency of the topographic error.", keywords = "Map lattice, Self-Organizing Map, topographic
error, topology preservation.", volume = "2", number = "9", pages = "3020-4", }
