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Self-organizing Maps

  • Marc M. Van Hulle

Abstract

A topographic map is a two-dimensional, nonlinear approximation of a potentially high-dimensional data manifold, which makes it an appealing instrument for visualizing and exploring high-dimensional data. The self-organizing map (SOM) is the most widely used algorithm, and it has led to thousands of applications in very diverse areas. In this chapter we introduce the SOM algorithm, discuss its properties and applications, and also discuss some of its extensions and new types of topographic map formation, such as those that can be used for processing categorical data, time series, and tree-structured data.

Keywords

Input Space Travel Salesman Problem Neighborhood Function Voronoi Region Neighborhood Range 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The author is supported by the Excellence Financing program (EF 2005) and the CREA Financing program (CREA/07/027) of K.U.Leuven, the Belgian Fund for Scientific Research – Flanders (G.0234.04 and G.0588.09), the Flemish Regional Ministry of Education (Belgium) (GOA 2000/11), the Belgian Science Policy (IUAP P6/29), and the European Commission (NEST-2003-012963, STREP-2002-016276, IST-2004-027017, and ICT-2007-217077).

References

  1. Abrantes AJ, Marques JS (1995) Unified approach to snakes, elastic nets, and Kohonen maps. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’95). IEEE Computer Society, Los Alamitos, CA, vol 5, pp 3427–3430Google Scholar
  2. Ahalt SC, Krishnamurthy AK, Chen P, Melton DE (1990) Competitive learning algorithms for vector quantization. Neural Netw 3:277–290CrossRefGoogle Scholar
  3. Alahakoon D, Halgamuge SK, Srinivasan B (2000) Dynamic self organising maps with controlled growth for knowledge discovery. IEEE Trans Neural Netw (Special issue on knowledge discovery and data mining) 11(3):601–614Google Scholar
  4. Axelson D, Bakken IJ, Gribbestad IS, Ehrnholm B, Nilsen G, Aasly J (2002) Applications of neural network analyses to in vivo 1H magnetic resonance spectroscopy of Parkinson disease patients. J Magn Reson Imaging 16(1):13–20CrossRefGoogle Scholar
  5. Ball KD, Erman B, Dill KA (2002) The elastic net algorithm and protein structure prediction. J Comput Chem 23(1):77–83CrossRefGoogle Scholar
  6. Barreto G, Araújo A (2001) Time in self-organizing maps: an overview of models. Int J Comput Res 10(2):139–179Google Scholar
  7. Bauer H-U, Villmann T (1997) Growing a hypercubical output space in a self-organizing feature map. IEEE Trans Neural Netw 8(2):218–226CrossRefGoogle Scholar
  8. Bauer H-U, Der R, Herrmann M (1996) Controlling the magnification factor of self-organizing feature maps. Neural Comput 8:757–771CrossRefGoogle Scholar
  9. Benaim M, Tomasini L (1991) Competitive and self-organizing algorithms based on the minimization of an information criterion. In: Proceedings of 1991 international conference in artificial neural networks (ICANN'91). Espoo, Finland. Elsevier Science Publishers, North-Holland, pp 391–396Google Scholar
  10. Bishop CM (2006) Pattern recognition and machine learning. Springer, New YorkzbMATHGoogle Scholar
  11. Bishop CM, Svensén M, Williams CKI (1996) GTM: a principled alternative to the self-organizing map. In: Proceedings 1996 International Conference on Artificial Neural Networks (ICANN’96). Bochum, Germany, 16–19 July 1996. Lecture notes in computer science, vol 1112. Springer, pp 165–170Google Scholar
  12. Bishop CM, Hinton GE, and Strachan IGD (1997) In: Proceedings IEE fifth international conference on artificial neural networks. Cambridge UK, 7–9 July 1997, pp 111–116Google Scholar
  13. Bishop CM, Svensén M, Williams CKI (1998) GTM: the generative topographic mapping. Neural Comput 10:215–234CrossRefGoogle Scholar
  14. Blackmore J, Miikkulainen R (1993) Incremental grid growing: encoding high-dimensional structure into a two-dimensional feature map. In: Proceedings of IEEE international conference on neural networks. San Francisco, CA. IEEE Press, Piscataway, NJ, vol 1, pp 450–455Google Scholar
  15. Bruske J, Sommer G (1995) Dynamic cell structure learns perfectly topology preserving map. Neural Comput 7(4):845–865CrossRefGoogle Scholar
  16. Carreira-Perpiñán MÁ, Renals S (1998) Dimensionality reduction of electropalatographic data using latent variable models. Speech Commun 26(4):259–282CrossRefGoogle Scholar
  17. Centre NNR (2003) Bibliography on the Self-Organizing Map (SOM) and Learning Vector Quantization (LVQ), Helsinki University of Technology. http://liinwww.ira.uka.de/bibliography/Neural/SOM.LVQ.html
  18. Chappell G, Taylor J (1993) The temporal Kohonen map. Neural Netw 6:441–445CrossRefGoogle Scholar
  19. Chinrungrueng C, Séquin CH (1995) Optimal adaptive k-means algorithm with dynamic adjustment of learning rate. IEEE Trans Neural Netw 6:157–169CrossRefGoogle Scholar
  20. Cottrell M, Fort JC (1987) Etude d’un processus d’auto-organization. Ann Inst Henri Poincaré 23:1–20MathSciNetzbMATHGoogle Scholar
  21. D’Alimonte D, Lowe D, Nabney IT, Sivaraksa M (2005) Visualising uncertain data. In: Proceedings European conference on emergent aspects in clinical data analysis (EACDA2005). Pisa, Italy, 28–30 September 2005. http://ciml.di.unipi.it/EACDA2005/papers.html
  22. Deleus FF, Van Hulle MM (2001) Science and technology interactions discovered with a new topographic map-based visualization tool. In: Proceedings of 7th ACM SIGKDD international conference on knowledge discovery in data mining. San Francisco, 26–29 August 2001. ACM Press, New York, pp 42–50Google Scholar
  23. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood for incomplete data via the EM algorithm. J R Stat Soc B 39:1–38MathSciNetzbMATHGoogle Scholar
  24. Der R, Herrmann M (1993) Phase transitions in self-organizing feature maps. In: Proceedings of 1993 international conference on artificial neuron networks (ICANN'93). Amsterdam, The Netherlands, 13–16 September 1993, Springer, New York, pp 597–600Google Scholar
  25. DeSieno D (1988) Adding a conscience to competitive learning. In: Proceedings of IEEE international conference on neural networks. San Diego, CA, IEEE Press, New York, vol I, pp 117–124Google Scholar
  26. Durbin R, Willshaw D (1987) An analogue approach to the travelling salesman problem using an elastic net method. Nature 326:689–691CrossRefGoogle Scholar
  27. Durbin R, Szeliski R, Yuille AL (1989) An analysis of the elastic net approach to the traveling salesman problem. Neural Comput 1:348–358CrossRefGoogle Scholar
  28. Erwin E, Obermayer K, Schulten K (1992) Self-organizing maps: ordering, convergence properties and energy functions. Biol Cybern 67:47–55CrossRefzbMATHGoogle Scholar
  29. Euliano NR, Principe JC (1999). A spatiotemporal memory based on SOMs with activity diffusion. In: Oja E, Kaski S (eds) Kohonen maps. Elsevier, Amsterdam, The Netherlands, pp 253–266CrossRefGoogle Scholar
  30. Fritzke B (1994) Growing cell structures – a self-organizing network for unsupervised and supervised learning. Neural Netw 7(9):1441–1460CrossRefGoogle Scholar
  31. Fritzke B (1995a) A growing neural gas network learns topologies. In: Tesauro G, Touretzky DS, Leen TK (eds) Advances in neural information proceedings systems 7 (NIPS 1994). MIT Press, Cambridge, MA, pp 625–632Google Scholar
  32. Fritzke B (1995b) Growing grid – a self-organizing network with constant neighborhood range and adaptation strength. Neural Process Lett 2(5):9–13CrossRefGoogle Scholar
  33. Fritzke B (1996) Growing self-organizing networks – why? In: European symposium on artificial neural networks (ESANN96). Bruges, Belgium, 1996. D Facto Publications, Brussels, Belgium, pp 61–72Google Scholar
  34. Gautama T, Van Hulle MM (2006) Batch map extensions of the kernel-based maximum entropy learning rule. IEEE Trans Neural Netw 16(2):529–532CrossRefGoogle Scholar
  35. Gersho A, Gray RM (1991) Vector quantization and signal compression. Kluwer, Boston, MA/DordrechtGoogle Scholar
  36. Geszti T (1990) Physical models of neural networks. World Scientific Press, SingaporezbMATHGoogle Scholar
  37. Gilson SJ, Middleton I, Damper RI (1997) A localised elastic net technique for lung boundary extraction from magnetic resonance images. In: Proceedings of fifth international conference on artificial neural networks. Cambridge, UK, 7–9 July 1997. Mascarenhas Publishing, Stevenage, UK, pp 199–204Google Scholar
  38. Gorbunov S, Kisel I (2006) Elastic net for stand-alone RICH ring finding. Nucl Instrum Methods Phys Res A 559:139–142CrossRefGoogle Scholar
  39. Graepel T, Burger M, Obermayer K (1997) Phase transitions in stochastic self-organizing maps. Phys Rev E 56(4):3876–3890CrossRefGoogle Scholar
  40. Graepel T, Burger M, Obermayer K (1998) Self-organizing maps: generalizations and new optimization techniques. Neurocomputing 21:173–190CrossRefzbMATHGoogle Scholar
  41. Grossberg S (1976) Adaptive pattern classification and universal recoding: I. Parallel development and coding of neural feature detectors. Biol Cybern 23:121–134MathSciNetCrossRefzbMATHGoogle Scholar
  42. Günter S, Bunke H (2002) Self-organizing map for clustering in the graph domain, Pattern Recog Lett 23:415–417CrossRefGoogle Scholar
  43. Hagenbuchner M, Sperduti A, Tsoi AC (2003) A self-organizing map for adaptive processing of structured data. IEEE Trans Neural Netw 14(3):491–505CrossRefGoogle Scholar
  44. Hammer B, Micheli A, Strickert M, Sperduti A (2004) A general framework for unsupervised processing of structured data. Neurocomputing 57:3–35CrossRefGoogle Scholar
  45. Hammer B, Micheli A, Neubauer N, Sperduti A, Strickert M (2005) Self organizing maps for time series. In: Proceedings of WSOM 2005. Paris, France, 5–8 September 2005, pp 115–122Google Scholar
  46. Heskes T (2001) Self-organizing maps, vector quantization, and mixture modeling. IEEE Trans Neural Netw 12(6):1299–1305CrossRefGoogle Scholar
  47. Heskes TM, Kappen B (1993) Error potentials for self-organization. In: Proceedings of IEEE international conference on neural networks. San Francisco, CA. IEEE Press, Piscataway, NJ, pp 1219–1223Google Scholar
  48. Jin B, Zhang Y-Q, Wang B (2005) Evolutionary granular kernel trees and applications in drug activity comparisons, In: Proceedings of the 2005 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB’05). San Diego, CA, 14–15 November 2005, IEEE Press, Piscataway, NY, pp 1–6Google Scholar
  49. Kabán A (2005) A scalable generative topographic mapping for sparse data sequences. In: Proceedings of the International Conference on Information Technology: Coding and Computing (ITCC’05). Las Vegas, NV, 4–6 April 2005. IEEE Computer Society, vol 1, pp 51–56Google Scholar
  50. Kass M, Witkin A, Terzopoulos D (1987) Active contour models. Int J Comput Vis 1(4):321–331CrossRefGoogle Scholar
  51. Kangas J (1990) Time-delayed self-organizing maps. In: Proceedings IEEE/INNS international Joint Conference on neural networks 1990. San Diego, CA, IEEE, New York, vol 2, pp 331–336Google Scholar
  52. Kaski S, Honkela T, Lagus K, Kohonen T (1998) WEBSOM – self-organizing maps of document collections. Neurocomputing 21:101–117CrossRefzbMATHGoogle Scholar
  53. Kim YK, Ra JB (1995) Adaptive learning method in self-organizing map for edge preserving vector quantization. IEEE Trans Neural Netw 6:278–280CrossRefGoogle Scholar
  54. Kitamoto A (2002) Evolution map: modeling state transition of typhoon image sequences by spatio-temporal clustering. Lect Notes Comput Sci 2534/2002: 283–290CrossRefGoogle Scholar
  55. Kohonen T (1982) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59–69MathSciNetCrossRefzbMATHGoogle Scholar
  56. Kohonen T (1984) Self-organization and associative memory. Springer, HeidelbergzbMATHGoogle Scholar
  57. Kohonen T (1991) Self-organizing maps: optimization approaches. In: Kohonen T, Mäkisara K, Simula O, Kangas J (eds) Artificial neural networks. North-Holland, Amsterdam, pp 981–990Google Scholar
  58. Kohonen T (1995) Self-organizing maps, 2nd edn. Springer, HeidelbergGoogle Scholar
  59. Kohonen T (1997) Self-organizing maps. SpringerGoogle Scholar
  60. Kohonen T, Somervuo P (1998) Self-organizing maps on symbol strings. Neurocomputing 21:19–30CrossRefzbMATHGoogle Scholar
  61. Kohonen T, Kaski S, Salojärvi J, Honkela J, Paatero V, Saarela A (1999) Self organization of a massive document collection. IEEE Trans Neural Netw 11(3): 574–585CrossRefGoogle Scholar
  62. Koskela T, Varsta M, Heikkonen J, Kaski K (1998) Recurrent SOM with local linear models in time series prediction. In: Verleysen M (ed) Proceedings of 6th European symposium on artificial neural networks (ESANN 1998). Bruges, Belgium, April 22–24, 1998. D-Facto, Brussels, Belgium, pp 167–172Google Scholar
  63. Kostiainen T, Lampinen J (2002) Generative probability density model in the self-organizing map. In: Seiffert U, Jain L (eds) Self-organizing neural networks: Recent advances and applications. Physica-Verlag, Heidelberg, pp 75–94Google Scholar
  64. Laaksonen J, Koskela M, Oja E (2002) PicSOM–self-organizing image retrieval with MPEG-7 content descriptors. IEEE Trans Neural Netw 13(4):841–853CrossRefGoogle Scholar
  65. Lin JK, Grier DG, Cowan JD (1997) Faithful representation of separable distributions. Neural Comput 9:1305–1320CrossRefGoogle Scholar
  66. Linsker R (1988) Self-organization in a perceptual network. Computer 21:105–117CrossRefGoogle Scholar
  67. Linsker R (1989) How to generate ordered maps by maximizing the mutual information between input and output signals. Neural Comput 1:402–411CrossRefGoogle Scholar
  68. Luttrell SP (1989) Self-organization: a derivation from first principles of a class of learning algorithms. In: Proceedings IEEE international joint conference on neural networks (IJCNN89). Washington, DC, Part I, IEEE Press, Piscataway, NJ, pp 495–498Google Scholar
  69. Luttrell SP (1990) Derivation of a class of training algorithms. IEEE Trans Neural Netw 1:229–232CrossRefGoogle Scholar
  70. Luttrell SP (1991) Code vector density in topographic mappings: scalar case. IEEE Trans Neural Netw 2:427–436CrossRefGoogle Scholar
  71. Martinetz TM (1993) Competitive Hebbian learning rule forms perfectly topology preserving maps. In: Proceedings of international conference on artificial neural networks (ICANN93). Amsterdam, The Netherlands, 13–16 September 1993. Springer, London, pp 427–434Google Scholar
  72. Martinetz T, Schulten K (1991) A “neural-gas” network learns topologies. In: Kohonen T, Mäkisara K, Simula O, Kangas J (eds) Proceedings of International Conference on Artificial Neural Networks (ICANN-91). Espoo, Finland, 24–28 June 1991, vol I, North-Holland, Amsterdam, The Netherlands, pp 397–402Google Scholar
  73. Martinetz T, Berkovich S, Schulten K (1993) “Neural-gas” network for vector quantization and its application to time-series prediction. IEEE Trans Neural Netw 4(4):558–569CrossRefGoogle Scholar
  74. Merkl D, He S, Dittenbach M, Rauber A (2003) Adaptive hierarchical incremental grid growing: an architecture for high-dimensional data visualization. In: Proceedings of 4th workshop on self-organizing maps (WSOM03). Kitakyushu, Japan, 11–14 September 2003Google Scholar
  75. Mulier F, Cherkassky V (1995) Self-organization as an iterative kernel smoothing process. Neural Comput 7:1165–1177CrossRefGoogle Scholar
  76. Rauber A, Merkl D, Dittenbach M (2002) The growing hierarchical self-organizing map: exploratory analysis of high-dimensional data. IEEE Trans Neural Netw 13(6):1331–1341CrossRefGoogle Scholar
  77. Redner RA, Walker HF (1984) Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev 26(2):195–239MathSciNetCrossRefzbMATHGoogle Scholar
  78. Risi S, Mörchen F, Ultsch A, Lewark P (2007) Visual mining in music collections with emergent SOM. In: Proceedings of workshop on self-organizing maps (WSOM ’07). Bielefeld, Germany, September 3–6, 2007, ISBN: 978-3-00-022473-7, CD ROM, available online at http://biecoll.ub.uni-bielefeld.de
  79. Ritter H (1991) Asymptotic level density for a class of vector quantization processes. IEEE Trans Neural Netw 2(1):173–175MathSciNetCrossRefGoogle Scholar
  80. Ritter H (1998) Self-organizing maps in non-Euclidean spaces. In: Oja E, Kaski S (eds) Kohonen maps. Elsevier, Amsterdam, pp 97–108Google Scholar
  81. Ritter H, Kohonen T (1989) Self-organizing semantic maps. Biol Cybern 61:241–254CrossRefGoogle Scholar
  82. Ritter H, Schulten K (1986) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99–106CrossRefzbMATHGoogle Scholar
  83. Ritter H, Schulten K (1988) Kohonen's self-organizing maps: exploring their computational capabilities, In: Proceedings of IEEE international conference on neural networks (ICNN). San Diego, CA. IEEE, New York, vol I, pp 109–116Google Scholar
  84. Ritter H, Martinetz T, Schulten K (1992) Neural computation and self-organizing maps: an introduction. Addison-Wesley, Reading, MAzbMATHGoogle Scholar
  85. Rose K, Gurewitz E, Fox GC (1993) Constrained clustering as an optimization method. IEEE Trans Pattern Anal Mach Intell 15(8):785–794CrossRefGoogle Scholar
  86. Schulz R, Reggia JA (2004) Temporally asymmetric learning supports sequence processing in multi-winner self-organizing maps. Neural Comput 16(3):535–561CrossRefzbMATHGoogle Scholar
  87. Seo S, Obermayer K (2004) Self-organizing maps and clustering methods for matrix data. Neural Netw 17(8–9):1211–1229CrossRefzbMATHGoogle Scholar
  88. Shawe-Taylor J, Cristianini N (2004) Kernel methods in computational biology. MIT Press, Cambridge, MAGoogle Scholar
  89. Simon G, Lendasse A, Cottrell M, Fort J-C, Verleysen M (2003) Double SOM for long-term time series prediction. In: Proceedings of the workshop on self-organizing maps (WSOM 2003). Hibikino, Japan, September 11–14, 2003, pp 35–40Google Scholar
  90. Somervuo PJ (2004) Online algorithm for the self-organizing map of symbol strings. Neural Netw 17(8–9):1231–1240CrossRefGoogle Scholar
  91. Steil JJ, Sperduti A (2007) Indices to evaluate self-organizing maps for structures. In: Proceedings of the workshop on self-organizing maps (WSOM07) Bielefeld, Germany, 3–6 September 2007. CD ROM, 2007, available online at http://biecoll.ub.uni-bielefeld.de
  92. Strickert M, Hammer B (2003a) Unsupervised recursive sequence processing, In: Verleysen M (ed) European Symposium on Artificial Neural Networks (ESANN 2003). Bruges, Belgium, 23–25 April 2003. D-Side Publications, Evere, Belgium, pp 27–32Google Scholar
  93. Strickert M, Hammer B (2003b) Neural gas for sequences. In: Proceedings of the workshop on self-organizing maps (WSOM’03). Hibikino, Japan, September 2003, pp 53–57Google Scholar
  94. Strickert M, Hammer B (2005) Merge SOM for temporal data. Neurocomputing 64:39–72CrossRefGoogle Scholar
  95. Tiňo P, Nabney I (2002) Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way. IEEE Trans Pattern Anal Mach Intell 24(5):639–656CrossRefGoogle Scholar
  96. Tiňo P, Kabán A, Sun Y (2004) A generative probabilistic approach to visualizing sets of symbolic sequences. In: Kohavi R, Gehrke J, DuMouchel W, Ghosh J (eds) Proceedings of the tenth ACM SIGKDD international conference on knowledge discovery and data mining (KDD-2004), Seattle, WA, 22–25 August 2004. ACM Press, New York, pp 701–706Google Scholar
  97. Tolat V (1990) An analysis of Kohonen's self-organizing maps using a system of energy functions. Biol Cybern 64:155–164CrossRefzbMATHGoogle Scholar
  98. Ultsch A, Siemon HP (1990) Kohonen's self organizing feature maps for exploratory data analysis. In: Proceedings international neural networks. Kluwer, Paris, pp 305–308Google Scholar
  99. Ultsch A, Mörchen F (2005) ESOM-Maps: Tools for clustering, visualization, and classification with emergent SOM. Technical Report No. 46, Department of Mathematics and Computer Science, University of Marburg, GermanyGoogle Scholar
  100. Ueda N, Nakano R (1993) A new learning approach based on equidistortion principle for optimal vector quantizer design. In: Proceedings of IEEE NNSP93, Linthicum Heights, MD. IEEE, Piscataway, NJ, pp 362–371Google Scholar
  101. Van den Bout DE, Miller TK III (1989) TInMANN: the integer Markovian artificial neural network. In: Proceedings of international joint conference on neural networks (IJCNN89). Washington, DC, 18–22 June 1989, Erlbaum, Englewood Chifts, NJ, pp II205–II211Google Scholar
  102. Van Hulle MM (1997a) Topology-preserving map formation achieved with a purely local unsupervised competitive learning rule. Neural Netw 10(3): 431–446CrossRefGoogle Scholar
  103. Van Hulle MM (1997b) Nonparametric density estimation and regression achieved with topographic maps maximizing the information-theoretic entropy of their outputs. Biol Cybern 77:49–61CrossRefzbMATHGoogle Scholar
  104. Van Hulle MM (1998) Kernel-based equiprobabilistic topographic map formation. Neural Comput 10(7):1847–1871CrossRefGoogle Scholar
  105. Van Hulle MM (2000) Faithful representations and topographic maps: from distortion- to information-based self-organization. Wiley, New YorkGoogle Scholar
  106. Van Hulle MM (2002a) Kernel-based topographic map formation by local density modeling. Neural Comput 14(7):1561–1573CrossRefzbMATHGoogle Scholar
  107. Van Hulle MM (2002b) Joint entropy maximization in kernel-based topographic maps. Neural Comput 14(8):1887–1906Google Scholar
  108. Van Hulle MM (2005a) Maximum likelihood topographic map formation. Neural Comput 17(3):503–513CrossRefzbMATHGoogle Scholar
  109. Van Hulle MM (2005b) Edgeworth-expanded topographic map formation. In: Proceedings of workshop on self-organizing maps (WSOM05). Paris, France, 5–8 September 2005, pp 719–724Google Scholar
  110. Van Hulle MM (2009) Kernel-based topographic maps: theory and applications. In: Wah BW (ed) Encyclopedia of computer science and engineering. Wiley, Hoboken, vol 3, pp 1633–1650Google Scholar
  111. Van Hulle MM, Gautama T (2004) Optimal smoothing of kernel-based topographic maps with application to density-based clustering of shapes. J VLSI Signal Proces Syst Signal, Image, Video Technol 37:211–222CrossRefGoogle Scholar
  112. Verbeek JJ, Vlassis N, Kröse BJA (2005) Self-organizing mixture models. Neurocomputing 63:99–123CrossRefGoogle Scholar
  113. Vesanto J (1997) Using the SOM and local models in time-series prediction. In: Proceedings of workshop on self-organizing maps (WSOM 1997). Helsinki, Finland, 4–6 June 1997. Helsinki University of Technology, Espoo, Finland, pp 209–214Google Scholar
  114. Voegtlin T (2002) Recursive self-organizing maps. Neural Netw 15(8–9):979–992CrossRefGoogle Scholar
  115. Wiemer JC (2003) The time-organized map algorithm: extending the self-organizing map to spatiotemporal signals. Neural Comput 15(5):1143–1171CrossRefzbMATHGoogle Scholar
  116. Willshaw DJ, von der Malsburg C (1976) How patterned neural connections can be set up by self-organization. Proc Roy Soc Lond B 194:431–445Google Scholar
  117. Yin H, Allinson NM (2001) Self-organizing mixture networks for probability density estimation. IEEE Trans Neural Netw 12:405–411CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marc M. Van Hulle
    • 1
  1. 1.Laboratorium voor NeurofysiologieK.U. LeuvenLeuvenBelgium

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