Skip to main content
SpringerLink
Log in

Clustering I: Basic Clustering Models and Algorithms

  • Chapter
  • First Online:
Book cover Neural Networks and Statistical Learning

  • 4319 Accesses

Abstract

Clustering is an unsupervised classification technique that identifies some inherent structure present in a set of objects based on a similarity measure. Clustering methods can be derived from statistical models or competitive learning and correspondingly they can be classified into generative (or model-based) and discriminative (or similarity-based) approaches. A clustering problem can also be modeled as a COP. Clustering neural networks are statistical models, where a probability density function (pdf) for data is estimated by learning its parameters. In this chapter, our emphasis is placed on a number of competitive learning-based neural networks and clustering algorithms. We describe the SOM, learning vector quantization (LVQ), and ART models, as well as C-means, subtractive, and fuzzy clustering algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The link distance between two points A and B inside a polygon P is defined to be the minimum number of edges required to connect A and B inside P.

References

  1. Alex, N., Hasenfuss, A., & Hammer, B. (2009). Patch clustering for massive data sets. Neurocomputing, 72, 1455–1469.

    Article  Google Scholar 

  2. Anderson, I. A., Bezdek, J. C., & Dave, R. (1982). Polygonal shape description of plane boundaries. In L. Troncale (Ed.), Systems science and science (Vol. 1, pp. 295–301). Louisville, KY: SGSR.

    Google Scholar 

  3. Angelov, P. P., & Filev, D. P. (2004). An approach to online identification of Takagi–Sugeno fuzzy models. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34(1), 484–498.

    Article  Google Scholar 

  4. Aoki, T., & Aoyagi, T. (2007). Self-organizing maps with asymmetric neighborhood function. Neural Computation, 19, 2515–2535.

    Article  MathSciNet  MATH  Google Scholar 

  5. Atiya, A. F. (2005). Estimating the posterior probabilities using the \(K\)-nearest neighbor rule. Neural Computation, 17, 731–740.

    Article  MATH  Google Scholar 

  6. Baraldi, A., & Blonda, P. (1999). A survey of fuzzy clustering algorithms for pattern recognition—Part II. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 29(6), 786–801.

    Article  Google Scholar 

  7. Bax, E. (2012). Validation of \(k\)-nearest neighbor classifiers. IEEE Transactions on Information Theory, 58(5), 3225–3234.

    Article  MathSciNet  MATH  Google Scholar 

  8. Beliakov, G., & Li, G. (2012). Improving the speed and stability of the \(k\)-nearest neighbors method. Pattern Recognition Letters, 33, 1296–1301.

    Article  Google Scholar 

  9. Beni, G., & Liu, X. (1994). A least biased fuzzy clustering method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(9), 954–960.

    Article  Google Scholar 

  10. Berglund, E., & Sitte, J. (2006). The parameterless self-organizing map algorithm. IEEE Transactions on Neural Networks, 17(2), 305–316.

    Article  Google Scholar 

  11. Bermejo, S. (2006). The regularized LVQ1 algorithm. Neurocomputing, 70, 475–488.

    Article  Google Scholar 

  12. Bezdek, J. (1981). Pattern recognition with fuzzy objective function algorithms. New York: Plenum Press.

    Google Scholar 

  13. Biehl, M., Ghosh, A., & Hammer, B. (2007). Dynamics and generalization ability of LVQ algorithms. Journal of Machine Learning Research, 8, 323–360.

    MathSciNet  MATH  Google Scholar 

  14. Brodal, A. (1981). Neurological anatomy in relation to clinical medicine (3rd ed.). New York: Oxford University Press.

    Google Scholar 

  15. Burke, L. I. (1991). Clustering characterization of adaptive resonance. Neural Networks, 4(4), 485–491.

    Article  Google Scholar 

  16. Carpenter, G., Grossberg, S., & Rosen, D. B. (1991). ART 2-A: An adaptive resonance algorithm for rapid category learning and recognition. Neural Networks, 4, 493–504.

    Article  Google Scholar 

  17. Carpenter, G. A., & Grossberg, S. (1987). A massively parallel architecture for a self-organizing neural pattern recognition machine. Computer Vision, Graphics, and Image Processing, 37, 54–115.

    Article  MATH  Google Scholar 

  18. Carpenter, G. A., & Grossberg, S. (1987). ART 2: Self-organization of stable category recognition codes for analog input patterns. Applied Optics, 26, 4919–4930.

    Article  Google Scholar 

  19. Carpenter, G. A., Grossberg, S., & Reynolds, J. H. (1991). ARTMAP: Supervised real-time learning and classification of nonstationary data by a self-organizing neural network. Neural Networks, 4(5), 565–588.

    Article  Google Scholar 

  20. Carpenter, G. A., Grossberg, S., Markuzon, N., Reynolds, J. H., & Rosen, D. B. (1992). Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps. IEEE Transactions on Neural Networks, 3, 698–713.

    Article  Google Scholar 

  21. Cheng, Y. (1997). Convergence and ordering of Kohonen’s batch map. Neural Computation, 9, 1667–1676.

    Article  Google Scholar 

  22. Chinrunrueng, C., & Sequin, C. H. (1995). Optimal adaptive \(k\)-means algorithm with dynamic adjustment of learning rate. IEEE Transactions on Neural Networks, 6(1), 157–169.

    Article  Google Scholar 

  23. Chiu, S. (1994). Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems, 2(3), 267–278.

    Google Scholar 

  24. Chiu, S. L. (1994). A cluster estimation method with extension to fuzzy model identification. In Proceedings of the IEEE International Conference on Fuzzy Systems (Vol. 2, pp. 1240–1245). Orlando, FL.

    Google Scholar 

  25. Cho, J., Paiva, A. R. C., Kim, S.-P., Sanchez, J. C., & Principe, J. C. (2007). Self-organizing maps with dynamic learning for signal reconstruction. Neural Networks, 20, 274–284.

    Article  MATH  Google Scholar 

  26. Choy, C. S. T., & Siu, W. C. (1998). Fast sequential implementation of “neural-gas” network for vector quantization. IEEE Transactions on Communications, 46(3), 301–304.

    Article  Google Scholar 

  27. Cottrell, M., Hammer, B., Hasenfuss, A., & Villmann, T. (2006). Batch and median neural gas. Neural Networks, 19, 762–771.

    Article  MATH  Google Scholar 

  28. Cover, T. M., & Hart, P. E. (1967). Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13, 21–27.

    Article  MATH  Google Scholar 

  29. Dave, R. N., & Krishnapuram, R. (1997). Robust clustering methods: A unified view. IEEE Transactions on Fuzzy Systems, 5(2), 270–293.

    Article  Google Scholar 

  30. de Carvalho, F. A. T., Tenorio, C. P., & Cavalcanti, N. L, Jr. (2006). Partitional fuzzy clustering methods based on adaptive quadratic distances. Fuzzy Sets and Systems, 157, 2833–2857.

    Article  MathSciNet  MATH  Google Scholar 

  31. Demartines, P., & Herault, J. (1997). Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transactions on Neural Networks, 8(1), 148–154.

    Article  Google Scholar 

  32. Desarbo, W. S., Carroll, J. D., Clark, L. A., & Green, P. E. (1984). Synthesized clustering: A method for amalgamating clustering bases with differential weighting variables. Psychometrika, 49, 57–78.

    Article  MathSciNet  MATH  Google Scholar 

  33. Ding, C., & He, X. (2004). Cluster structure of \(k\)-means clustering via principal component analysis. In Proceedings of the 8th Pacific-Asia Conference on Advances in Knowledge Discovery and Data Mining (PAKDD), LNCS (Vol. 3056, pp. 414–418). Sydney, Australia; Berlin: Springer.

    Google Scholar 

  34. Du, K.-L., & Swamy, M. N. S. (2006). Neural networks in a softcomputing framework. London: Springer.

    Google Scholar 

  35. Du, K.-L. (2010). Clustering: A neural network approach. Neural Networks, 23(1), 89–107.

    Article  MATH  Google Scholar 

  36. Dunn, J. C. (1974). Some recent investigations of a new fuzzy partitioning algorithm and its applicatiopn to pattern classification problems. Journal of Cybernetics, 4, 1–15.

    Article  Google Scholar 

  37. Estevez, P. A., & Figueroa, C. J. (2006). Online data visualization using the neural gas network. Neural Networks, 19, 923–934.

    Article  MATH  Google Scholar 

  38. Fayed, H. A., & Atiya, A. F. (2009). A novel template reduction approach for the \(K\)-nearest neighbor method. IEEE Transactions on Neural Networks, 20(5), 890–896.

    Article  Google Scholar 

  39. Fix, E., & Hodges, J. L., Jr. (1951). Discriminatory analysis—Nonparametric discrimination: Consistency properties. Project No. 2-49-004, Report No. 4. Randolph Field, TX: USAF School of Aviation (Reprinted in International Statistical Review, 57(3), 238–247 (1989)).

    Google Scholar 

  40. Flanagan, J. A. (1996). Self-organization in Kohonen’s SOM. Neural Networks, 9(7), 1185–1197.

    Article  Google Scholar 

  41. Fort, J. C. (1988). Solving a combinatorial problem via self-organizing process: An application of Kohonen-type neural networks to the travelling salesman problem. Biology in Cybernetics, 59, 33–40.

    Article  MATH  Google Scholar 

  42. Fritzke, B. (1997). The LBG-U method for vector quantization—An improvement over LBG inspired from neural networks. Neural Processing Letters, 5(1), 35–45.

    Article  Google Scholar 

  43. Fukushima, K. (2010). Neocognitron trained with winner-kill-loser rule. Neural Networks, 23, 926–938.

    Article  Google Scholar 

  44. Gath, I., & Geva, A. B. (1989). Unsupervised optimal fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 773–781.

    Article  MATH  Google Scholar 

  45. Gersho, A. (1979). Asymptotically optimal block quantization. IEEE Transactions on Information Theory, 25(4), 373–380.

    Article  MathSciNet  MATH  Google Scholar 

  46. Ghosh, A., Biehl, M., & Hammer, B. (2006). Performance analysis of LVQ algorithms: A statistical physics approach. Neural Networks, 19, 817–829.

    Article  MATH  Google Scholar 

  47. Gottlieb, L.-A., Kontorovich, A., & Krauthgamer, R. (2014). Efficient classification for metric data. IEEE Transactions on Information Theory, 60(9), 5750–5759.

    Article  MathSciNet  MATH  Google Scholar 

  48. Grossberg, S. (1976). Adaptive pattern classification and universal recording: I. Parallel development and coding of neural feature detectors; II. Feedback, expectation, olfaction, and illusions. Biological Cybernetics, 23, 121–34, 187–202.

    Article  MathSciNet  MATH  Google Scholar 

  49. Guo, Y., & Sengur, A. (2015). NCM: Neutrosophic \(c\)-means clustering algorithm. Pattern Recognition, 48(8), 2710–2724.

    Article  Google Scholar 

  50. Gustafson, D. E., & Kessel, W. (1979). Fuzzy clustering with a fuzzy covariance matrix. In Proceedings of the IEEE Conference on Decision and Control (pp. 761–766). San Diego, CA.

    Google Scholar 

  51. Hall, L. O., & Goldgof, D. B. (2011). Convergence of the single-pass and online fuzzy \(C\)-means algorithms. IEEE Transactions on Fuzzy Systems, 19(4), 792–794.

    Article  Google Scholar 

  52. Hammer, B., Strickert, M., & Villmann, T. (2005). Supervised neural gas with general similarity measure. Neural Processing Letters, 21(1), 21–44.

    Article  Google Scholar 

  53. Hara, T., & Hirose, A. (2004). Plastic mine detecting radar system using complex-valued self-organizing map that deals with multiple-frequency interferometric images. Neural Networks, 17, 1201–1210.

    Article  Google Scholar 

  54. Hart, P. E. (1968). The condensed nearest neighbor rule. IEEE Transactions on Information Theory, 14(3), 515–516.

    Article  Google Scholar 

  55. Hathaway, R. J., & Bezdek, J. C. (1993). Switching regression models and fuzzy clustering. IEEE Transactions on Fuzzy Systems, 1, 195–204.

    Article  Google Scholar 

  56. Hathaway, R. J., & Bezdek, J. C. (2000). Generalized fuzzy \(c\)-means clustering strategies using \(L_p\) norm distances. IEEE Transactions on Fuzzy Systems, 8(5), 576–582.

    Article  Google Scholar 

  57. He, J., Tan, A. H., & Tan, C. L. (2004). Modified ART 2A growing network capable of generating a fixed number of nodes. IEEE Transactions on Neural Networks, 15(3), 728–737.

    Article  Google Scholar 

  58. Huang, J. Z., Ng, M. K., Rong, H., & Li, Z. (2005). Automated variable weighting in k-means type clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(5), 657–668.

    Article  Google Scholar 

  59. Huang, P., & Zhang, D. (2010). Locality sensitive \(C\)-means clustering algorithms. Neurocomputing, 73, 2935–2943.

    Article  Google Scholar 

  60. Hueter, G. J. (1988). Solution of the traveling salesman problem with an adaptive ring. In Proceedings of International Conference on Neural Networks (pp. 85–92). San Diego, CA.

    Google Scholar 

  61. Hwang, C., & Rhee, F. (2007). Uncertain fuzzy clustering: Interval type-2 fuzzy approach to \(C\)-means. IEEE Transactions on Fuzzy Systems, 15(1), 107–120.

    Article  Google Scholar 

  62. Kanungo, T., Mount, D. M., Netanyahu, N. S., Piatko, C. D., Silverman, R., & Wu, A. Y. (2002). An efficient \(k\)-means clustering algorithm: Analysis and implementation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(7), 881–892.

    Article  MATH  Google Scholar 

  63. Karayiannis, N. B., & Randolph-Gips, M. M. (2003). Soft learning vector quantization and clustering algorithms based on non-Euclidean norms: Multinorm algorithms. IEEE Transactions on Neural Networks, 14(1), 89–102.

    Article  Google Scholar 

  64. Kaymak, U., & Setnes, M. (2002). Fuzzy clustering with volume prototypes and adaptive cluster merging. IEEE Transactions on Fuzzy Systems, 10(6), 705–712.

    Article  Google Scholar 

  65. Kersten, P. R. (1999). Fuzzy order statistics and their application to fuzzy clustering. IEEE Transactions on Fuzzy Systems, 7(6), 708–712.

    Article  Google Scholar 

  66. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43, 59–69.

    Article  MathSciNet  MATH  Google Scholar 

  67. Kohonen, T. (1989). Self-organization and associative memory. Berlin: Springer.

    Book  MATH  Google Scholar 

  68. Kohonen, T. (1990). The self-organizing map. Proceedings of the IEEE, 78, 1464–1480.

    Article  Google Scholar 

  69. Kohonen, T. (1990). Improved versions of learning vector quantization. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 1, pp. 545–550). San Diego, CA.

    Google Scholar 

  70. Kohonen, T. (1990). Derivation of a class of training algorithms. IEEE Transactions on Neural Networks, 1, 229–232.

    Article  Google Scholar 

  71. Kohonen, T. (1996). Emergence of invariant-feature detectors in the adaptive-subspace self-organizing map. Biological Cybernetics, 75, 281–291.

    Article  MATH  Google Scholar 

  72. Kohonen, T. (2001). Self-organizing maps (3rd ed.). Berlin: Springer.

    Book  MATH  Google Scholar 

  73. Kohonen, T., Kangas, J., Laaksonen, J., & Torkkola, K. (1992). LVQPAK: A program package for the correct application of learning vector quantization algorithms. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 1, pp. 725–730). Baltimore, MD.

    Google Scholar 

  74. Kohonen, T. (2006). Self-organizing neural projections. Neural Networks, 19, 723–733.

    Article  MATH  Google Scholar 

  75. Kolen, J., & Hutcheson, T. (2002). Reducing the time complexity of the fuzzy \(C\)-means algorithm. IEEE Transactions on Fuzzy Systems, 10(2), 263–267.

    Article  Google Scholar 

  76. Krishnapuram, R., & Kim, J. (2000). Clustering algorithms based on volume criteria. IEEE Transactions on Fuzzy Systems, 8(2), 228–236.

    Article  Google Scholar 

  77. Kusumoto, H., & Takefuji, Y. (2006). \(O(\log _2 M)\) self-organizing map algorithm without learning of neighborhood vectors. IEEE Transactions on Neural Networks, 17(6), 1656–1661.

    Article  Google Scholar 

  78. Laaksonen, J., & Oja, E. (1996). Classification with learning k-nearest neighbors. In Proceedings of the International Conference on Neural Networks (pp. 1480–1483). Washington, DC.

    Google Scholar 

  79. Lai, J. Z. C., Huang, T.-J., & Liaw, Y.-C. (2009). A fast \(k\)-means clustering algorithm using cluster center displacement. Pattern Recognition, 42, 2551–2556.

    Article  MATH  Google Scholar 

  80. Leski, J. (2003). Towards a robust fuzzy clustering. Fuzzy Sets and Systems, 137, 215–233.

    Article  MathSciNet  MATH  Google Scholar 

  81. Li, M. J., Ng, M. K., Cheung, Y.-M., & Huang, J. Z. (2008). Agglomerative fuzzy \(K\)-means clustering algorithm with selection of number of clusters. IEEE Transactions on Knowledge and Data Engineering, 20(11), 1519–1534.

    Article  Google Scholar 

  82. Likas, A., Vlassis, N., & Verbeek, J. J. (2003). The global \(k\)-means clustering algorithm. Pattern Recognition, 36(2), 451–461.

    Article  Google Scholar 

  83. Linda, O., & Manic, M. (2012). General type-2 fuzzy \(C\)-means algorithm for uncertain fuzzy clustering. IEEE Transactions on Fuzzy Systems, 20(5), 883–897.

    Article  Google Scholar 

  84. Linde, Y., Buzo, A., & Gray, R. M. (1980). An algorithm for vector quantizer design. IEEE Transactions on Communications, 28, 84–95.

    Article  Google Scholar 

  85. Lippman, R. P. (1987). An introduction to computing with neural nets. IEEE ASSP Magazine, 4(2), 4–22.

    Article  MathSciNet  Google Scholar 

  86. Lo, Z. P., & Bavarian, B. (1991). On the rate of convergence in topology preserving neural networks. Biological Cybernetics, 65, 55–63.

    Article  MathSciNet  MATH  Google Scholar 

  87. Luttrell, S. P. (1990). Derivation of a class of training algorithms. IEEE Transactions on Neural Networks, 1, 229–232.

    Article  Google Scholar 

  88. Luttrell, S. P. (1994). A Bayesian analysis of self-organizing maps. Neural Computation, 6, 767–794.

    Article  MATH  Google Scholar 

  89. MacQueen, J. B. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability (pp. 281–297). Berkeley, CA: University of California Press.

    Google Scholar 

  90. Martinetz, T. M. (1993). Competitive Hebbian learning rule forms perfectly topology preserving maps. In Proceedings of the International Conference on Artificial Neural Networks (ICANN) (pp. 427–434). Amsterdam, The Netherlands.

    Google Scholar 

  91. Martinetz, T. M., Berkovich, S. G., & Schulten, K. J. (1993). Neural-gas network for vector quantization and its application to time-series predictions. IEEE Transactions on Neural Networks, 4(4), 558–569.

    Article  Google Scholar 

  92. Martinetz, T. M., & Schulten, K. J. (1994). Topology representing networks. Neural Networks, 7, 507–522.

    Article  Google Scholar 

  93. Moore, B. (1988). ART and pattern clustering. In D. Touretzky, G. Hinton & T. Sejnowski (Eds.), Proceedings of the 1988 Connectionist Model Summer School (pp. 174–183). San Mateo, CA: Morgan Kaufmann.

    Google Scholar 

  94. Mulder, S. A., & Wunsch, D. C, I. I. (2003). Million city traveling salesman problem solution by divide and conquer clustering with adaptive resonance neural networks. Neural Networks, 16, 827–832.

    Article  Google Scholar 

  95. Obermayer, K., Ritter, H., & Schulten, K. (1991). Development and spatial structure of cortical feature maps: A model study. In R. P. Lippmann, J. E. Moody, & D. S. Touretzky (Eds.), Advances in neural information processing systems (Vol. 3, pp. 11–17). San Mateo, CA: Morgan Kaufmann.

    Google Scholar 

  96. Odorico, R. (1997). Learning vector quantization with training count (LVQTC). Neural Networks, 10(6), 1083–1088.

    Article  Google Scholar 

  97. Pal, N. R., Bezdek, J. C., & Tsao, E. C. K. (1993). Generalized clustering networks and Kohonen’s self-organizing scheme. IEEE Transactions on Neural Networks, 4(2), 549–557.

    Article  Google Scholar 

  98. Pal, N. R., & Chakraborty, D. (2000). Mountain and subtractive clustering method: Improvements and generalizations. International Journal of Intelligent Systems, 15, 329–341.

    Article  MATH  Google Scholar 

  99. Patane, G., & Russo, M. (2001). The enhanced LBG algorithm. Neural Networks, 14(9), 1219–1237.

    Article  Google Scholar 

  100. Pedrycz, W., & Waletzky, J. (1997). Fuzzy clustering with partial supervision. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 27(5), 787–795.

    Article  Google Scholar 

  101. Richardt, J., Karl, F., & Muller, C. (1998). Connections between fuzzy theory, simulated annealing, and convex duality. Fuzzy Sets and Systems, 96, 307–334.

    Article  MathSciNet  Google Scholar 

  102. Rodriguez, A., & Laio, A. (2014). Clustering by fast search and find of density peaks. Science, 344(6191), 1492–1496.

    Article  Google Scholar 

  103. Rose, K. (1998). Deterministic annealing for clustering, compression, classification, regression, and related optimization problems. Proceedings of the IEEE, 86(11), 2210–2239.

    Article  Google Scholar 

  104. Rose, K., Gurewitz, E., & Fox, G. C. (1990). A deterministic annealing approach to clustering. Pattern Recognition Letters, 11(9), 589–594.

    Article  MATH  Google Scholar 

  105. Sato, A., & Yamada, K. (1995). Generalized learning vector quantization. In G. Tesauro, D. Touretzky, & T. Leen (Eds.), Advances in neural information processing systems (Vol. 7, pp. 423–429). Cambridge, MA: MIT Press.

    Google Scholar 

  106. Serrano-Gotarredona, T., & Linares-Barranco, B. (1996). A modified ART 1 algorithm more suitable for VLSI implementations. Neural Networks, 9(6), 1025–1043.

    Article  Google Scholar 

  107. Seo, S., & Obermayer, K. (2003). Soft learning vector quantization. Neural Computation, 15, 1589–1604.

    Article  MATH  Google Scholar 

  108. Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding machine learning: From theory to algorithms. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  109. Su, Z., & Denoeux, T. (2019). BPEC: Belief-peaks evidential clustering. IEEE Transactions on Fuzzy Systems, 27(1), 111–123.

    Article  Google Scholar 

  110. Tsekouras, G., Sarimveis, H., Kavakli, E., & Bafas, G. (2004). A hierarchical fuzzy-clustering approach to fuzzy modeling. Fuzzy Sets and Systems, 150(2), 245–266.

    Article  MathSciNet  MATH  Google Scholar 

  111. Verzi, S. J., Heileman, G. L., Georgiopoulos, M., & Anagnostopoulos, G. C. (2003). Universal approximation with fuzzy ART and fuzzy ARTMAP. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 3, pp. 1987–1892). Portland, OR.

    Google Scholar 

  112. von der Malsburg, C. (1973). Self-organizing of orientation sensitive cells in the striata cortex. Kybernetik, 14, 85–100.

    Article  Google Scholar 

  113. Wilson, D. L. (1972). Asymptotic properties of nearest neighbor rules using edited data. IEEE Transactions on Systems, Man, and Cybernetics, 2(3), 408–420.

    Article  MathSciNet  MATH  Google Scholar 

  114. Witoelar, A., Biehl, M., Ghosh, A., & Hammer, B. (2008). Learning dynamics and robustness of vector quantization and neural gas. Neurocomputing, 71, 1210–1219.

    Article  Google Scholar 

  115. Wu, K.-L., Yang, M.-S., & Hsieh, J.-N. (2010). Mountain \(c\)-regressions method. Pattern Recognition, 43, 86–98.

    Article  MATH  Google Scholar 

  116. Yager, R. R., & Filev, D. (1994). Approximate clustering via the mountain method. IEEE Transactions on Systems, Man, and Cybernetics, 24(8), 1279–1284.

    Article  Google Scholar 

  117. Yair, E., Zeger, K., & Gersho, A. (1992). Competitive learning and soft competition for vector quantizer design. IEEE Transactions on Signal Processing, 40(2), 294–309.

    Article  Google Scholar 

  118. Yang, M. S. (1993). On a class of fuzzy classification maximum likelihood procedures. Fuzzy Sets and Systems, 57, 365–375.

    Article  MathSciNet  MATH  Google Scholar 

  119. Yang, M.-S., Wu, K.-L., Hsieh, J.-N., & Yu, J. (2008). Alpha-cut implemented fuzzy clustering algorithms and switching regressions. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 38(3), 588–603.

    Article  Google Scholar 

  120. Yin, H., & Allinson, N. M. (1995). On the distribution and convergence of the feature space in self-organizing maps. Neural Computation, 7(6), 1178–1187.

    Google Scholar 

  121. Yu, J., Cheng, Q., & Huang, H. (2004). Analysis of the weighting exponent in the FCM. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34(1), 634–639.

    Article  Google Scholar 

  122. Yu, J., & Yang, M. S. (2005). Optimality test for generalized FCM and its application to parameter selection. IEEE Transactions on Fuzzy Systems, 13(1), 164–176.

    Article  Google Scholar 

  123. Zhao, W.-L., Deng, C.-H., & Ngo, C.-W. (2018). \(k\)-means: A revisit. Neurocomputing, 291, 195–206.

    Article  Google Scholar 

  124. Zheng, H., Lefebvre, G., & Laurent, C. (2008). Fast-learning adaptive-subspace self-organizing map: An application to saliency-based invariant image feature construction. IEEE Transactions on Neural Networks, 19(5), 746–757.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada

    Ke-Lin Du & M. N. S. Swamy

  2. Xonlink Inc., Hangzhou, China

    Ke-Lin Du

Authors
  1. Ke-Lin Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. N. S. Swamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ke-Lin Du .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer-Verlag London Ltd., part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Du, KL., Swamy, M.N.S. (2019). Clustering I: Basic Clustering Models and Algorithms. In: Neural Networks and Statistical Learning. Springer, London. https://doi.org/10.1007/978-1-4471-7452-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation