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Fuzzy clustering with Minkowski distance

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  • Groenen, P.J.F.
  • Kaymak, U.
  • van Rosmalen, J.M.

Abstract

Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L_1-distance and Bobrowski and Bezdek (1991) also used the L_infty-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski distance, Groenen and Jajuga (2001) introduced a majorization algorithm to minimize the error. One of the advantages of iterative majorization is that it is a guaranteed descent algorithm, so that every iteration reduces the error until convergence is reached. However, their algorithm was limited to the case of Minkowski parameter between 1 and 2, that is, between the L_1-distance and the Euclidean distance. Here, we extend their majorization algorithm to any Minkowski distance with Minkowski parameter greater than (or equal to) 1. This extension also includes the case of the L_infty-distance. We also investigate how well this algorithm performs and present an empirical application.

Suggested Citation

  • Groenen, P.J.F. & Kaymak, U. & van Rosmalen, J.M., 2006. "Fuzzy clustering with Minkowski distance," Econometric Institute Research Papers EI 2006-24, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:7873
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    References listed on IDEAS

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    1. Patrick Groenen & Willem Heiser, 1996. "The tunneling method for global optimization in multidimensional scaling," Psychometrika, Springer;The Psychometric Society, vol. 61(3), pages 529-550, September.
    2. Hunter D.R. & Lange K., 2004. "A Tutorial on MM Algorithms," The American Statistician, American Statistical Association, vol. 58, pages 30-37, February.
    3. P. J. F. Groenen & W. J. Heiser & J. J. Meulman, 1999. "Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing," Journal of Classification, Springer;The Classification Society, vol. 16(2), pages 225-254, July.
    4. Kiers, Henk A. L., 2002. "Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 157-170, November.
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