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Abstrakty
In this paper, we consider the l1-clustering problem for a finite data-point set which should be partitioned into k disjoint nonempty subsets. In that case, the objective function does not have to be either convex or differentiable, and generally it may have many local or global minima. Therefore, it becomes a complex global optimization problem. A method of searching for a locally optimal solution is proposed in the paper, the convergence of the corresponding iterative process is proved and the corresponding algorithm is given. The method is illustrated by and compared with some other clustering methods, especially with the l2-clustering method, which is also known in the literature as a smooth k-means method, on a few typical situations, such as the presence of outliers among the data and the clustering of incomplete data. Numerical experiments show in this case that the proposed l1-clustering algorithm is faster and gives significantly better results than the l2-clustering algorithm.
Rocznik
Tom
Strony
151--163
Opis fizyczny
Bibliogr. 47 poz., rys., tab., wykr.
Twórcy
autor
- Department of Mathematics, University of Osijek, Trg Lj. Gaja 6, HR 31 000 Osijek, Croatia
Bibliografia
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- [12] Cupec, R., Grbić, R., Sabo, K. and Scitovski, R. (2009). Three points method for searching the best least absolute deviations plane, Applied Mathematics and Computation 215(3): 983–994.
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- [33] Simiński, K. (2012). Neuro-rough-fuzzy approach for regression modelling from missing data, International Journal of Applied Mathematics and Computer Science 22(2): 461–476, DOI: 10.2478/v10006-012-0035-4.
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- [40] Sabo, K. and Scitovski, R. (2008). The best least absolute deviations line—properties and two efficient methods, ANZIAM Journal 50(2): 185–198.
- [41] Sabo, K., Scitovski, R. and Vazler, I. (2011). Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data, Journal of Optimization Theory and Applications 149(2): 293–314.
- [42] Sabo, K., Scitovski, R. and Vazler, I. (2012). One-dimensional center-based l1-clustering method, Optimization Letters 7(1): 5–22.
- [43] Sabo, K., Scitovski, R., Vazler, I. and Zekić-Sušac, M. (2011). Mathematical models of natural gas consumption, Energy Conversion and Management 52(3): 1721–1727.
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- [45] Vardi, Y., Zhang, C. H. (2000). The multivariate L1-median and associated data depth, Proceedings of the National Academy of Sciences, United States of America 97(4): 1423–1426.
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- [47] Zhang, J., Peng, L., Zhao, X. and Kuruoglu E.E. (2012 ). Robust data clustering by learning multi-metric lq-norm distances, Expert Systems with Applications 39(1): 335–349.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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