Abstract
This paper proposes a cluster validation procedure allowing to obtain the optimal number of clusters on a set of fuzzy partitions. Such a procedure is established considering fuzzy classification systems endowed with a dissimilarity function that, in turn, generates a dissimilarity matrix. Establishing a dissimilarity matrix for the case of a crisp partition, we propose an optimization problem comparing the characteristic polynomials of the fuzzy partition and crisp partition. Based on the above, we propose a definition for the optimal number of fuzzy classes in a fuzzy partition. Our approach is illustrated through an example on image analysis by the fuzzy c-means algorithm.
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Notes
- 1.
Reference [12] A recursive rule \(\rho \) is a family of aggregation operators
$$\{\rho _{n}:[0,1]^{n}\rightarrow [0,1]\}_{n>1}$$such that there exist an ordering rule \(\pi \) and two sequences of binary operators \(\{L_{n}:[0,1]^{2}\rightarrow [0,1]\}_{n>1}\) and \(\{R_{n}:[0,1]^{2}\rightarrow [0,1]\}_{n>1}\) such that for each n and for each \((a_{1},\ldots ,a_{n})\in [0,1]^{n}\), \(\rho _{n}(a_{\pi (1)},\ldots ,a_{\pi (n)})= L_{n}(\rho _{n-1}(a_{\pi (1)},\ldots ,a_{\pi (n-1)}),a_{\pi (n)})= R_{n}(a_{\pi (1)},\rho _{n-1}(a_{\pi (2)},\ldots ,a_{\pi (n)}).\)
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Acknowledgements
This research has been partially supported by the Government of Spain (grant PGC2018-096509-B-I00) Complutense University (UCM Research Group 910149) and Gran Colombia University (grant JCG2019-FCEM-01).
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Castiblanco, F., Franco, C., Rodriguez, J.T., Montero, J. (2022). Optimal Number of Classes in Fuzzy Partitions. In: Bede, B., Ceberio, M., De Cock, M., Kreinovich, V. (eds) Fuzzy Information Processing 2020. NAFIPS 2020. Advances in Intelligent Systems and Computing, vol 1337. Springer, Cham. https://doi.org/10.1007/978-3-030-81561-5_12
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