Abstract
In this paper, we define I-fuzzy partitions (or intuitionistic fuzzy partitions as called by Atanassov or interval-valued fuzzy partitions). As our ultimate goal is to compare the results of standard fuzzy clustering algorithms (e.g. fuzzy c-means), we define a method to construct them from a set of fuzzy clusters obtained from several executions of fuzzy c-means. From a practical point of view, the approach presented here tries to solve the difficulty of comparing the results of fuzzy clustering methods and, in particular, the difficulty of finding the global optimal.
Similar content being viewed by others
References
Atanassov KT (1983) Intuitionistic fuzzy sets. VII ITKR’s session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84) (in Bulgarian)
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Atanassov KT (1999) Intuitionistic fuzzy sets. Physica-Verlag, Heidelberg
Atanassov K (2005) Answer to Dubois D, Gottwald S, Hajek P, Kacprzyk J, Prade H’s paper Terminological difficulties in fuzzy set theory: the case of Intuitionistic fuzzy sets. Fuzzy Sets Syst 156(3):496–499
Bezdek JC (1981) Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York
Bhutani KR, Chaudhuri BB, Rosenfeld A (2002) Corrigendum to “A modified Hausdorff distance between fuzzy sets”. Inf Sci 148:233–234
Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78(3):305–316
Bustince H, Pagola M, Barrenechea E (2007) Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images. Inf Sci 177(3):906–929
Chaudhuri BB, Rosenfeld A (1999) A modified Hausdorff distance between fuzzy sets. Inf Sci 118:159–171
Domingo-Ferrer J, Torra V (2001) Disclosure control methods and information loss for microdata. In: Doyle P, Lane JI, Theeuwes JJM, Zayatz LM (eds) Confidentiality, disclosure, and data access: theory and practical applications for statistical agencies. Elsevier, pp 91–110
Dubois D, Gottwald S, Hajek P, Kacprzyk J, Prade H (2005) Terminological difficulties in fuzzy set theory—the case of intuitionistic fuzzy sets. Fuzzy Sets Syst 156(3): 485–491
Hasegawa Y, Endo Y, Hamasuna Y, Miyamoto S (2007) Fuzzy c-means for data with tolerance defined as hyper-rectangle. In: Proceedings of MDAI 2007, lecture notes in artificial intelligence, vol 4617, pp 237–248
Hastie T, Tibshirani R, Friedman J, (2001) The elements of statistical learning. Springer, Berlin
Höppner F, Klawonn F, Kruse R, Runkler T (1999) Fuzzy cluster analysis. Wiley
Ladra S, Torra V (2008) On the comparison of generic information loss measures and cluster-specific ones. Int J Unc Fuzzy Knowl Based Syst 16(1):107–120
Ladra S, Torra V (2010) Information loss for synthetic data through fuzzy clustering. Int J Unc Fuzzy Knowl Based Syst 18(1):25–37
Lane J, Heus P, Mulcahy T (2008) Data access in a cyber world: making use of cyberinfrastructure. Trans Data Priv 1(1):2–16
Lopez de Mantaras R (1991) A distance-based attribute selection. Measure for decision tree induction. Mach Learn 6: 81–92
Miyamoto S (1999) Introduction to fuzzy clustering (in Japanese). Ed. Morikita, Tokyo
Miyamoto S, Umayahara K (2000) Methods in hard and fuzzy clustering. In: Liu Z-Q, Miyamoto S (eds) Soft computing and human-centered machines. Springer, Tokyo, pp 85–129
Murata R, Endo Y, Haruyama H, Miyamoto S (2006) On fuzzy c-means for data with tolerance. J Adv Comput Intell Intell Inform 10(5):673–681
Pappis CP, Karacapilidis NI (1993) A comparative assessment of measures of similarity of fuzzy values. Fuzzy Sets Syst 56:171–174
Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets and Syst 114(3):505–518
Templ M, Petelin T (2009) A graphical user interface for microdata protection which provides reproducibility and interactions: the sdcMicro GUI. Trans Data Priv 2(3):207–224
Torra V, Endo Y, Miyamoto S (2009) On the comparison of some fuzzy clustering methods for privacy preserving data mining: towards the development of specific information loss measures. Kybernetika 45(3):548–560
Torra V, Endo Y, Miyamoto S (2009) Computationally intensive parameter selection for clustering algorithms: the case of fuzzy c-means with tolerance (submitted)
Acknowledgments
Partial support by the Spanish MEC (projects ARES – CONSOLIDER INGENIO 2010 CSD2007-00004 – and eAEGIS – TSI2007-65406-C03-02) is acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found at http://dx.doi.org/10.1007/s00500-010-0635-6
Rights and permissions
About this article
Cite this article
Torra, V., Miyamoto, S. A definition for I-fuzzy partitions. Soft Comput 15, 363–369 (2011). https://doi.org/10.1007/s00500-010-0605-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-010-0605-z