×

zbMATH — the first resource for mathematics

Fuzzy \(c\)-ordered-means clustering. (English) Zbl 06840609
Summary: Fuzzy clustering helps to find natural vague boundaries in data. The fuzzy \(c\)-means method is one of the most popular clustering methods based on minimization of a criterion function. However, one of the greatest disadvantages of this method is its sensitivity to the presence of noise and outliers in data. This paper introduces a new robust fuzzy clustering method named fuzzy \(C\)-ordered-means (FCOM) clustering. This method uses both the Huber’s M-estimators and the Yager’s OWA operators to obtain its robustness. The proposed method is compared to many other ones, e.g.: the fuzzy \(C\)-means (FCM), the possibilistic clustering (PC), the fuzzy noise clustering method (NCM), the \(L_p\) norm clustering (\(L_p\) FCM) (\(0 < p < 1\)), the \(L_1\) norm clustering (\(L_1\) FCM), the fuzzy clustering with polynomial fuzzifier (PFCM) and the \(\varepsilon\)-insensitive fuzzy \(C\)-means (\(\beta\)FCM). To this end, experiments on synthetic data with outliers are performed as well as on data with heavy-tailed and overlapping groups of points in background noise.

MSC:
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H86 Multivariate analysis and fuzziness
Software:
robustbase
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bezdek, J. C., Pattern recognition with fuzzy objective function algorithms, (1982), Plenum Press New York
[2] De Carvalho, F.; Tenorio, C. P., Fuzzy k-means clustering algorithms for interval-valued data based on adaptive quadratic distances, Fuzzy Sets Syst., 161, 3, 2978-2999, (2010) · Zbl 1204.62106
[3] Coppi, R.; D’Urso, P.; Giordani, P., Fuzzy and possibilistic clustering models for fuzzy data, Comput. Stat. Data Anal., 56, 915-927, (2012) · Zbl 1243.62089
[4] Davé, R. N., Characterization and detection of noise in clustering, Pattern Recognit. Lett., 12, 11, 657-664, (1991)
[5] Davé, R. N.; Krishnapuram, R., Robust clustering methods: a unified view, IEEE Trans. Fuzzy Syst., 5, 2, 270-293, (1997)
[6] D’Urso, P., Fuzzy clustering for data time arrays with inlier and outlier time trajectories, IEEE Trans. Fuzzy Syst., 13, 5, 583-604, (2005)
[7] D’Urso, P.; Giordani, P., A robust fuzzy k-means clustering model for interval valued data, Comput. Stat., 21, 251-269, (2006) · Zbl 1113.62076
[8] D’Urso, P.; Massari, R., Fuzzy clustering of human activity patterns, Fuzzy Sets Syst., 215, 29-54, (2013)
[9] D’Urso, P.; De Giovanni, L.; Massari, R.; Di Lallo, D., Noise fuzzy clustering of time series by the autoregressive metric, Metron, 71, 217-243, (2013) · Zbl 1302.62207
[10] D’Urso, P.; De Giovanni, L., Robust clustering of imprecise data, Chemom. Intell. Lab. Syst., 136, 58-80, (2014)
[11] D’Urso, P.; De Giovanni, L.; Massari, R., Trimmed fuzzy clustering for interval-valued data, Adv. Data Anal. Classif., (2014), in press
[12] D’Urso, P., Fuzzy clustering, (Hennig, C.; Meila, M.; Murtagh, F.; Rocci, R., Handbook of Cluster Analysis, (2015), Chapman & Hall), in press
[13] Duda, R. O.; Hart, P. E., Pattern classification and scene analysis, (1973), John Wiley & Sons New York · Zbl 0277.68056
[14] Dunn, J. C., A fuzzy relative of the ISODATA process and its use in detecting compact well-separated cluster, J. Cybern., 3, 3, 32-57, (1973) · Zbl 0291.68033
[15] Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (1994), Kluwer Dordrecht · Zbl 0827.90002
[16] Fukunaga, K., Introduction to statistical pattern recognition, (1990), Academic Press San Diego · Zbl 0711.62052
[17] Hathaway, R. J.; Bezdek, J. C., Generalized fuzzy c-means clustering strategies using \(L_p\) norm distances, IEEE Trans. Fuzzy Syst., 8, 5, 576-582, (2000)
[18] Henning, C., Dissolution point and isolation robustness: robustness criteria for general cluster analysis methods, J. Multivar. Anal., 99, 1154-1176, (2008) · Zbl 1141.62052
[19] Horta, D.; de Andrade, I. C.; Campello, R. J., Evolutionary fuzzy clustering of relational data, Comput. Sci., 412, 42, 5854-5870, (2011) · Zbl 1223.68094
[20] Huber, P. J., Robust statistics, (1981), Wiley New York · Zbl 0536.62025
[21] Jajuga, K., \(L_1\)-norm based fuzzy clustering, Fuzzy Sets Syst., 39, 1, 43-50, (1991) · Zbl 0714.62052
[22] Kersten, P. R., Fuzzy order statistics and their application to fuzzy clustering, IEEE Trans. Fuzzy Syst., 7, 6, 708-712, (1999)
[23] Krishnapuram, R.; Keller, J. M., A possibilistic approach to clustering, IEEE Trans. Fuzzy Syst., 1, 1, 98-110, (1993)
[24] Leski, J. M., Towards a robust fuzzy clustering, Fuzzy Sets Syst., 137, 2, 215-233, (2003)
[25] Leski, J. M., Neuro-fuzzy system with learning tolerant to imprecision, Fuzzy Sets Syst., 138, 2, 427-439, (2003)
[26] Leski, J. M., Generalized weighted conditional fuzzy clustering, IEEE Trans. Fuzzy Syst., 11, 6, 709-715, (2003)
[27] Leski, J. M., An ε-margin nonlinear classifier based on if-then rules, IEEE Trans. Syst. Man Cybern., Part B, Cybern., 34, 1, 68-76, (2004)
[28] Leski, J. M.; Henzel, N., ECG baseline wander and powerline interference reduction using nonlinear filter bank, Signal Process., 85, 2, 781-793, (2005) · Zbl 1148.94382
[29] Maronna, R. A.; Martin, R. D.; Yohai, V. J., Robust statistics: theory and methods, (2006), John Wiley and Sons New York · Zbl 1094.62040
[30] Pedrycz, W., Conditional fuzzy clustering in the design of radial basis function neural network, IEEE Trans. Neural Netw., 9, 4, 601-612, (1998)
[31] Ruspini, E. H., A new approach to clustering, Inf. Control, 15, 1, 22-32, (1969) · Zbl 0192.57101
[32] Tou, J. T.; Gonzalez, R. C., Pattern recognition principles, (1974), Adison-Wesley London · Zbl 0299.68058
[33] Winkler, R.; Flawonn, F.; Kruse, R., Fuzzy clustering with polynomial fuzzifier connection with M-estimators, Appl. Comput. Math., 10, 146-163, (2011) · Zbl 1208.62105
[34] Xu, C.; Zhang, P.; Li, B.; Wu, D.; Fun, H., Vague c-means clustering algorithm, Pattern Recognit. Lett., 34, 5, 505-510, (2013)
[35] Yager, R. R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern., 18, 1, 183-190, (1988) · Zbl 0637.90057
[36] Yager, R. R., OWA operators in regression problems, IEEE Trans. Fuzzy Syst., 18, 1, 106-113, (2010)
[37] Vapnik, V., Statistical learning theory, (1998), Wiley New York · Zbl 0935.62007
[38] Zhao, F.; Liu, H.; Jiao, L., Spectral clustering with fuzzy similarity measure, Digit. Signal Process., 21, 6, 701-709, (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.