Abstract
In this paper, following a partitioning around medoids approach, a fuzzy clustering model for interval-valued data, i.e., FCMd-ID, is introduced. Successively, for avoiding the disruptive effects of possible outlier interval-valued data in the clustering process, a robust fuzzy clustering model with a trimming rule, called Trimmed Fuzzy \(C\)-medoids for interval-valued data (TrFCMd-ID), is proposed. In order to show the good performances of the robust clustering model, a simulation study and two applications are provided.
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Anderson DT, Bezdek JC, Popescu M, Keller JM (2010) Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Trans Fuzzy Syst 18(5):906–918
Billard L, Diday E (2006) Symbolic data analysis: conceptual statistics and data mining. Wiley, England
Brown J, Broderick AJ, Lee N (2007) Word of mouth communication within online communities: conceptualizing the online social network. J Interact Mark 21(3):2–20
Campello RJGB (2007) A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognit Lett 28(7):833–841
Campello RJGB, Hruschka ER (2006) A fuzzy extension of the silhouette width criterion for cluster analysis. Fuzzy Sets Syst 157(21):2858–2875
de Carvalho FdAT (2007) Fuzzy \(c\)-means clustering methods for symbolic interval data. Pattern Recognit Lett 28(4):423–437
de Carvalho FdAT, Tenório CP (2010) Fuzzy \(k\)-means clustering algorithms for interval-valued data based on adaptive quadratic distances. Fuzzy Sets Syst 161(23):2978–2999
de Carvalho FdAT, Csernel M, Lechevallier Y (2009) Clustering constrained symbolic data. Pattern Recognit Lett 30(11):1037–1045
de Souza RMCR, de Carvalho FdAT (2004) Clustering of interval data based on city-block distances. Pattern Recognit Lett 25(3):353–365
Cazes P, Chouakria A, Diday E, Schektrman Y (1997) Entension de l’analyse en composantes principales à des données de type intervalle. Revue Stat Appl 45(3):5–24
Coppi R, D’Urso P, Giordani P (2012) Fuzzy and possibilistic clustering for fuzzy data. Comput Stat Data Anal 56(4):915–927
D’Urso P, De Giovanni L (2011) Midpoint radius self-organizing maps for interval-valued data with telecommunications application. Appl Soft Comput 11(5):3877–3886
D’Urso P, Giordani P (2004) A least squares approach to principal component analysis for interval valued data. Chemom Intell Lab Syst 70(2):179–192
D’Urso P, Giordani P (2006a) A weighted fuzzy \(c\)-means clustering model for fuzzy data. Comput Stat Data Anal 50(6):1496–1523
D’Urso P, Giordani P (2006b) A robust fuzzy k-means clustering model for interval valued data. Comput Stat 21(2):251–269
D’Urso P, De Giovanni L, Massari R (2014) Self-organizing maps for imprecise data. Fuzzy Sets Syst 237(16):63–89
El-Sonbaty Y, Ismail MA (1998) Fuzzy clustering for symbolic data. IEEE Trans Fuzzy Syst 6(2):195–204
Everitt BS, Landau S, Leese M (2001) Cluster analysis, 4th edn. Arnold Press, London
Fu KS (1977) Syntactic pattern recognition, applications. Springer, New York
García-Escudero LA, Gordaliza A (1999) Robustness properties of k-means and trimmed k-means. J Am Stat Assoc 94(447):956–969
García-Escudero LA, Gordaliza A (2005) A proposal for robust curve clustering. J Classif 22(2):185–201
García-Escudero LA, Gordaliza A, Matrán C, Mayo-Iscar A (2010) A review of robust clustering methods. Adv Data Anal Classif 4(2):89–109
Gowda K, Ravi T (1995) Agglomerative clustering of symbolic objects using the concepts of both similarity and dissimilarity. Pattern Recognit Lett 16(6):647–652
Guru DS, Kiranagi BB, Nagabhushan P (2004) Multivalued type proximity measure and concept of mutual similarity value useful for clustering symbolic patterns. Pattern Recognit Lett 25(10):1203–1213
Heiser WJ, Groenen PJF (1997) Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima. Psychometrika 62(1):63–83
Ichino M, Yaguchi H (1994) Generalized Minkowski metrics for mixed feature-type data analysis. IEEE Trans Syst Man Cybern 24(4):698–708
Jeng JT, Chuang CC, Tseng CC, Juan CJ (2010) Robust interval competitive agglomeration clustering algorithm with outliers. Int J Fuzzy Syst 12(3):227–236
Kamdar T, Joshi A (2000) On creating adaptive Web servers using Weblog Mining. Technical Report TR-CS-00-05, Department of Computer Science and Electrical Engineering, University of Maryland, Baltimore County
Katona Z, Zubcsek PP, Sarvary M (2011) Network effects and personal influences: the diffusion of an online social network. J Mark Res 48(3):425–443
Kaufman L, Rousseeuw PJ (1987) Clustering by means of medoids. In: Dodge Y (ed) Statistical data analysis based on the L1-norm and related methods. North-Holland, Amsterdam, pp 405–416
Kaufman L, Rousseeuw PJ (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York
Kim J, Krishnapuram R, Davé R (1996) Application of the least trimmed squares technique to prototype-based clustering. Pattern Recognit Lett 17(6):633–641
Kohonen T (1989) Self-organization and associative memory, 3rd edn. Springer, New York
Krishnapuram R, Joshi A, Yi L (1999) A fuzzy relative of the k-medoids algorithm with application to web document and snippet clustering. In: IEEE international fuzzy systems conference (FUZZIEEE99), vol 3, IEEE, Seoul, pp 1281–1286
Krishnapuram R, Joshi A, Nasraoui O, Yi L (2001) Low-complexity fuzzy relational clustering algorithms for web mining. IEEE Trans Fuzzy Syst 9(4):595–607
Masson MH, Denœux T (2004) Clustering interval-valued proximity data using belief functions. Pattern Recognit Lett 25(2):163–171
McBratney AB, Moore AW (1985) Application of fuzzy sets to climatic classification. Agric For Meteorol 35(1):165–185
Palmer A, Koenig-Lewis N (2009) An experiential, social network-based approach to direct marketing. Direct Mark Int J 3(3):162–176
Qualman E (2012) Socialnomics: How social media transforms the way we live and do business. Wiley, New Jersey
Runkler T, Bezdek J (1999) Alternating cluster estimation: a new tool for clustering and function approximation. IEEE Trans Fuzzy Syst 7(4):377–393
Vinod HD (1969) Integer programming and the theory of grou**. J Am Stat Assoc 64(326):506–519
Webb Young J, Burgoyne B (2009) You’ve got a friend: measuring the value of brand friending on social networks. In: Market research study annual conference, Market Research Study
Wedel M, Kamakura WA (1998) Market segmentation: conceptual and methodological foundations. Kluwer Academic, Boston
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The authors thank the editors and the three referees for their useful comments and suggestions which helped to improve the quality and presentation of this manuscript.
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D’Urso, P., De Giovanni, L. & Massari, R. Trimmed fuzzy clustering for interval-valued data. Adv Data Anal Classif 9, 21–40 (2015). https://doi.org/10.1007/s11634-014-0169-3
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DOI: https://doi.org/10.1007/s11634-014-0169-3
Keywords
- Interval-valued data
- Partitioning around medoids
- Fuzzy clustering
- Robust clustering
- Trimming
- Web advertising