# American Institute of Mathematical Sciences

July  2016, 1(2&3): 247-259. doi: 10.3934/bdia.2016007

## Scalable clustering by truncated fuzzy $c$-means

 1 Department of Mathematics, University of Connecticut, 341 Mans eld Road, Storrs, CT 06269-1009, United States 2 Business School, Hunan University, Changsha, Hunan 410082, China 3 Columbian College of Arts & Sciences, George Washington University, Washington, D.C., 20052, United States

Received  July 2016 Revised  August 2016 Published  September 2016

Most existing clustering algorithms are slow for dividing a large dataset into a large number of clusters. In this paper, we propose a truncated FCM algorithm to address this problem. The main idea behind our proposed algorithm is to keep only a small number of cluster centers during the iterative process of the FCM algorithm. Our numerical experiments on both synthetic and real datasets show that the proposed algorithm is much faster than the original FCM algorithm and the accuracy is comparable to that of the original FCM algorithm.
Citation: Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007
##### References:
 [1] C. C. Aggarwal and C. K. Reddy (eds.), Data Clustering: Algorithms and Applications,, CRC Press, (2014). [2] D. Arthur and S. Vassilvitskii, $k$-means++: The advantages of careful seeding,, in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, (2007), 1027. [3] J. C. Bezdek, R. Ehrlich and W. Full, FCM: The fuzzy c-means clustering algorithm,, Computers & Geosciences, 10 (1984), 191. doi: 10.1016/0098-3004(84)90020-7. [4] J. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms,, Kluwer Academic Publishers, (1981). [5] A. Broder, L. Garcia-Pueyo, V. Josifovski, S. Vassilvitskii and S. Venkatesan, Scalable k-means by ranked retrieval,, in Proceedings of the 7th ACM International Conference on Web Search and Data Mining, (2014), 233. doi: 10.1145/2556195.2556260. [6] R. L. Cannon, J. V. Dave and J. Bezdek, Efficient implementation of the fuzzy c-means clustering algorithms,, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8 (1986), 248. doi: 10.1109/TPAMI.1986.4767778. [7] T. W. Cheng, D. B. Goldgof and L. O. Hall, Fast fuzzy clustering,, Fuzzy Sets and Systems, 93 (1998), 49. doi: 10.1016/S0165-0114(96)00232-1. [8] M. de Souto, I. Costa, D. de Araujo, T. Ludermir and A. Schliep, Clustering cancer gene expression data: A comparative study,, BMC Bioinformatics, 9 (2008). doi: 10.1186/1471-2105-9-497. [9] J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters,, Journal of Cybernetics, 3 (1973), 32. doi: 10.1080/01969727308546046. [10] G. Gan, Data Clustering in C++: An Object-Oriented Approach,, Data Mining and Knowledge Discovery Series, (2011). doi: 10.1201/b10814. [11] G. Gan, Application of data clustering and machine learning in variable annuity valuation,, Insurance: Mathematics and Economics, 53 (2013), 795. doi: 10.1016/j.insmatheco.2013.09.021. [12] G. Gan, A multi-asset Monte Carlo simulation model for the valuation of variable annuities,, in Proceedings of the Winter Simulation Conference, (2015), 3162. doi: 10.1109/WSC.2015.7408450. [13] G. Gan and S. Lin, Valuation of large variable annuity portfolios under nested simulation: A functional data approach,, Insurance: Mathematics and Economics, 62 (2015), 138. doi: 10.1016/j.insmatheco.2015.02.007. [14] G. Gan and M. K.-P. Ng, Subspace clustering using affinity propagation,, Pattern Recognition, 48 (2015), 1455. doi: 10.1016/j.patcog.2014.11.003. [15] G. Gan and M. K.-P. Ng, Subspace clustering with automatic feature grouping,, Pattern Recognition, 48 (2015), 3703. doi: 10.1016/j.patcog.2015.05.016. [16] G. Gan, Y. Zhang and D. K. Dey, Clustering by propagating probabilities between data points,, Applied Soft Computing, 41 (2016), 390. doi: 10.1016/j.asoc.2016.01.034. [17] R. J. Hathaway and J. C. Bezdek, Extending fuzzy and probabilistic clustering to very large data sets,, Computational Statistics & Data Analysis, 51 (2006), 215. doi: 10.1016/j.csda.2006.02.008. [18] T. Havens, J. Bezdek, C. Leckie, L. Hall and M. Palaniswami, Fuzzy c-means algorithms for very large data,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1130. doi: 10.1109/TFUZZ.2012.2201485. [19] M.-C. Hung and D.-L. Yang, An efficient fuzzy c-means clustering algorithm,, in Proceedings IEEE International Conference on Data Mining, (2001), 225. [20] Z.-X. Ji, Q.-S. Sun and D.-S. Xia, A modified possibilistic fuzzy c-means clustering algorithm for bias field estimation and segmentation of brain MR image,, Computerized Medical Imaging and Graphics, 35 (2011), 383. doi: 10.1016/j.compmedimag.2010.12.001. [21] D. Jiang, C. Tang and A. Zhang, Cluster analysis for gene expression data: A survey,, IEEE Transactions on Knowledge and Data Engineering, 16 (2004), 1370. [22] F. Klawonn, Fuzzy clustering: Insights and a new approach,, Mathware & Soft Computing, 11 (2004), 125. [23] J. F. Kolen and T. Hutcheson, Reducing the time complexity of the fuzzy c-means algorithm,, IEEE Transactions on Fuzzy Systems, 10 (2002), 263. doi: 10.1109/91.995126. [24] T. Kwok, K. Smith, S. Lozano and D. Taniar, Parallel fuzzy c- means clustering for large data sets,, in Euro-Par 2002 Parallel Processing (eds. B. Monien and R. Feldmann), (2002), 365. doi: 10.1007/3-540-45706-2_48. [25] H. Liu, F. Zhao and L. Jiao, Fuzzy spectral clustering with robust spatial information for image segmentation,, Applied Soft Computing, 12 (2012), 3636. doi: 10.1016/j.asoc.2012.05.026. [26] J. D. MacCuish and N. E. MacCuish, Clustering in Bioinformatics and Drug Discovery,, CRC Press, (2010). doi: 10.1201/b10331. [27] J. Macqueen, Some methods for classification and analysis of multivariate observations,, in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics andProbability (eds. L. LeCam and J. Neyman), 1 (1967), 281. [28] S. A. A. Shalom, M. Dash and M. Tue, Graphics hardware based efficient and scalable fuzzy c-means clustering,, in Proceedings of the 7th Australasian Data Mining Conference, 87 (2008), 179. [29] A. Stetco, X.-J. Zeng and J. Keane, Fuzzy c-means++: Fuzzy c-means with effective seeding initalization,, Expert Systems with Applications, 42 (2015). doi: 10.1016/j.eswa.2015.05.014.

show all references

##### References:
 [1] C. C. Aggarwal and C. K. Reddy (eds.), Data Clustering: Algorithms and Applications,, CRC Press, (2014). [2] D. Arthur and S. Vassilvitskii, $k$-means++: The advantages of careful seeding,, in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, (2007), 1027. [3] J. C. Bezdek, R. Ehrlich and W. Full, FCM: The fuzzy c-means clustering algorithm,, Computers & Geosciences, 10 (1984), 191. doi: 10.1016/0098-3004(84)90020-7. [4] J. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms,, Kluwer Academic Publishers, (1981). [5] A. Broder, L. Garcia-Pueyo, V. Josifovski, S. Vassilvitskii and S. Venkatesan, Scalable k-means by ranked retrieval,, in Proceedings of the 7th ACM International Conference on Web Search and Data Mining, (2014), 233. doi: 10.1145/2556195.2556260. [6] R. L. Cannon, J. V. Dave and J. Bezdek, Efficient implementation of the fuzzy c-means clustering algorithms,, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8 (1986), 248. doi: 10.1109/TPAMI.1986.4767778. [7] T. W. Cheng, D. B. Goldgof and L. O. Hall, Fast fuzzy clustering,, Fuzzy Sets and Systems, 93 (1998), 49. doi: 10.1016/S0165-0114(96)00232-1. [8] M. de Souto, I. Costa, D. de Araujo, T. Ludermir and A. Schliep, Clustering cancer gene expression data: A comparative study,, BMC Bioinformatics, 9 (2008). doi: 10.1186/1471-2105-9-497. [9] J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters,, Journal of Cybernetics, 3 (1973), 32. doi: 10.1080/01969727308546046. [10] G. Gan, Data Clustering in C++: An Object-Oriented Approach,, Data Mining and Knowledge Discovery Series, (2011). doi: 10.1201/b10814. [11] G. Gan, Application of data clustering and machine learning in variable annuity valuation,, Insurance: Mathematics and Economics, 53 (2013), 795. doi: 10.1016/j.insmatheco.2013.09.021. [12] G. Gan, A multi-asset Monte Carlo simulation model for the valuation of variable annuities,, in Proceedings of the Winter Simulation Conference, (2015), 3162. doi: 10.1109/WSC.2015.7408450. [13] G. Gan and S. Lin, Valuation of large variable annuity portfolios under nested simulation: A functional data approach,, Insurance: Mathematics and Economics, 62 (2015), 138. doi: 10.1016/j.insmatheco.2015.02.007. [14] G. Gan and M. K.-P. Ng, Subspace clustering using affinity propagation,, Pattern Recognition, 48 (2015), 1455. doi: 10.1016/j.patcog.2014.11.003. [15] G. Gan and M. K.-P. Ng, Subspace clustering with automatic feature grouping,, Pattern Recognition, 48 (2015), 3703. doi: 10.1016/j.patcog.2015.05.016. [16] G. Gan, Y. Zhang and D. K. Dey, Clustering by propagating probabilities between data points,, Applied Soft Computing, 41 (2016), 390. doi: 10.1016/j.asoc.2016.01.034. [17] R. J. Hathaway and J. C. Bezdek, Extending fuzzy and probabilistic clustering to very large data sets,, Computational Statistics & Data Analysis, 51 (2006), 215. doi: 10.1016/j.csda.2006.02.008. [18] T. Havens, J. Bezdek, C. Leckie, L. Hall and M. Palaniswami, Fuzzy c-means algorithms for very large data,, IEEE Transactions on Fuzzy Systems, 20 (2012), 1130. doi: 10.1109/TFUZZ.2012.2201485. [19] M.-C. Hung and D.-L. Yang, An efficient fuzzy c-means clustering algorithm,, in Proceedings IEEE International Conference on Data Mining, (2001), 225. [20] Z.-X. Ji, Q.-S. Sun and D.-S. Xia, A modified possibilistic fuzzy c-means clustering algorithm for bias field estimation and segmentation of brain MR image,, Computerized Medical Imaging and Graphics, 35 (2011), 383. doi: 10.1016/j.compmedimag.2010.12.001. [21] D. Jiang, C. Tang and A. Zhang, Cluster analysis for gene expression data: A survey,, IEEE Transactions on Knowledge and Data Engineering, 16 (2004), 1370. [22] F. Klawonn, Fuzzy clustering: Insights and a new approach,, Mathware & Soft Computing, 11 (2004), 125. [23] J. F. Kolen and T. Hutcheson, Reducing the time complexity of the fuzzy c-means algorithm,, IEEE Transactions on Fuzzy Systems, 10 (2002), 263. doi: 10.1109/91.995126. [24] T. Kwok, K. Smith, S. Lozano and D. Taniar, Parallel fuzzy c- means clustering for large data sets,, in Euro-Par 2002 Parallel Processing (eds. B. Monien and R. Feldmann), (2002), 365. doi: 10.1007/3-540-45706-2_48. [25] H. Liu, F. Zhao and L. Jiao, Fuzzy spectral clustering with robust spatial information for image segmentation,, Applied Soft Computing, 12 (2012), 3636. doi: 10.1016/j.asoc.2012.05.026. [26] J. D. MacCuish and N. E. MacCuish, Clustering in Bioinformatics and Drug Discovery,, CRC Press, (2010). doi: 10.1201/b10331. [27] J. Macqueen, Some methods for classification and analysis of multivariate observations,, in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics andProbability (eds. L. LeCam and J. Neyman), 1 (1967), 281. [28] S. A. A. Shalom, M. Dash and M. Tue, Graphics hardware based efficient and scalable fuzzy c-means clustering,, in Proceedings of the 7th Australasian Data Mining Conference, 87 (2008), 179. [29] A. Stetco, X.-J. Zeng and J. Keane, Fuzzy c-means++: Fuzzy c-means with effective seeding initalization,, Expert Systems with Applications, 42 (2015). doi: 10.1016/j.eswa.2015.05.014.
 [1] Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial & Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565 [2] Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2017, 2 (5) : 1-20. doi: 10.3934/bdia.2017018 [3] Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems & Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022 [4] Yongbin Ou, Cun-Quan Zhang. A new multimembership clustering method. Journal of Industrial & Management Optimization, 2007, 3 (4) : 619-624. doi: 10.3934/jimo.2007.3.619 [5] Elissar Nasreddine. Two-dimensional individual clustering model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 307-316. doi: 10.3934/dcdss.2014.7.307 [6] Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647 [7] Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025 [8] Jinyuan Zhang, Aimin Zhou, Guixu Zhang, Hu Zhang. A clustering based mate selection for evolutionary optimization. Big Data & Information Analytics, 2017, 2 (1) : 77-85. doi: 10.3934/bdia.2017010 [9] Gurkan Ozturk, Mehmet Tahir Ciftci. Clustering based polyhedral conic functions algorithm in classification. Journal of Industrial & Management Optimization, 2015, 11 (3) : 921-932. doi: 10.3934/jimo.2015.11.921 [10] Guojun Gan, Kun Chen. A soft subspace clustering algorithm with log-transformed distances. Big Data & Information Analytics, 2016, 1 (1) : 93-109. doi: 10.3934/bdia.2016.1.93 [11] Richard L Buckalew. Cell cycle clustering and quorum sensing in a response / signaling mediated feedback model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 867-881. doi: 10.3934/dcdsb.2014.19.867 [12] Ruiqi Yang, Dachuan Xu, Yicheng Xu, Dongmei Zhang. An adaptive probabilistic algorithm for online k-center clustering. Journal of Industrial & Management Optimization, 2019, 15 (2) : 565-576. doi: 10.3934/jimo.2018057 [13] Jiangchuan Fan, Xinyu Guo, Jianjun Du, Weiliang Wen, Xianju Lu, Brahmani Louiza. Analysis of the clustering fusion algorithm for multi-band color image. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1233-1249. doi: 10.3934/dcdss.2019085 [14] Daniel Roggen, Martin Wirz, Gerhard Tröster, Dirk Helbing. Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods. Networks & Heterogeneous Media, 2011, 6 (3) : 521-544. doi: 10.3934/nhm.2011.6.521 [15] Willem Mélange, Herwig Bruneel, Bart Steyaert, Dieter Claeys, Joris Walraevens. A continuous-time queueing model with class clustering and global FCFS service discipline. Journal of Industrial & Management Optimization, 2014, 10 (1) : 193-206. doi: 10.3934/jimo.2014.10.193 [16] Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465 [17] Chao Mi, Jun Wang, Weijian Mi, Youfang Huang, Zhiwei Zhang, Yongsheng Yang, Jun Jiang, Postolache Octavian. Research on regional clustering and two-stage SVM method for container truck recognition. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1117-1133. doi: 10.3934/dcdss.2019077 [18] Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014 [19] Feyza Gürbüz, Panos M. Pardalos. A decision making process application for the slurry production in ceramics via fuzzy cluster and data mining. Journal of Industrial & Management Optimization, 2012, 8 (2) : 285-297. doi: 10.3934/jimo.2012.8.285 [20] Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems & Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27

Impact Factor: