Abstract：In the existing subspace clustering algorithms, it is assumed that the data is derived from a union of multiple linear subspace, and these algorithms cannot deal with problems of nonlinear and time warping in time series clustering. To overcome these issues, elastic kernel low rank representation subspace clustering(EKLRR) and elastic kernel least squares regression subspace clustering(EKLSR) are proposed by introducing kernel tricks and elastic distance, and they are called elastic kernel subspace clustering(EKSC). Moreover, the grouping effect of EKLSR and the convergence of EKLRR are proved theoretically. The experimental results on five UCR datasets show the effectiveness of the proposed algorithms.
[1] WANG X, LIN J, PATEL N, et al. A Self-learning and Online Algorithm for Time Series Anomaly Detection, with Application in CPU Manufacturing // Proc of the 25th ACM International Conference on Information and Knowledge Management. New York, USA: ACM, 2016: 1823-1832. [2] MA Y X, ZHANG Z J. Subsequence Pattern Discovery in Chinese University Financial Data // Proc of the IEEE International Confe-rence on Logistics, Informatics and Service Sciences. Washington, USA: IEEE, 2016. DOI: 10.1109/LISS.2016.7854526. [3] DONG L, WANG L J, KHAHRO S F, et al. Wind Power Day-Ahead Prediction with Cluster Analysis of NWP. Renewable and Sustainable Energy Reviews, 2016, 60: 1206-1212. [4] AGHABOZORGI S, SHIRKHORSHIDI A S, WAH T Y. Time-Series Clustering-A Decade Review. Information Systems, 2015, 53: 16-38. [5] LIAO T W. Clustering of Time Series Data-A Survey. Pattern Re-cognition, 2005, 38(11): 1857-1874. [6] HUANG X H, YE Y M, XIONG L Y, et al. Time Series k-means: A New k-means Type Smooth Subspace Clustering for Time Series Data. Information Sciences, 2016, 367/368: 1-13. [7] KATE R J. Using Dynamic Time Warping Distances as Features for Improved Time Series Classification. Data Mining and Knowledge Discovery, 2016, 30(2): 283-312. [8] BERNDT D J, CLIHORD J. Using Dynamic Time Warping to Find Patterns in Sequences[C/OL]. [2017-03-02]. http://www.aaai.org/Papers/Workshops/1994/WS-94-03/WS94-03-031.pdf. [9] CHEN L, NG R. On the Marriage of Lp-norms and Edit Distance // Proc of the 30th International Conference on Very Large Data Bases. Berlin, Germany: Springer, 2004: 792-803. [10] SIM K, GOPALKRISHNAN V, ZIMEK A, et al. A Survey on Enhanced Subspace Clustering. Data Mining and Knowledge Discovery, 2013, 26(2): 332-397. [11] ELHAMIFAR E, VIDAL R. Sparse Subspace Clustering // Proc of the IEEE Conference on Computer Vision and Pattern Recognition. Washington, USA: IEEE, 2009: 2790-2797. [12] LIU G C, LIN Z C, YU Y. Robust Subspace Segmentation by Low-Rank Representation // Proc of the 27th International Conference on Machine Learning. New York, USA: ACM, 2010: 663-670. [13] LU C Y, MIN H, ZHAO Z Q, et al. Robust and Efficient Subspace Segmentation via Least Squares Regression // Proc of the European Conference on Computer Vision. Berlin, Germany: Springer, 2012: 347-360. [14] ZHANG H Y, LIN Z C, ZHANG C, et al. Robust Latent Low Rank Representation for Subspace Clustering. Neurocomputing, 2014, 145: 369-373. [15] 刘展杰,陈晓云.局部子空间聚类.自动化学报, 2016, 42(8): 1238-1247. (LIU Z J, CHEN X Y. Local Subspace Clustering. Acta Automatica Sinica, 2016, 42(8): 1238-1247.) [16] NGUYEN H, YANG W K, SHEN F M, et al. Kernel Low-Rank Representation for Face Recognition. Neurocomputing, 2015, 155: 32-42. [17] PAN Y Q, JIANG M Y, LI F. Kernel Locality Preserving Low-Rank Representation. Journal of Information Science & Enginee-ring, 2016, 32: 1561-1574. [18] BAHADORI M T, KALE D, FAN Y Y, et al. Functional Subspace Clustering with Application to Time Series // Proc of the 32nd International Conference on Machine Learning. New York, USA: ACM, 2015: 228-237. [19] ZHANG D Y, ZUO W M, ZHANG D, et al. Time Series Classification Using Support Vector Machine with Gaussian Elastic Metric Kernel // Proc of the 20th IEEE International Conference on Pa-ttern Recognition. Washington, USA: IEEE, 2010: 29-32. [20] CHEN Z H, ZUO W M, HU Q H, et al. Kernel Sparse Representation for Time Series Classification. Information Sciences, 2015, 292: 15-26. [21] SHI J B, MALIK J. Normalized Cuts and Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000, 22(8): 888-905. [22] MORI U, MENDIBURU A, LOZANO J A. Similarity Measure Selection for Clustering Time Series Databases. IEEE Transactions on Knowledge and Data Engineering, 2016, 28(1): 181-195. [23] LIN Z C, CHEN M M, MA Y. The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices[J/OL]. [2017-02-21]. http://perception.csl.illinois.edu/matrix-rank/Files/ALM-v5.pdf. [24] CAI J F, CAND S E J, SHEN Z W. A Singular Value Threshol- ding Algorithm for Matrix Completion. SIAM Journal on Optimization, 2010, 20(4): 1956-1982. [25] TIERNEY S, GAO J B, GUO Y, et al. Low-Rank Representation over the Manifold of Curves[J/OL]. [2017-02-21]. https://128.84.21.199/pdf/1601.00732.pdf. [26] CHEN X Y, JIAN C R. Gene Expression Data Clustering Based on Graph Regularized Subspace Segmentation. Neurocomputing, 2014, 143: 44-50. [27] CAI D, HE X F, WANG X H, et al. Locality Preserving Nonnegative Matrix Factorization // Proc of the 21st International Joint Conference on Artificial Intelligence. New York, USA: ACM, 2009: 1010-1015.
2017, Vol. 30 
