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Wavelets-based clustering of multivariate time series. (English) Zbl 1237.62079

Summary: Crisp and fuzzy clustering methods based on a combination of univariate and multivariate wavelet features are considered for the clustering of multivariate time series. The performance of each of these methods is evaluated for stationary and variance nonstationary multivariate time series with different error correlation structures.

The main outcomes of the simulation studies are are as follows: the superior performance of this approach for both the crisp and fuzzy cluster methods compared to some of the other approaches for clustering multivariate time series; and the very good performance of the fuzzy relational method, overall, to cluster longer time series when all of them do not appear to group exclusively into well separated clusters. We consider an application to multivariate greenhouse gases time series and show that the crisp and fuzzy clustering methods considered are well validated.

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistics) 62H86 Multivariate analysis and fuzziness 62M10 Time series, auto-correlation, regression, etc. (statistics) 65T60 Wavelets (numerical methods) 65C60 Computational problems in statistics 62P12 Applications of statistics to environmental and related topics
##### References:
 [1] Keogh, E. J.; Kasetty, S.: On the need for time series data mining benchmarks: a survey and empirical demonstration, (2002) [2] Liao, T. W.: Clustering of time series data — a survey, Pattern recognition 38, 1857-1874 (2005) · Zbl 1077.68803 · doi:10.1016/j.patcog.2005.01.025 [3] Caiado, J.; Crato, N.; Peña, D.: A periodogram-based metric for time series classification, Comput. stat. Data anal. 50, 2668-2684 (2006) [4] Kalpakis, K.; Gada, D.; Puttagunta, V.: Distance measures for the effective clustering of ARIMA time-series, , 273-280 (2001) [5] Savvides, A.; Promponas, V. J.; Fokianos, K.: Clustering of biological time series by cepstral coefficients based distances, Pattern recognition 41, 2398-2412 (2008) · Zbl 1138.68515 · doi:10.1016/j.patcog.2008.01.002 [6] Wang, X.; Smith, K.; Hyndman, R.: Characteristics-based clustering for time series data, Data MIN knowl. Discovery 13, No. 3, 335-364 (2006) [7] D’urso, P.; Maharaj, E. A.: Autocorrelation-based fuzzy clustering of time series, Fuzzy sets syst. 160, 3565-3589 (2009) [8] Maharaj, E. A.; D’urso, P.; Galagedera, D. U. A.: Wavelets-based fuzzy clustering of time series, J. classification 27, 231-275 (2010) [9] Maharaj, E. A.; D’urso, P.: Fuzzy clustering of time series in the frequency domain, Fuzzy clustering of time series in the frequency domain 181, 1187-1211 (2011) · Zbl 1215.62061 · doi:10.1016/j.ins.2010.11.031 [10] Kakizawa, Y.; Shumway, H.; Taniguchi, M.: Discriminant and clustering for multivariate time series, J. am. Stat. assoc. 93, No. 441, 328-340 (1998) · Zbl 0906.62060 · doi:10.2307/2669629 [11] Maharaj, E. A.: The comparison and classification of stationary multivariate time series, Pattern recognition 32, No. 7, 1129-1138 (1999) [12] D’urso, P.: Fuzzy C-means clustering models for multivariate time-varying data: different approaches, Int. J. Uncertainty fuzziness knowl. Based syst. 12, No. 3, 287-326 (2004) · Zbl 1046.62061 · doi:10.1142/S0218488504002849 [13] D’urso, P.: Fuzzy clustering for data time array with inlier and outlier time trajectories, IEEE trans. Fuzzy syst. 13, No. 5, 583-604 (2005) [14] Singhal, A.; Seborg, D.: Clustering multivariate time series data, J. chemometrics 19, 427-438 (2005) [15] Abonyi, J.; Feil, B.; Nemeth, S.; Arva, P.: Modified gath – geva clustering for fuzzy segmentation of multivariate time-series, Fuzzy sets syst. 149, 39-56 (2005) · Zbl 1071.68543 · doi:10.1016/j.fss.2004.07.008 [16] Wu, E. H. C.; Li, P. L. H.: Independent component analysis for clustering multivariate time series data, ADMA 2005, lecture notes in artificial intelligence 3384, 474-482 (2005) [17] Coppi, R.; D’urso, P.: Fuzzy unsupervised classification of multivariate time trajectories with the Shannon entropy regularization, Comput. stat. Data anal. 50, No. 6, 1452-1477 (2006) [18] Coppi, R.; D’urso, P.; Giordani, P.: A fuzzy clustering model for multivariate spatial time series, Journal of classification 27, 54-88 (2010) [19] Liao, T. W.: A clustering procedure for exploratory mining of vector time series, Pattern recognition 40, 2250-2562 (2007) · Zbl 1118.68632 · doi:10.1016/j.patcog.2007.01.005 [20] Tokushige, S.; Yadohisa, H.; Inada, K.: Crisp and fuzzy K-means clustering algorithms for multivariate functional data, Comput. stat. 22, 1-16 (2007) · Zbl 1196.62089 · doi:10.1007/s00180-006-0013-0 [21] Wang, X.; Wirth, A.; Wang, L.: Structure-based statistical features and multivariate time series clustering, (2007) [22] Fröhwirth-Schnatter, S.; Kaufmann, S.: J. bus. Econ. stat., J. bus. Econ. stat. 26, No. 1, 78-89 (2008) [23] Percival, D. B.; Walden, A. T.: Wavelet methods for time series analysis, (2000) [24] Coppi, R.; D’urso, P.: The geometric approach to the comparison of multivariate time trajectories, Advances in data science and classification, 93-100 (2001) [25] D’urso, P.: Dissimilarity measures for time trajectories, J. ital. Stat. soc. 1 – 3, 53-83 (2000) [26] Kaufman, L.; Rousseeuw, P. J.: Finding groups in data: an introduction to cluster analysis, (1990) [27] Hubert, L.; Arabie, P.: Comparing partitions, J. classification, 193-218 (1985) [28] Priestley, M. B.: Evolutionary spectra and non-stationary processes, J. R. Stat. soc. B 27, 204-237 (1965) · Zbl 0144.41001