Abstract
This paper presents DivClusFD, a new divisive hierarchical method for the non-supervised classification of functional data. Data of this type present the peculiarity that the differences among clusters may be caused by changes as well in level as in shape. Different clusters can be separated in different subregion and there may be no subregion in which all clusters are separated. In each step of division, the DivClusFD method explores the functions and their derivatives at several fixed points, seeking the subregion in which the highest number of clusters can be separated. The number of clusters is estimated via the gap statistic. The functions are assigned to the new clusters by combining the k-means algorithm with the use of functional boxplots to identify functions that have been incorrectly classified because of their atypical local behavior. The DivClusFD method provides the number of clusters, the classification of the observed functions into the clusters and guidelines that may be for interpreting the clusters. A simulation study using synthetic data and tests of the performance of the DivClusFD method on real data sets indicate that this method is able to classify functions accurately.
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Abraham C, Cornillon PA, Matzner-Lber E, Molinari N (2003) Unsupervised curves clustering using B-Splines. Scand J Stat 30:581–595. doi:10.1111/1467-9469.00350
Alonso AM, Casado D, Romo J (2012) Supervised classification for functional data: a weighted distance aprproach. Comput Stat Data Anal 56:2334–2346. doi:10.1016/j.csda.2012.01.013
Berrendero JR, Justel A, Svarc M (2011) Principal components for multivariate functional data. Comput Stat Data Anal 55:2619–2634. doi:10.1016/j.csda.2011.03.011
Bouveyron C, Jacques J (2011) Model-based clustering of time series in group-specific functional subspaces. Adv Data Anal Classif 5:281–300. doi:10.1007/s11634-011-0095-6
Chen Y, Keogh E, Hu B, Begum N, Bagnall A, Mueen A, Batista G (2015) The UCR time series classification archive. http://www.cs.ucr.edu/~eamonn/time_series_data/
Chiou JM, Li PL (2011) Functional clustering and identifying substructures of longitudinal data. J R Stat Soc B 69:679–699. doi:10.1111/j.1467-9868.2007.00605.x
Febrero-Bande M, Oviedo de la Fuente M (2012) Statistical computing in functional data analysis: the R Package fda.usc. J Stat Softw 51(4):1–28. doi:10.18637/jss.v051.i04
Fraiman R, Justel A, Svarc M (2008) Selection of variables for cluster analysis and classification rules. J Am Stat Assoc 103:1294–1303. doi:10.1198/016214508000000544
Ieva F, Paganoni AM, Pigoli D, Vitelli V (2013) Multivariate functional clustering for the morphological analysis of electrocardiograph curves. J R Stat Soc Ser C (Appl Stat) 62:401–418. doi:10.1111/j.1467-9876.2012.01062.x
Jacques J, Preda C (2013) Funclust: a curves clustering method using functional random variable density approximation. Neurocomputing 171:112–164. doi:10.1016/j.neucom.2012.11.042
Jacques J, Preda C (2014a) Model-based clustering for multivariate functional data. Comput Stat Data Anal 71:92–106. doi:10.1016/j.csda.2012.12.004
Jacques J, Preda C (2014b) Functional data clustering: a survey. Adv Data Anal Classif 8(3):231–255. doi:10.1007/s11634-013-0158-y
James G, Sugar C (2003) Clustering for sparsely sampled functional data. J Am Stat Assoc 98:397–408. doi:10.1198/016214503000189
Keogh E, Pazzani M (2001) Dynamic time warping with higher order features. In: First SIAM international conference on data mining (SDM’2001), Chicago, IL, USA. doi:10.1007/s10618-015-0418-x
López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104:718–734. doi:10.1198/jasa.2009.0108
López-Pintado S, Sun Y, Lin JK, Genton MG (2014) Simplicial band depth for multivariate functional data. Adv Data Anal Classif 8:321–338. doi:10.1007/s11634-014-0166-6
Mosler K (2013) Depth statistics. In: Becker C, Fried R, Kuhnt S (eds) Robustness and complex data structures. Springer, Berlin, pp 17–34. doi:10.1007/978-3-642-35494-6_2
Ray S, Mallick B (2006) Functional clustering by Bayesian wavelet methods. J R Stat Soc Ser B Stat Methodol 68(2):305–332. doi:10.1111/j.1467-9868.2006.00545.x
Ramsay J, Hooker G, Graves S (2009) Functional data analysis with R and Matlab. Springer, Berlin. doi:10.1007/978-0-387-98185-7
Ramsay J, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New York. doi:10.1111/j.1541-0420.2007.00743_1.x
Sangalli LM, Secchi P, Vantini S, Vitelli V (2010a) \(k\)-means alignment for curve clustering. Comput Stat Data Anal 54:1219–1233. doi:10.1016/j.csda.2009.12.008
Sangalli LM, Secchi P, Vantini S, Vitelli V (2010b) Functional clustering and alignment methods with applications. Commun Appl Ind Math 1:205–224. doi:10.1685/2010CAIM486
Serban N, Wasserman L (2005) CATS: cluster analysis by transformation and smoothing. J Am Stat Assoc 100:990–999. doi:10.1198/016214504000001574
Sun Y, Genton MG (2011) Functional boxplots. J Comput Graph Stat 20:316–334. doi:10.1198/jcgs.2011.09224
Tarpey T, Kinateder KKJ (2003) Clustering functional data. J Classif 20:93–114. doi:10.1007/s11634-013-0158-y
Tibshirani R, Walther G, Hastie T (2001) Estimating the number of data clusters via the gap statistic. J R Stat Soc B 63:411–423. doi:10.1111/1467-9868.00293
Tokushige S, Yadohisa H, Inada K (2007) Crisp and Fuzzy \(k\)-means clustering algorithms for multivariate functional data. Comput Stat 21:1–16. doi:10.1007/s00180-006-0013-0
Tuddenham RD, Snyder MM (1954) Physical growth of California boys and girls from birth to eighteen years. Tech. Rep. 1, University of California Publications in Child Development
Tukey J (1977) Exploratory data analysis. Addison-Westley, Boston. doi:10.1002/bimj.4710230408
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We wish to thank the editors and four anonymous referees who have carefully reviewed the paper. Their suggestions and comments have helped us to improve the quality of this paper.
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This work was supported by the Spanish Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER), Grant CTM2016-79741-R for MICROAIPOLAR project (to A. Justel and M. Svarc) and Spanish Ministerio de Economía y Competitividad, Grant CTM2011-28736 (to A. Justel).
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Justel, A., Svarc, M. A divisive clustering method for functional data with special consideration of outliers. Adv Data Anal Classif 12, 637–656 (2018). https://doi.org/10.1007/s11634-017-0290-1
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DOI: https://doi.org/10.1007/s11634-017-0290-1