Multiscale Clustering for Functional Data

  • Yaeji Lim
  • Hee-Seok OhEmail author
  • Ying Kuen Cheung


In an era of massive and complex data, clustering is one of the most important procedures for understanding and analyzing unstructured multivariate data. Classical methods such as K-means and hierarchical clustering, however, are not efficient in grouping data that are high dimensional and have inherent multiscale structures. This paper presents new clustering procedures that can adapt to multiscale characteristics and high dimensionality of data. The proposed methods are based on a novel combination of multiresolution analysis and functional data analysis. As the core of the methodology, a clustering approach using the concept of multiresolution analysis may reflect both the global trend and local activities of data, and functional data analysis handles the high-dimensional data efficiently. Practical algorithms to implement the proposed methods are further discussed. The empirical performance of the proposed methods is evaluated through numerical studies including a simulation study and real data analysis, which demonstrates promising results of the proposed clustering.


Empirical mode decomposition Functional data High-dimensional data Multiresolution analysis Wavelet transform 



We thank the Editor and referees for comments which led to a substantially improved manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korea government (NRF- 2016R1C1B1006572 and NRF-2018R1D1A1B07042933) and by NIH grants (R01HL111195 and R01MH109496).


  1. Antoniadis, A., Brossat, X., Cugliari, J., Poggi, J. M. (2013). Clustering functional data using wavelets. International Journal of Wavelets, Multiresolution and Information Processing, 11(01), 1350003.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chiou, J. M., & Li, P. L. (2007). Functional clustering and identifying substructures of longitudinal data. Journal of the Royal Statistical Society Series B, 69, 679–699.MathSciNetCrossRefGoogle Scholar
  3. Floriello, D., & Vitelli, V. (2017). Sparse clustering of functional data. Journal of Multivariate Analysis, 154, 1–18.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Giacofci, M., Lambert–Lacroix, S., Marot, G., Picard, F. (2013). Wavelet–based clustering for mixed–effects functional models in high dimension. Biometrics, 69, 31–40.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Hansen, J., Ruedy, R., Sato, M., Lo, K. (2010). Global surface temperature change. Reviews of Geophysics, 48, RG4004,
  6. Huang, N. E., & Shen, S. S. P. (2005). Hilbert-Huang transform and its applications. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar
  7. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N. C., Tung, C. C., Liu, H. H. (1998). The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society of London A, 454, 903–995.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.CrossRefzbMATHGoogle Scholar
  9. James, G. M., & Sugar, C. A. (2003). Clustering for sparsely sampled functional data. Journal of the American Statistical Association, 98, 397–408.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Jaques, J., & Preda, C. (2013). Functional data clustering: a survey. Advances in Data Analysis and Classification, 8, 231–255.MathSciNetCrossRefGoogle Scholar
  11. Lee, T. C. M. (2004). Improved smoothing spline regression by combining estimates of different smoothness. Statistics & Probability Letters, 67, 133–140.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Mallat, S. (2009). A wavelet tour of signal processing, 3rd. New York: Academic Press.zbMATHGoogle Scholar
  13. Morris, J. S., & Carroll, R. J. (2006). Wavelet-based functional mixed models. Journal of the Royal Statistical Society, Series B, 68, 179–199.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846–850.CrossRefGoogle Scholar
  15. Ray, S., & Mallick, B. (2006). Functional clustering by Bayesian wavelet methods. Journal of the Royal Statistical Society, Series B, 68, 305–332.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Tibshirani, R., Walther, G., Hastie, T. (2001). Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society, Series B, 63, 411–423.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Wand, M. P. (2000). A comparison of regression spline smoothing procedures. Computational Statistics, 15, 443–462.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Wakefield, J., Zhou, C., Self, S. (2003). Modelling gene expression over time: curve clustering with informative prior distributions. Bayesian Statistics, 7, 721–732.MathSciNetGoogle Scholar
  19. Witten, D. M., & Tibshirani, R. (2010). A framework for feature selection in clustering. Journal of the American Statistical Association, 105, 713–726.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Classification Society 2019

Authors and Affiliations

  1. 1.Department of Applied StatisticsChung-Ang UniversitySeoulKorea
  2. 2.Department of StatisticsSeoul National UniversitySeoulRepublic of Korea
  3. 3.Department of BiostatisticsColumbia UniversityNew YorkUSA

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