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Exploratory functional data analysis

Author

Listed:
  • Zhuo Qu

    (King Abdullah University of Science and Technology
    Department of Biostatistics, St. Jude Children’s Research Hospital)

  • Wenlin Dai

    (Institute of Statistics and Big Data, Renmin University of China)

  • Carolina Euan

    (Lancaster University)

  • Ying Sun

    (King Abdullah University of Science and Technology)

  • Marc G. Genton

    (King Abdullah University of Science and Technology)

Abstract

With the advance of technology, functional data are being recorded more frequently, whether over one-dimensional or multi-dimensional domains. Due to the high dimensionality and complex features of functional data, exploratory data analysis (EDA) faces significant challenges. To meet the demands of practical applications, researchers have developed various EDA tools, including visualization tools, outlier detection techniques, and clustering methods that can handle diverse types of functional data. This paper offers a comprehensive overview of recent procedures for exploratory functional data analysis (EFDA). It begins by introducing fundamental statistical concepts, such as mean and covariance functions, as well as robust statistics such as the median and quantiles in multivariate functional data. Then, the paper reviews popular visualization methods for functional data, such as the rainbow plot, and various versions of the functional boxplot, each designed to accommodate different features of functional data. In addition to visualization tools, the paper also reviews outlier detection methods, which are commonly integrated with visualization methods to identify anomalous patterns within the data. Finally, the paper focuses on functional data clustering techniques which provide another set of practical tools for EFDA. The paper concludes with a brief discussion of future directions for EFDA. All the reviewed methods have been implemented in an R package named EFDA .

Suggested Citation

  • Zhuo Qu & Wenlin Dai & Carolina Euan & Ying Sun & Marc G. Genton, 2025. "Exploratory functional data analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 34(2), pages 459-482, June.
    • Handle: RePEc:spr:testjl:v:34:y:2025:i:2:d:10.1007_s11749-024-00952-8
    DOI: 10.1007/s11749-024-00952-8
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    References listed on IDEAS

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    1. Stefano Castruccio & Marc G. Genton & Ying Sun, 2019. "Visualizing spatiotemporal models with virtual reality: from fully immersive environments to applications in stereoscopic view," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 182(2), pages 379-387, February.
    2. C. Abraham & P. A. Cornillon & E. Matzner‐Løber & N. Molinari, 2003. "Unsupervised Curve Clustering using B‐Splines," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(3), pages 581-595, September.
    3. Mari Myllymäki & Tomáš Mrkvička & Pavel Grabarnik & Henri Seijo & Ute Hahn, 2017. "Global envelope tests for spatial processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 381-404, March.
    4. Jeng‐Min Chiou & Pai‐Ling Li, 2007. "Functional clustering and identifying substructures of longitudinal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 679-699, September.
    5. Wenceslao González‐Manteiga & Rosa M. Crujeiras & Ying Sun & Marc G. Genton, 2012. "Adjusted functional boxplots for spatio‐temporal data visualization and outlier detection," Environmetrics, John Wiley & Sons, Ltd., vol. 23(1), pages 54-64, February.
    6. Gerda Claeskens & Mia Hubert & Leen Slaets & Kaveh Vakili, 2014. "Multivariate Functional Halfspace Depth," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 411-423, March.
    7. Zihao Wu & Carolina Euan & Rosa M. Crujeiras & Ying Sun, 2023. "Estimation and Clustering of Directional Wave Spectra," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 28(3), pages 502-525, September.
    8. Zhuo Qu & Wenlin Dai & Marc G. Genton, 2021. "Robust functional multivariate analysis of variance with environmental applications," Environmetrics, John Wiley & Sons, Ltd., vol. 32(1), February.
    9. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    10. Weiyi Xie & Sebastian Kurtek & Karthik Bharath & Ying Sun, 2017. "A Geometric Approach to Visualization of Variability in Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 979-993, July.
    11. Naveen N. Narisetty & Vijayan N. Nair, 2016. "Extremal Depth for Functional Data and Applications," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1705-1714, October.
    12. Francesca Ieva & Anna M. Paganoni & Davide Pigoli & Valeria Vitelli, 2013. "Multivariate functional clustering for the morphological analysis of electrocardiograph curves," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(3), pages 401-418, May.
    13. Pierre Lafaye de Micheaux & Pavlo Mozharovskyi & Myriam Vimond, 2021. "Depth for Curve Data and Applications," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1881-1897, October.
    14. Zhenhua Lin & Jane-Ling Wang, 2022. "Mean and Covariance Estimation for Functional Snippets," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(537), pages 348-360, January.
    15. Carlo Sguera & Sara López-Pintado, 2021. "A notion of depth for sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 630-649, September.
    16. Tianbo Chen & Ying Sun & Carolina Euan & Hernando Ombao, 2021. "Clustering Brain Signals: a Robust Approach Using Functional Data Ranking," Journal of Classification, Springer;The Classification Society, vol. 38(3), pages 425-442, October.
    17. Pedro C. Alvarez‐Esteban & Carolina Euán & Joaquín Ortega, 2016. "Time series clustering using the total variation distance with applications in oceanography," Environmetrics, John Wiley & Sons, Ltd., vol. 27(6), pages 355-369, September.
    18. Cuesta-Albertos, Juan Antonio & Fraiman, Ricardo, 2007. "Impartial trimmed k-means for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4864-4877, June.
    19. Cristian F. Jiménez‐Varón & Fouzi Harrou & Ying Sun, 2024. "Pointwise data depth for univariate and multivariate functional outlier detection," Environmetrics, John Wiley & Sons, Ltd., vol. 35(5), August.
    20. Amandine Schmutz & Julien Jacques & Charles Bouveyron & Laurence Chèze & Pauline Martin, 2020. "Clustering multivariate functional data in group-specific functional subspaces," Computational Statistics, Springer, vol. 35(3), pages 1101-1131, September.
    21. Antonio Cuevas & Manuel Febrero & Ricardo Fraiman, 2007. "Robust estimation and classification for functional data via projection-based depth notions," Computational Statistics, Springer, vol. 22(3), pages 481-496, September.
    22. Serban, Nicoleta & Wasserman, Larry, 2005. "CATS: Clustering After Transformation and Smoothing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 990-999, September.
    23. Ricardo Fraiman & Graciela Muniz, 2001. "Trimmed means for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 419-440, December.
    24. Shang, Han Lin & Haberman, Steven, 2017. "Grouped multivariate and functional time series forecasting:An application to annuity pricing," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 166-179.
    25. Carolina Euán & Hernando Ombao & Joaquín Ortega, 2018. "The Hierarchical Spectral Merger Algorithm: A New Time Series Clustering Procedure," Journal of Classification, Springer;The Classification Society, vol. 35(1), pages 71-99, April.
    26. López-Pintado, Sara & Romo, Juan, 2011. "A half-region depth for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1679-1695, April.
    27. Clara Happ & Sonja Greven, 2018. "Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 649-659, April.
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    1. Rob J. Hyndman, 2025. "Comments on: Exploratory functional data analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 34(2), pages 483-487, June.

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