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Depth-based nonparametric description of functional data, with emphasis on use of spatial depth

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  • Serfling, Robert
  • Wijesuriya, Uditha

Abstract

Statistical depth and related quantile functions, originally introduced for nonparametric description and analysis of multivariate data in a way sensitive to inherent geometry, are in active development for functional data and in this setting offer special options since the data may be visualized regardless of dimension. This paper provides depth-based methods for revealing the structure of a functional data set in terms of relevant sample quantile curves displayed at selected levels, and for constructing and displaying confidence bands for corresponding “population” versions. Also, the usual functional boxplot is enhanced, by adding inner fences to flag possible shape outliers, along with the outer fences that flag location outliers. This enables the boxplot to serve as a stand-alone tool for functional data, as with univariate and multivariate data. Further, the spatial depth approach, well-established for multivariate data, is investigated for nonparametric description of functional data along these lines. In comparison with four other commonly used depth approaches for functional data, over a range of actual and simulated data sets, the spatial depth approach is seen to offer a very competitive combination of robustness, efficiency, computational ease, simplicity, and versatility.

Suggested Citation

  • Serfling, Robert & Wijesuriya, Uditha, 2017. "Depth-based nonparametric description of functional data, with emphasis on use of spatial depth," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 24-45.
    • Handle: RePEc:eee:csdana:v:105:y:2017:i:c:p:24-45
    DOI: 10.1016/j.csda.2016.07.007
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    References listed on IDEAS

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    1. Carlo Sguera & Pedro Galeano & Rosa Lillo, 2014. "Spatial depth-based classification for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 725-750, December.
    2. 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.
    3. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    4. Hervé Cardot & Etienne Josserand, 2011. "Horvitz--Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling," Biometrika, Biometrika Trust, vol. 98(1), pages 107-118.
    5. 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.
    6. Robert Serfling, 2010. "Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 915-936.
    7. Mohamed Chaouch & Camelia Goga, 2012. "Using Complex Surveys to Estimate the L 1 ‐Median of a Functional Variable: Application to Electricity Load Curves," International Statistical Review, International Statistical Institute, vol. 80(1), pages 40-59, April.
    8. Fraiman, Ricardo & Svarc, Marcela, 2013. "Resistant estimates for high dimensional and functional data based on random projections," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 326-338.
    9. 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.
    10. 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.
    11. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    12. Hongtu Zhu & Jianqing Fan & Linglong Kong, 2014. "Spatially Varying Coefficient Model for Neuroimaging Data With Jump Discontinuities," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1084-1098, September.
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    Cited by:

    1. Nagy, Stanislav, 2017. "Monotonicity properties of spatial depth," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 373-378.
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    3. Elías, Antonio & Jiménez, Raúl & Shang, Han Lin, 2022. "On projection methods for functional time series forecasting," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    4. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.

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