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Simplicial band depth for multivariate functional data

Author

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  • Sara López-Pintado
  • Ying Sun
  • Juan Lin
  • Marc Genton

Abstract

We propose notions of simplicial band depth for multivariate functional data that extend the univariate functional band depth. The proposed simplicial band depths provide simple and natural criteria to measure the centrality of a trajectory within a sample of curves. Based on these depths, a sample of multivariate curves can be ordered from the center outward and order statistics can be defined. Properties of the proposed depths, such as invariance and consistency, can be established. A simulation study shows the robustness of this new definition of depth and the advantages of using a multivariate depth versus the marginal depths for detecting outliers. Real data examples from growth curves and signature data are used to illustrate the performance and usefulness of the proposed depths. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Sara López-Pintado & Ying Sun & Juan Lin & Marc Genton, 2014. "Simplicial band depth for multivariate functional data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(3), pages 321-338, September.
    • Handle: RePEc:spr:advdac:v:8:y:2014:i:3:p:321-338
    DOI: 10.1007/s11634-014-0166-6
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    References listed on IDEAS

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    1. Tatiyana V. Apanasovich & Marc G. Genton & Ying Sun, 2012. "A Valid Matérn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 180-193, March.
    2. 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.
    3. Lopez-Pintado, Sara & Romo, Juan, 2007. "Depth-based inference for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4957-4968, June.
    4. 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.
    5. 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.
    6. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
    7. Gneiting, Tilmann & Kleiber, William & Schlather, Martin, 2010. "Matérn Cross-Covariance Functions for Multivariate Random Fields," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1167-1177.
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    Citations

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    Cited by:

    1. Gijbels, Irène & Nagy, Stanislav, 2015. "Consistency of non-integrated depths for functional data," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 259-282.
    2. Qiu, Zhiping & Chen, Jianwei & Zhang, Jin-Ting, 2021. "Two-sample tests for multivariate functional data with applications," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    3. Nagy, Stanislav & Ferraty, Frédéric, 2019. "Data depth for measurable noisy random functions," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 95-114.
    4. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    5. 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.
    6. Qiu, Zhiping & Fan, Jiangyuan & Zhang, Jin-Ting & Chen, Jianwei, 2024. "Tests for equality of several covariance matrix functions for multivariate functional data," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    7. 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.
    8. Sara López-Pintado, 2015. "Discussion of Multivariate functional outlier detection by M. Hubert, P. Rousseeuw and P. Segaert," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 253-256, July.
    9. Zhu, Tianming & Zhang, Jin-Ting & Cheng, Ming-Yen, 2022. "One-way MANOVA for functional data via Lawley–Hotelling trace test," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    10. Yuan Yan & Marc Genton, 2015. "Discussion of “Multivariate functional outlier detection” by Mia Hubert, Peter Rousseeuw and Pieter Segaert," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 245-251, July.
    11. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.
    12. Kevin Leckey & Dennis Malcherczyk & Melanie Horn & Christine H. Müller, 2023. "Simple powerful robust tests based on sign depth," Statistical Papers, Springer, vol. 64(3), pages 857-882, June.
    13. Francesca Ieva & Anna Paganoni, 2015. "Discussion of “multivariate functional outlier detection” by M. Hubert, P. Rousseeuw and P. Segaert," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 217-221, July.
    14. Francesca Ieva & Anna Maria Paganoni, 2020. "Component-wise outlier detection methods for robustifying multivariate functional samples," Statistical Papers, Springer, vol. 61(2), pages 595-614, April.
    15. Kuhnt, Sonja & Rehage, André, 2016. "An angle-based multivariate functional pseudo-depth for shape outlier detection," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 325-340.
    16. Ana Justel & Marcela Svarc, 2018. "A divisive clustering method for functional data with special consideration of outliers," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(3), pages 637-656, September.
    17. Elías, Antonio & Jiménez, Raúl & Shang, Han Lin, 2022. "On projection methods for functional time series forecasting," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
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