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A half-region depth for functional data

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  • López-Pintado, Sara
  • Romo, Juan

Abstract

A new definition of depth for functional observations is introduced based on the notion of "half-region" determined by a curve. The half-region depth provides a simple and natural criterion to measure the centrality of a function within a sample of curves. It has computational advantages relative to other concepts of depth previously proposed in the literature which makes it applicable to the analysis of high-dimensional data. Based on this depth a sample of curves can be ordered from the center-outward and order statistics can be defined. The properties of the half-region depth, such as consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally, real data examples are analyzed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:4:p:1679-1695
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Hernández Banadik, Nicolás Jorge & Muñoz García, Alberto, 2017. "Kernel depth funcions for functional data," DES - Working Papers. Statistics and Econometrics. WS 24615, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Gijbels, Irène & Nagy, Stanislav, 2015. "Consistency of non-integrated depths for functional data," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 259-282.
    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. Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    6. Kuelbs, James & Zinn, Joel, 2016. "Convergence of quantile and depth regions," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3681-3700.
    7. Nagy, Stanislav & Gijbels, Irène & Hlubinka, Daniel, 2016. "Weak convergence of discretely observed functional data with applications," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 46-62.
    8. Ana Arribas-Gil & Juan Romo, 2015. "Discussion of “Multivariate functional outlier detection”," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 263-267, July.
    9. Oluwasegun Taiwo Ojo & Antonio Fernández Anta & Rosa E. Lillo & Carlo Sguera, 2022. "Detecting and classifying outliers in big 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. 16(3), pages 725-760, September.
    10. 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.
    11. Kuelbs, James & Zinn, Joel, 2015. "Half-region depth for stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 86-105.
    12. Alba M. Franco-Pereira & Rosa E. Lillo, 2020. "Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations," 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. 14(3), pages 651-676, September.
    13. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    14. Alonso, Andrés M. & Casado, David & Romo, Juan, 2012. "Supervised classification for functional data: A weighted distance approach," Computational Statistics & Data Analysis, Elsevier, vol. 56(7), pages 2334-2346.
    15. Alicia Nieto-Reyes & Heather Battey & Giacomo Francisci, 2021. "Functional Symmetry and Statistical Depth for the Analysis of Movement Patterns in Alzheimer’s Patients," Mathematics, MDPI, vol. 9(8), pages 1-17, April.
    16. Lucas Fernandez-Piana & Marcela Svarc, 2022. "An integrated local depth measure," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(2), pages 175-197, June.
    17. Anirvan Chakraborty & Probal Chaudhuri, 2014. "On data depth in infinite dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(2), pages 303-324, April.
    18. Ojo, Oluwasegun Taiwo & Fernández Anta, Antonio & Genton, Marc G. & Lillo Rodríguez, Rosa Elvira, 2022. "Multivariate Functional Outlier Detection using the FastMUOD Indices," DES - Working Papers. Statistics and Econometrics. WS 35665, Universidad Carlos III de Madrid. Departamento de Estadística.
    19. 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.
    20. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.
    21. Cleveland, Jason & Zhao, Weilong & Wu, Wei, 2018. "Robust template estimation for functional data with phase variability using band depth," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 10-26.
    22. Valencia García, Dalia Jazmin & Lillo Rodríguez, Rosa Elvira & Romo, Juan, 2013. "Spearman coefficient for functions," DES - Working Papers. Statistics and Econometrics. WS ws133329, Universidad Carlos III de Madrid. Departamento de Estadística.
    23. Jiménez Recaredo, Raúl José & Elías Fernández, Antonio, 2017. "Prediction Bands for Functional Data Based on Depth Measures," DES - Working Papers. Statistics and Econometrics. WS 24606, Universidad Carlos III de Madrid. Departamento de Estadística.
    24. Epifanio, Irene & Ventura-Campos, Noelia, 2011. "Functional data analysis in shape analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2758-2773, September.

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