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A topologically valid construction of depth for functional data

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  • Nieto-Reyes, Alicia
  • Battey, Heather

Abstract

Numerous problems remain in the construction of statistical depth for functional data. Issues stem largely from the absence of a well-conceived notion of symmetry. The present paper proposes a topologically valid notion of symmetry for distributions on functional metric spaces and a corresponding notion of depth. The latter is shown to satisfy the axiomatic definition of functional depth introduced by Nieto-Reyes and Battey (2016).

Suggested Citation

  • Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    • Handle: RePEc:eee:jmvana:v:184:y:2021:i:c:s0047259x21000166
    DOI: 10.1016/j.jmva.2021.104738
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    References listed on IDEAS

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    1. Carrizosa, Emilio, 1996. "A Characterization of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 58(1), pages 21-26, July.
    2. 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.
    3. 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.
    4. Cuevas, Antonio & Fraiman, Ricardo, 2009. "On depth measures and dual statistics. A methodology for dealing with general data," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 753-766, April.
    5. Peter Hall & Mohammad Hosseini‐Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126, February.
    6. 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.
    7. 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.
    8. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    9. 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.
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    Cited by:

    1. Alicia Nieto-Reyes & Rafael Duque & Giacomo Francisci, 2021. "A Method to Automate the Prediction of Student Academic Performance from Early Stages of the Course," Mathematics, MDPI, vol. 9(21), pages 1-14, October.
    2. 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.
    3. Luis González-De La Fuente & Alicia Nieto-Reyes & Pedro Terán, 2022. "Properties of Statistical Depth with Respect to Compact Convex Random Sets: The Tukey Depth," Mathematics, MDPI, vol. 10(15), pages 1-23, August.
    4. 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.

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