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On data depth in infinite dimensional spaces

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  • Anirvan Chakraborty
  • Probal Chaudhuri

Abstract

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Suggested Citation

  • 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.
  • Handle: RePEc:spr:aistmt:v:66:y:2014:i:2:p:303-324
    DOI: 10.1007/s10463-013-0416-y
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    References listed on IDEAS

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    Cited by:

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    2. 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.
    3. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.
    4. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    5. Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast DD-classification of functional data," Statistical Papers, Springer, vol. 58(4), pages 1055-1089, December.
    6. Dai, Wenlin & Mrkvička, Tomáš & Sun, Ying & Genton, Marc G., 2020. "Functional outlier detection and taxonomy by sequential transformations," Computational Statistics & Data Analysis, Elsevier, vol. 149(C).
    7. 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.
    8. Kuelbs, James & Zinn, Joel, 2015. "Half-region depth for stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 86-105.
    9. 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.
    10. 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.
    11. Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    12. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.

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