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On the concept of depth for functional data

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  • López Pintado, Sara
  • Romo, Juan

Abstract

The statistical analysis of functional data is a growing need in many research areas. We propose a new depth notion for functional observations based on the graphic representation of the curves. Given a collection of functions, it allows to establish the centrality of a function and provides a natural center-outward ordering of the sample curves. Robust statistics such as the median function or a trimmed mean function can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally very fast and turns out to be convenient to study high-dimensional observations. The natural properties are established for the new depth and the uniform consistency of the sample depth is proved. Simulation results show that the trimmed mean presents a better behavior than the mean for contaminated models. Several real data sets are considered to illustrate this new concept of depth. Finally, we use this new depth to generalize to functions the Wilcoxon rank sum test. It allows to decide whether two groups of curves come from the same population. This functional rank test is applied to girls and boys growth curves concluding that they present different growth patterns.

Suggested Citation

  • López Pintado, Sara & Romo, Juan, 2006. "On the concept of depth for functional data," DES - Working Papers. Statistics and Econometrics. WS ws063012, Universidad Carlos III de Madrid. Departamento de Estadística.
  • Handle: RePEc:cte:wsrepe:ws063012
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    File URL: https://e-archivo.uc3m.es/bitstream/handle/10016/243/ws063012.pdf?sequence=1
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    References listed on IDEAS

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    1. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    2. 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.
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    Cited by:

    1. López Pintado, Sara & Romo, Juan, 2006. "Depth-based inference for functional data," DES - Working Papers. Statistics and Econometrics. WS ws063113, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Rob J. Hyndman & Han Lin Shang, 2008. "Rainbow plots, Bagplots and Boxplots for Functional Data," Monash Econometrics and Business Statistics Working Papers 9/08, Monash University, Department of Econometrics and Business Statistics.
    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.

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