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A Geometric Approach to Visualization of Variability in Functional Data

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  • Weiyi Xie
  • Sebastian Kurtek
  • Karthik Bharath
  • Ying Sun

Abstract

We propose a new method for the construction and visualization of boxplot-type displays for functional data. We use a recent functional data analysis framework, based on a representation of functions called square-root slope functions, to decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of functional data is a very complex concept. Thus, we propose to identify outliers based on any of the three main components after decomposition. We provide a variety of visualization tools for the proposed boxplot-type displays including surface plots. We evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface temperature functions, electrocardiogram functions, and growth curves. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:519:p:979-993
    DOI: 10.1080/01621459.2016.1256813
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    References listed on IDEAS

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    Cited by:

    1. Moritz Herrmann & Fabian Scheipl, 2021. "A Geometric Perspective on Functional Outlier Detection," Stats, MDPI, vol. 4(4), pages 1-41, November.
    2. Marc G. Genton & Ying Sun, 2019. "Comments on: Data science, big data and statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 338-341, June.
    3. 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).
    4. 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.
    5. Jorge R. Sosa Donoso & Miguel Flores & Salvador Naya & Javier Tarrío-Saavedra, 2023. "Local Correlation Integral Approach for Anomaly Detection Using Functional Data," Mathematics, MDPI, vol. 11(4), pages 1-18, February.
    6. Ma, Yijia & Zhou, Xinyu & Wu, Wei, 2024. "A stochastic process representation for time warping functions," Computational Statistics & Data Analysis, Elsevier, vol. 194(C).
    7. Derek Tucker, J. & Shand, Lyndsay & Chowdhary, Kenny, 2021. "Multimodal Bayesian registration of noisy functions using Hamiltonian Monte Carlo," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).

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