IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v125y2018icp10-26.html
   My bibliography  Save this article

Robust template estimation for functional data with phase variability using band depth

Author

Listed:
  • Cleveland, Jason
  • Zhao, Weilong
  • Wu, Wei

Abstract

Registration, or alignment, of functional observations has been a fundamental problem in functional data analysis. The creation of a template was the key step for alignment of a group of functions. Recent studies have defined the template with the notion of “mean” in the given observations. However, the mean can be sensitive to the, commonly observed, outlier functions in the data. To deal with this problem, a new approach is proposed to adopt the notion of “median” using the time warping functions in the alignment process, based on the recently developed band depth in functional data. A semi-parametric model is provided with an algorithm that yields a consistent estimator for the underlying median template. The robustness of this depth-based registration is illustrated using simulations and two real data sets. In addition, a new depth-based boxplot is proposed for outlier detection in functional data with phase variability.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:csdana:v:125:y:2018:i:c:p:10-26
    DOI: 10.1016/j.csda.2018.03.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947318300574
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2018.03.009?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rong Tang & Hans-Georg Müller, 2008. "Pairwise curve synchronization for functional data," Biometrika, Biometrika Trust, vol. 95(4), pages 875-889.
    2. Kneip, Alois & Ramsay, James O, 2008. "Combining Registration and Fitting for Functional Models," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1155-1165.
    3. Gerda Claeskens & Mia Hubert & Leen Slaets & Kaveh Vakili, 2014. "Multivariate Functional Halfspace Depth," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 411-423, March.
    4. 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.
    5. 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.
    6. Xueli Liu & Hans-Georg Muller, 2004. "Functional Convex Averaging and Synchronization for Time-Warped Random Curves," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 687-699, January.
    7. 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.
    8. 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.
    9. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    10. 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.
    11. Daniel Gervini & Theo Gasser, 2004. "Self‐modelling warping functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 959-971, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Nieto-Reyes, Alicia & Battey, Heather, 2021. "A topologically valid construction of depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    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. 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.
    5. Serfling, Robert & Wijesuriya, Uditha, 2017. "Depth-based nonparametric description of functional data, with emphasis on use of spatial depth," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 24-45.
    6. 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.
    7. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    8. repec:cte:wsrepe:24606 is not listed on IDEAS
    9. 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.
    10. repec:cte:wsrepe:24615 is not listed on IDEAS
    11. Elías, Antonio & Jiménez, Raúl & Shang, Han Lin, 2022. "On projection methods for functional time series forecasting," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    12. Arribas-Gil, Ana & Müller, Hans-Georg, 2014. "Pairwise dynamic time warping for event data," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 255-268.
    13. Miguel Flores & Salvador Naya & Rubén Fernández-Casal & Sonia Zaragoza & Paula Raña & Javier Tarrío-Saavedra, 2020. "Constructing a Control Chart Using Functional Data," Mathematics, MDPI, vol. 8(1), pages 1-26, January.
    14. 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).
    15. repec:cte:wsrepe:ws133329 is not listed on IDEAS
    16. 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.
    17. 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.
    18. Gijbels, Irène & Nagy, Stanislav, 2015. "Consistency of non-integrated depths for functional data," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 259-282.
    19. 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.
    20. Zhuo Qu & Wenlin Dai & Marc G. Genton, 2021. "Robust functional multivariate analysis of variance with environmental applications," Environmetrics, John Wiley & Sons, Ltd., vol. 32(1), February.
    21. Jason Cleveland & Wei Wu & Anuj Srivastava, 2016. "Norm-preserving constraint in the Fisher--Rao registration and its application in signal estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 338-359, June.
    22. 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.
    23. Tucker, J. Derek & Wu, Wei & Srivastava, Anuj, 2013. "Generative models for functional data using phase and amplitude separation," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 50-66.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:125:y:2018:i:c:p:10-26. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.