IDEAS home Printed from https://ideas.repec.org/a/bla/jorssb/v77y2015i4p777-801.html
   My bibliography  Save this article

Components and completion of partially observed functional data

Author

Listed:
  • David Kraus

Abstract

type="main" xml:id="rssb12087-abs-0001"> Functional data are traditionally assumed to be observed on the same domain. Motivated by a data set of heart rate temporal profiles, we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere. We formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework: estimation of parameters (mean and covariance operator) and principal component analysis. Principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory. Next, we propose a functional completion procedure that recovers the missing part by using the observed part of the curve. We construct prediction intervals for principal scores and bands for missing parts of trajectories. The prediction problems are seen to be ill-posed inverse problems; regularization techniques are used to obtain a stable solution. A simulation study shows the good performance of our methods. We illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results.

Suggested Citation

  • David Kraus, 2015. "Components and completion of partially observed functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(4), pages 777-801, September.
  • Handle: RePEc:bla:jorssb:v:77:y:2015:i:4:p:777-801
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1111/rssb.2015.77.issue-4
    Download Restriction: Access to full text is restricted to subscribers.
    ---><---

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

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Nengxiang Ling & Rui Kan & Philippe Vieu & Shuyu Meng, 2019. "Semi-functional partially linear regression model with responses missing at random," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(1), pages 39-70, January.
    2. Kraus, David & Stefanucci, Marco, 2020. "Ridge reconstruction of partially observed functional data is asymptotically optimal," Statistics & Probability Letters, Elsevier, vol. 165(C).
    3. Valentina Masarotto & Victor M. Panaretos & Yoav Zemel, 2019. "Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 172-213, February.
    4. Antonio Elías & Raúl Jiménez & J. E. Yukich, 2023. "Localization processes for functional data analysis," 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. 17(2), pages 485-517, June.
    5. A. Delaigle & P. Hall, 2016. "Approximating fragmented functional data by segments of Markov chains," Biometrika, Biometrika Trust, vol. 103(4), pages 779-799.
    6. Konrad Menzel, 2023. "Transfer Estimates for Causal Effects across Heterogeneous Sites," Papers 2305.01435, arXiv.org, revised Feb 2024.
    7. Mojirsheibani, Majid & Shaw, Crystal, 2018. "Classification with incomplete functional covariates," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 40-46.
    8. Elías, Antonio & Jiménez, Raúl & Shang, Han Lin, 2022. "On projection methods for functional time series forecasting," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    9. Kraus, David, 2019. "Inferential procedures for partially observed functional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 583-603.
    10. Antonio Elías & Raúl Jiménez & Han Lin Shang, 2023. "Depth-based reconstruction method for incomplete functional data," Computational Statistics, Springer, vol. 38(3), pages 1507-1535, September.
    11. Tao Zhang & Zhiwen Wang & Yanling Wan, 2021. "Functional test for high-dimensional covariance matrix, with application to mitochondrial calcium concentration," Statistical Papers, Springer, vol. 62(3), pages 1213-1230, June.
    12. Liebl, Dominik & Rameseder, Stefan, 2019. "Partially observed functional data: The case of systematically missing parts," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 104-115.
    13. Yao, Binhong & Li, Peixing, 2023. "Covariance estimation error of incomplete functional data under RKHS framework," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    14. Marco Stefanucci & Stefano Mazzuco, 2022. "Analysing cause‐specific mortality trends using compositional functional data analysis," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(1), pages 61-83, January.
    15. Marco Stefanucci & Laura M. Sangalli & Pierpaolo Brutti, 2018. "PCA‐based discrimination of partially observed functional data, with an application to AneuRisk65 data set," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 72(3), pages 246-264, August.
    16. Park, Yeonjoo & Kim, Hyunsung & Lim, Yaeji, 2023. "Functional principal component analysis for partially observed elliptical process," Computational Statistics & Data Analysis, Elsevier, vol. 184(C).

    More about this item

    Statistics

    Access and download statistics

    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:bla:jorssb:v:77:y:2015:i:4:p:777-801. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/rssssea.html .

    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.