ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Management 27 April 2022

Robust function-on-function regression model with nonparametric random effects

Cite this:
https://doi.org/10.52396/JUSTC-2022-0016
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  • Author Bio:

    Shanshan Wang is currently a master student under the supervision of Assoc. Prof. Zhanfeng Wang at the University of Science and Technology of China. Her research mainly focuses on functional data

    Hao Ding received his PhD degree from the University of Science and Technology of China (USTC). He is currently a postdoctoral fellow at USTC. His research focuses on robust estimation, functional data analysis

  • Corresponding author: Email: dinghao@ustc.edu.cn
  • Received Date: 21 January 2022
  • Accepted Date: 23 February 2022
  • Rev Recd Date: 23 February 2022
  • Available Online: 27 April 2022
  • Extended t-process is robust to outliers and inherits many attractive properties from the Gaussian process. In this paper, we provide a function-on-function nonparametric random-effects model using extended t-process priors in which we consider heterogeneity of individual effect, flexible mean function, nonparametric covariance function and robustness. A likelihood-based estimation procedure is constructed to estimate parameters involved in the model. Information consistency for the parameter estimation is provided. Simulation studies and a real data example are further investigated to evaluate the performance of the developed procedures.

    Extended t-process is robust to outliers and inherits many attractive properties from the Gaussian process. In this paper, we provide a function-on-function nonparametric random-effects model using extended t-process priors in which we consider heterogeneity of individual effect, flexible mean function, nonparametric covariance function and robustness. A likelihood-based estimation procedure is constructed to estimate parameters involved in the model. Information consistency for the parameter estimation is provided. Simulation studies and a real data example are further investigated to evaluate the performance of the developed procedures.

    • A function-on-function random effects model with extended t-process priors is considered.
    • The proposed model is general and flexible which includes various kinds of functional models as special cases.
    • The extended t-process model is robust to outliers and inherits almost all the good features for Gaussian process regression.

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  • [1]
    Wang Z, Noh M, Lee Y, et al. A general robust t-process regression model. Computational Statistics and Data Analysis, 2021, 154: 107093. doi: 10.1016/j.csda.2020.107093
    [2]
    Yuan M, Cai T T. A reproducing kernel Hilbert space approach to functional linear regression. The Annals of Statistics, 2010, 38 (6): 3412–3444. doi: 10.1214/09-AOS772
    [3]
    Wang Z, Shi J Q, Lee Y. Extended t-process regression models. Journal of Statistical Planning and Inference, 2017, 189: 38–60. doi: 10.1016/j.jspi.2017.05.006
    [4]
    Seeger M W, Kakade S M, Foster D P. Information consistency of nonparametric Gaussian process methods. IEEE Transactions on Information Theory, 2008, 54: 2376–2382. doi: 10.1109/TIT.2007.915707
    [5]
    Zhang Y, Yeung D Y. Multi-task learning using generalized t-process. In: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Cambridge, MA: PMLR, 2010: 964–971.
    [6]
    Yao F, Müller H G, Wang J L. Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 2005, 100: 577–590. doi: 10.1198/016214504000001745
    [7]
    Wang B, Shi J Q. Generalized gaussian process regression model for non-gaussian functional data. Journal of the American Statistical Association, 2014, 109: 1123–1133. doi: 10.1080/01621459.2014.889021
    [8]
    Shi J Q, Choi T. Gaussian Process Regression Analysis for Functional Data. Boca Raton, FL: CRC Press, 2011
    [9]
    Wang Z, Ding H, Chen Z, et al. Nonparametric random effects functional regression model using Gaussian process priors. Statistica Sinica, 2021, 31: 53–78. doi: 10.5705/ss.202018.0296
    [10]
    Yu S, Tresp V, Yu K. Robust multi-task learning with t-processes. In: Proceedings of the 24th International Conference on Machine Learning. New York: ACM, 2007: 1103–1110.
    [11]
    Berlinet A, Thomas-Agnan C. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Berlin: Springer Science & Business Media, 2011.
    [12]
    Malfait N, Ramsay J O. The historical functional linear model. Canadian Journal of Statistics, 2003, 31: 115–128. doi: 10.2307/3316063
    [13]
    Sun X, Du P, Wang X, et al. Optimal penalized function-on-function regression under a reproducing kernel Hilbert space framework. Journal of the American Statistical Association, 2018, 113 (524): 1601–1611. doi: 10.1080/01621459.2017.1356320
    [14]
    Shah A, Wilson A, Ghahramani Z. Student-t processes as alternatives to Gaussian processes. In: Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics. Cambridge, MA: PMLR, 2014: 877–885.
    [15]
    Gervini D. Dynamic retrospective regression for functional data. Technometrics, 2015, 57: 26–34. doi: 10.1080/00401706.2013.879076
    [16]
    Ramsay J O, Silverman B W. Functional Data Analysis. New York: Springer, 2005.
    [17]
    Ramsay J O, Dalzell C. Some tools for functional data analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1991, 53: 539–572. doi: 10.1111/j.2517-6161.1991.tb01844.x
    [18]
    Yao F, Müller H G, Wang J L. Functional linear regression analysis for longitudinal data. The Annals of Statistics, 2005, 33: 2873–2903. doi: 10.1214/009053605000000660
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    Figure  1.  Random and fixed effects of model using ETPR for Arctic and Atlantic.

    Figure  2.  Random and fixed effects of model using ETPR for Continental and Pacific.

    [1]
    Wang Z, Noh M, Lee Y, et al. A general robust t-process regression model. Computational Statistics and Data Analysis, 2021, 154: 107093. doi: 10.1016/j.csda.2020.107093
    [2]
    Yuan M, Cai T T. A reproducing kernel Hilbert space approach to functional linear regression. The Annals of Statistics, 2010, 38 (6): 3412–3444. doi: 10.1214/09-AOS772
    [3]
    Wang Z, Shi J Q, Lee Y. Extended t-process regression models. Journal of Statistical Planning and Inference, 2017, 189: 38–60. doi: 10.1016/j.jspi.2017.05.006
    [4]
    Seeger M W, Kakade S M, Foster D P. Information consistency of nonparametric Gaussian process methods. IEEE Transactions on Information Theory, 2008, 54: 2376–2382. doi: 10.1109/TIT.2007.915707
    [5]
    Zhang Y, Yeung D Y. Multi-task learning using generalized t-process. In: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Cambridge, MA: PMLR, 2010: 964–971.
    [6]
    Yao F, Müller H G, Wang J L. Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 2005, 100: 577–590. doi: 10.1198/016214504000001745
    [7]
    Wang B, Shi J Q. Generalized gaussian process regression model for non-gaussian functional data. Journal of the American Statistical Association, 2014, 109: 1123–1133. doi: 10.1080/01621459.2014.889021
    [8]
    Shi J Q, Choi T. Gaussian Process Regression Analysis for Functional Data. Boca Raton, FL: CRC Press, 2011
    [9]
    Wang Z, Ding H, Chen Z, et al. Nonparametric random effects functional regression model using Gaussian process priors. Statistica Sinica, 2021, 31: 53–78. doi: 10.5705/ss.202018.0296
    [10]
    Yu S, Tresp V, Yu K. Robust multi-task learning with t-processes. In: Proceedings of the 24th International Conference on Machine Learning. New York: ACM, 2007: 1103–1110.
    [11]
    Berlinet A, Thomas-Agnan C. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Berlin: Springer Science & Business Media, 2011.
    [12]
    Malfait N, Ramsay J O. The historical functional linear model. Canadian Journal of Statistics, 2003, 31: 115–128. doi: 10.2307/3316063
    [13]
    Sun X, Du P, Wang X, et al. Optimal penalized function-on-function regression under a reproducing kernel Hilbert space framework. Journal of the American Statistical Association, 2018, 113 (524): 1601–1611. doi: 10.1080/01621459.2017.1356320
    [14]
    Shah A, Wilson A, Ghahramani Z. Student-t processes as alternatives to Gaussian processes. In: Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics. Cambridge, MA: PMLR, 2014: 877–885.
    [15]
    Gervini D. Dynamic retrospective regression for functional data. Technometrics, 2015, 57: 26–34. doi: 10.1080/00401706.2013.879076
    [16]
    Ramsay J O, Silverman B W. Functional Data Analysis. New York: Springer, 2005.
    [17]
    Ramsay J O, Dalzell C. Some tools for functional data analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1991, 53: 539–572. doi: 10.1111/j.2517-6161.1991.tb01844.x
    [18]
    Yao F, Müller H G, Wang J L. Functional linear regression analysis for longitudinal data. The Annals of Statistics, 2005, 33: 2873–2903. doi: 10.1214/009053605000000660

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