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Open AccessOpen Access JUSTC Research Articles: Mathematics

A manifold extended t-process regression

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0026
  • Received Date: 24 January 2021
  • Rev Recd Date: 12 April 2021
  • Publish Date: 31 May 2021
    Abstract

  • Abstract

    A manifold extended t-process regression (meTPR) model is developed to fit functional data with a complicated input space. A manifold method is used to transform covariate data from input space into a feature space, and then an extended t-process regression is used to map feature from feature space into observation space. An estimation procedure is constructed to estimate parameters in the model. Numerical studies are investigated with both synthetic data and real data, and results show that the proposed meTPR model performs well.

    Abstract

    A manifold extended t-process regression (meTPR) model is developed to fit functional data with a complicated input space. A manifold method is used to transform covariate data from input space into a feature space, and then an extended t-process regression is used to map feature from feature space into observation space. An estimation procedure is constructed to estimate parameters in the model. Numerical studies are investigated with both synthetic data and real data, and results show that the proposed meTPR model performs well.
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    • References(25)
  • [1]
    Williams C, Rasmussen C. Gaussian processes for regression. Advances in Neural Information Processing Systems, 1995, 8: 514-520.
    [2]
    Rasmussen C E. Gaussian processes in machine learning. In: Summer School on Machine Learning. Berlin: Springer, 2004: 63-71.
    [3]
    Shi J Q, Choi T.Gaussian Process Regression Analysis for Functional data. Boca Raton, FL: CRC Press, 2011.
    [4]
    Sun B, Yao H, Liu T. Short-term wind speed forecasting based on Gaussian process regression model. Proceedings of the Chinese Society for Electrical Engineering, 2012, 32(29): 104-109.
    [5]
    Liu K Y, Fang Y, Liu B G, et al. Intelligent deformation prediction model of tunnel surrounding rock based on genetic-Gaussian process regression coupling algorithm. Journal of the China Railway Society, 2011, 33: 101-106. (In Chinese)
    [6]
    Smola A J, Bartlett P L. Sparse greedy Gaussian process regression. In: Advances in Neural Information Processing Systems 13. Cambridge, MA: MIT Press, 2001: 619-625.
    [7]
    Seiferth D, Chowdhary G, Mühlegg M, et al. Online Gaussian process regression with non-Gaussian likelihood. In 2017 American Control Conference (ACC). IEEE, 2017: 3134-3140.
    [8]
    Banerjee A, Dunson D B, Tokdar S T. Efficient Gaussian process regression for large datasets. Biometrika, 2013, 100: 75-89.
    [9]
    Wauthier F L, Jordan M I. Heavy-tailed process priors for selective shrinkage. In: Advances in Neural Information Processing Systems 23. Cambridge, MA: MIT Press, 2010: 2406-2414.
    [10]
    Yu S, Tresp V, Yu K, et al. Robust multi-task learning with t-processes. In: Proceedings of the 24th International Conference on Machine learning. New York: Association for Computing Machinery, 2007: 1103-1110.
    [11]
    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.
    [12]
    Jylänki P, Vanhatalo J, Vehtari A. Robust Gaussian process regression with a student-t likelihood. Journal of Machine Learning Research, 2011, 12: 3227-3257.
    [13]
    Wang Z, Shi J Q, Lee Y. Extended t-process regression models. Journal of Statistical Planning and Inference, 2017, 189: 38-60.
    [14]
    Lin Z, Yao F. Functional regression on manifold with contamination.https://arxiv.org/abs/1704.03005.
    [15]
    Sober B, Aizenbud Y, Levin D.Approximation of functions over manifolds: A moving least-squares approach.https://arxiv.org/abs/1711.00765.
    [16]
    Zhou Zhihua, Zhan Dechuan. A manifold learning-based multi-instance regression algorithm. Chinese Journal of Computers, 2006, 29(11): 1948-1955. (In Chinese)
    [17]
    Gao Y, Liu Y J. Diversity based discriminant muti-manifold learning for dimensionality reduction. Automation and Instrumentation, 2020(4): 30-34. (In Chinese)
    [18]
    Fan J F, Chen D C. Combining manifold learning and nonlinear regression for head pose estimation. Journal of Image and Graphics, 2012, 17(8): 1002-1010. (In Chinese)
    [19]
    Calandra R, Peters J, Rasmussen C E, et al. Manifold Gaussian processes for regression. In: 2016 International Joint Conference on Neural Networks (IJCNN). IEEE, 2016: 3338-3345.
    [20]
    Mallasto A, Feragen A. Wrapped Gaussian process regression on Riemannian manifolds. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. IEEE, 2018: 5580-5588.
    [21]
    Lee Y, Nelder J A. Double hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society: Series C (Applied Statistics), 2006, 55: 139-185.
    [22]
    Xu P, Lee Y, Shi J Q, et al. Automatic detection of significant areas for functional data with directional error control. Statistics in Medicine, 2019, 38: 376-397.
    [23]
    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.
    [24]
    Berlinet A, Thomas-Agnan C. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Berlin: Springer Science & Business Media, 2011.
    [25]
    Hampel F R, Ronchetti E M, Rousseeuw P J, et al. Robust Statistics: The Approach Based on Influence Functions. Hobooken, NJ: Wiley, 2011.
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    [1]
    Williams C, Rasmussen C. Gaussian processes for regression. Advances in Neural Information Processing Systems, 1995, 8: 514-520.
    [2]
    Rasmussen C E. Gaussian processes in machine learning. In: Summer School on Machine Learning. Berlin: Springer, 2004: 63-71.
    [3]
    Shi J Q, Choi T.Gaussian Process Regression Analysis for Functional data. Boca Raton, FL: CRC Press, 2011.
    [4]
    Sun B, Yao H, Liu T. Short-term wind speed forecasting based on Gaussian process regression model. Proceedings of the Chinese Society for Electrical Engineering, 2012, 32(29): 104-109.
    [5]
    Liu K Y, Fang Y, Liu B G, et al. Intelligent deformation prediction model of tunnel surrounding rock based on genetic-Gaussian process regression coupling algorithm. Journal of the China Railway Society, 2011, 33: 101-106. (In Chinese)
    [6]
    Smola A J, Bartlett P L. Sparse greedy Gaussian process regression. In: Advances in Neural Information Processing Systems 13. Cambridge, MA: MIT Press, 2001: 619-625.
    [7]
    Seiferth D, Chowdhary G, Mühlegg M, et al. Online Gaussian process regression with non-Gaussian likelihood. In 2017 American Control Conference (ACC). IEEE, 2017: 3134-3140.
    [8]
    Banerjee A, Dunson D B, Tokdar S T. Efficient Gaussian process regression for large datasets. Biometrika, 2013, 100: 75-89.
    [9]
    Wauthier F L, Jordan M I. Heavy-tailed process priors for selective shrinkage. In: Advances in Neural Information Processing Systems 23. Cambridge, MA: MIT Press, 2010: 2406-2414.
    [10]
    Yu S, Tresp V, Yu K, et al. Robust multi-task learning with t-processes. In: Proceedings of the 24th International Conference on Machine learning. New York: Association for Computing Machinery, 2007: 1103-1110.
    [11]
    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.
    [12]
    Jylänki P, Vanhatalo J, Vehtari A. Robust Gaussian process regression with a student-t likelihood. Journal of Machine Learning Research, 2011, 12: 3227-3257.
    [13]
    Wang Z, Shi J Q, Lee Y. Extended t-process regression models. Journal of Statistical Planning and Inference, 2017, 189: 38-60.
    [14]
    Lin Z, Yao F. Functional regression on manifold with contamination.https://arxiv.org/abs/1704.03005.
    [15]
    Sober B, Aizenbud Y, Levin D.Approximation of functions over manifolds: A moving least-squares approach.https://arxiv.org/abs/1711.00765.
    [16]
    Zhou Zhihua, Zhan Dechuan. A manifold learning-based multi-instance regression algorithm. Chinese Journal of Computers, 2006, 29(11): 1948-1955. (In Chinese)
    [17]
    Gao Y, Liu Y J. Diversity based discriminant muti-manifold learning for dimensionality reduction. Automation and Instrumentation, 2020(4): 30-34. (In Chinese)
    [18]
    Fan J F, Chen D C. Combining manifold learning and nonlinear regression for head pose estimation. Journal of Image and Graphics, 2012, 17(8): 1002-1010. (In Chinese)
    [19]
    Calandra R, Peters J, Rasmussen C E, et al. Manifold Gaussian processes for regression. In: 2016 International Joint Conference on Neural Networks (IJCNN). IEEE, 2016: 3338-3345.
    [20]
    Mallasto A, Feragen A. Wrapped Gaussian process regression on Riemannian manifolds. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. IEEE, 2018: 5580-5588.
    [21]
    Lee Y, Nelder J A. Double hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society: Series C (Applied Statistics), 2006, 55: 139-185.
    [22]
    Xu P, Lee Y, Shi J Q, et al. Automatic detection of significant areas for functional data with directional error control. Statistics in Medicine, 2019, 38: 376-397.
    [23]
    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.
    [24]
    Berlinet A, Thomas-Agnan C. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Berlin: Springer Science & Business Media, 2011.
    [25]
    Hampel F R, Ronchetti E M, Rousseeuw P J, et al. Robust Statistics: The Approach Based on Influence Functions. Hobooken, NJ: Wiley, 2011.
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