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Parsimonious Tensor Response Regression

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  • Lexin Li
  • Xin Zhang

Abstract

Aiming at abundant scientific and engineering data with not only high dimensionality but also complex structure, we study the regression problem with a multidimensional array (tensor) response and a vector predictor. Applications include, among others, comparing tensor images across groups after adjusting for additional covariates, which is of central interest in neuroimaging analysis. We propose parsimonious tensor response regression adopting a generalized sparsity principle. It models all voxels of the tensor response jointly, while accounting for the inherent structural information among the voxels. It effectively reduces the number of free parameters, leading to feasible computation and improved interpretation. We achieve model estimation through a nascent technique called the envelope method, which identifies the immaterial information and focuses the estimation based upon the material information in the tensor response. We demonstrate that the resulting estimator is asymptotically efficient, and it enjoys a competitive finite sample performance. We also illustrate the new method on two real neuroimaging studies. Supplementary materials for this article are available online.

Suggested Citation

  • Lexin Li & Xin Zhang, 2017. "Parsimonious Tensor Response Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1131-1146, July.
    • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:519:p:1131-1146
    DOI: 10.1080/01621459.2016.1193022
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    Citations

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    Cited by:

    1. May, Paul & Biesecker, Matthew & Rekabdarkolaee, Hossein Moradi, 2022. "Response envelopes for linear coregionalization models," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    2. Liang, Jinwen & Härdle, Wolfgang Karl & Tian, Maozai, 2023. "Imputed quantile tensor regression for near-sited spatial-temporal data," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).
    3. Bo Wei & Limin Peng & Ying Guo & Amita Manatunga & Jennifer Stevens, 2023. "Tensor response quantile regression with neuroimaging data," Biometrics, The International Biometric Society, vol. 79(3), pages 1947-1958, September.
    4. Minji Lee & Zhihua Su, 2020. "A Review of Envelope Models," International Statistical Review, International Statistical Institute, vol. 88(3), pages 658-676, December.
    5. Wei Hu & Tianyu Pan & Dehan Kong & Weining Shen, 2021. "Nonparametric matrix response regression with application to brain imaging data analysis," Biometrics, The International Biometric Society, vol. 77(4), pages 1227-1240, December.
    6. Yue Zhao & Ingrid Van Keilegom & Shanshan Ding, 2022. "Envelopes for censored quantile regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1562-1585, December.
    7. Yang, Yuehan & Xia, Siwei & Yang, Hu, 2023. "Multivariate sparse Laplacian shrinkage for joint estimation of two graphical structures," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    8. Yeonhee Park & Zhihua Su & Hongtu Zhu, 2017. "Groupwise envelope models for imaging genetic analysis," Biometrics, The International Biometric Society, vol. 73(4), pages 1243-1253, December.
    9. Giuseppe Brandi & T. Di Matteo, 2020. "A new multilayer network construction via Tensor learning," Papers 2004.05367, arXiv.org.
    10. Monica Billio & Roberto Casarin & Matteo Iacopini & Sylvia Kaufmann, 2023. "Bayesian Dynamic Tensor Regression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(2), pages 429-439, April.
    11. Inkoo Lee & Debajyoti Sinha & Qing Mai & Xin Zhang & Dipankar Bandyopadhyay, 2023. "Bayesian regression analysis of skewed tensor responses," Biometrics, The International Biometric Society, vol. 79(3), pages 1814-1825, September.
    12. Ghannam, Mai & Nkurunziza, Sévérien, 2023. "Tensor Stein-rules in a generalized tensor regression model," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    13. Luo, Chongliang & Liang, Jian & Li, Gen & Wang, Fei & Zhang, Changshui & Dey, Dipak K. & Chen, Kun, 2018. "Leveraging mixed and incomplete outcomes via reduced-rank modeling," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 378-394.
    14. Daniel Spencer & Rajarshi Guhaniyogi & Raquel Prado, 2020. "Joint Bayesian Estimation of Voxel Activation and Inter-regional Connectivity in fMRI Experiments," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 845-869, December.
    15. Kai Deng & Xin Zhang, 2022. "Tensor envelope mixture model for simultaneous clustering and multiway dimension reduction," Biometrics, The International Biometric Society, vol. 78(3), pages 1067-1079, September.
    16. Lan Liu & Wei Li & Zhihua Su & Dennis Cook & Luca Vizioli & Essa Yacoub, 2022. "Efficient estimation via envelope chain in magnetic resonance imaging‐based studies," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 481-501, June.

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