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Function-on-Function Linear Regression by Signal Compression

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  • Ruiyan Luo
  • Xin Qi

Abstract

We consider functional linear regression models with a functional response and multiple functional predictors, with the goal of finding the best finite-dimensional approximation to the signal part of the response function. Defining the integrated squared correlation coefficient between a random variable and a random function, we propose to solve a penalized generalized functional eigenvalue problem, whose solutions satisfy that projections on the original predictors generate new scalar uncorrelated variables and these variables have the largest integrated squared correlation coefficient with the signal function. With these new variables, we transform the original function-on-function regression model to a function-on-scalar regression model whose predictors are uncorrelated, and estimate the model by penalized least-square method. This method is also extended to models with both multiple functional and scalar predictors. We provide the asymptotic consistency and the corresponding convergence rates for our estimates. Simulation studies in various settings and for both one and multiple functional predictors demonstrate that our approach has good predictive performance and is very computational efficient. Supplementary materials for this article are available online.

Suggested Citation

  • Ruiyan Luo & Xin Qi, 2017. "Function-on-Function Linear Regression by Signal Compression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 690-705, April.
  • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:518:p:690-705
    DOI: 10.1080/01621459.2016.1164053
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    Citations

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    Cited by:

    1. Zhou, Zhiyang, 2021. "Fast implementation of partial least squares for function-on-function regression," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    2. Zhang, Xiaoke & Zhong, Qixian & Wang, Jane-Ling, 2020. "A new approach to varying-coefficient additive models with longitudinal covariates," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    3. Chang, Jinyuan & Chen, Cheng & Qiao, Xinghao & Yao, Qiwei, 2023. "An autocovariance-based learning framework for high-dimensional functional time series," LSE Research Online Documents on Economics 117910, London School of Economics and Political Science, LSE Library.
    4. Ruiyan Luo & Xin Qi, 2022. "Restricted function‐on‐function linear regression model," Biometrics, The International Biometric Society, vol. 78(3), pages 1031-1044, September.
    5. Fabio Centofanti & Antonio Lepore & Alessandra Menafoglio & Biagio Palumbo & Simone Vantini, 2023. "Adaptive smoothing spline estimator for the function-on-function linear regression model," Computational Statistics, Springer, vol. 38(1), pages 191-216, March.

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