Function-on-Function Linear Regression by Signal Compression

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 vari...

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Vydáno v:	Journal of the American Statistical Association Ročník 112; číslo 518; s. 690 - 705
Hlavní autoři:	Luo, Ruiyan, Qi, Xin
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Alexandria Taylor & Francis 01.06.2017 Taylor & Francis Group,LLC Taylor & Francis Ltd
Témata:	Americans Compression Convergence Correlation coefficients Eigenvalues Finite-dimensional approximation to coefficient kernel function Function Function-on-function linear regression functional response models Integrated squared correlation coefficient Integrated squared covariance least squares linear models objectives Penalized generalized functional eigenvalue problem Performance prediction Projections Random variables Regression analysis Regression models Response functions Scalarity (Semantics) Signal compression Simulation Statistics Theory and Methods
ISSN:	0162-1459, 1537-274X, 1537-274X
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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.
AbstractList	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. 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.
Author	Qi, Xin Luo, Ruiyan
Author_xml	– sequence: 1 givenname: Ruiyan surname: Luo fullname: Luo, Ruiyan email: rluo@gsu.edu organization: Division of Epidemiology and Biostatistics, School of Public Health, Georgia State University – sequence: 2 givenname: Xin surname: Qi fullname: Qi, Xin organization: Department of Mathematics and Statistics, Georgia State University
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SubjectTerms	Americans Compression Convergence Correlation coefficients Eigenvalues Finite-dimensional approximation to coefficient kernel function Function Function-on-function linear regression functional response models Integrated squared correlation coefficient Integrated squared covariance least squares linear models objectives Penalized generalized functional eigenvalue problem Performance prediction Projections Random variables Regression analysis Regression models Response functions Scalarity (Semantics) Signal compression Simulation Statistics Theory and Methods
Title	Function-on-Function Linear Regression by Signal Compression
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