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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Bibliographic Details
Published in:Journal of the American Statistical Association Vol. 112; no. 518; pp. 690 - 705
Main Authors: Luo, Ruiyan, Qi, Xin
Format: Journal Article
Language:English
Published: Alexandria Taylor & Francis 01.06.2017
Taylor & Francis Group,LLC
Taylor & Francis Ltd
Subjects:
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
Online Access:Get full text
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Description
Summary: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.
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ISSN:0162-1459
1537-274X
1537-274X
DOI:10.1080/01621459.2016.1164053