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 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.