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Regression models with a functional response and functional covariate have received significant attention recently. While various nonparametric and semiparametric models have been developed, there is an urgent need for model selection and diagnostic methods. In this article, we develop a unified framework for estimation and model selection in nonparametric function-on-function regression. We propose a general nonparametric functional regression model with the model space constructed through smoothing spline analysis of variance (SS ANOVA). The proposed model reduces to some of the existing models when selected components in the SS ANOVA decomposition are eliminated. We propose new estimation procedures under either L1 or L2 penalty and show that the combination of the SS ANOVA decomposition and L1 penalty provides powerful tools for model selection and diagnostics. We establish consistency and convergence rates for estimates of the regression function and each component in its decomposition under both the L1 and L2 penalties. Simulation studies and real examples show that the proposed methods perform well. Technical details and additional simulation results are available in online . [ABSTRACT FROM AUTHOR]

Smoothing splines have been used pervasively in nonparametric regressions. However, the computational burden of smoothing splines is significant when the sample size n is large. When the number of predictors d ≥ 2 , the computational cost for smoothing splines is at the order of O (n 3) using the standard approach. Many methods have been developed to approximate smoothing spline estimators by using q basis functions instead of n ones, resulting in a computational cost of the order O (n q 2) . These methods are called the basis selection methods. Despite algorithmic benefits, most of the basis selection methods require the assumption that the sample is uniformly distributed on a hypercube. These methods may have deteriorating performance when such an assumption is not met. To overcome the obstacle, we develop an efficient algorithm that is adaptive to the unknown probability density function of the predictors. Theoretically, we show the proposed estimator has the same convergence rate as the full-basis estimator when q is roughly at the order of O [ n 2 d / { (p r + 1) (d + 2) } ] , where p ∈ [ 1 , 2 ] and r ≈ 4 are some constants depend on the type of the spline. Numerical studies on various synthetic datasets demonstrate the superior performance of the proposed estimator in comparison with mainstream competitors. Supplementary files for this article are available online. [ABSTRACT FROM AUTHOR]

The current parameterization and algorithm used to fit a smoothing spline analysis of variance (SSANOVA) model are computationally expensive, making a generalized additive model (GAM) the preferred method for multivariate smoothing. In this article, we propose an efficient reparameterization of the smoothing parameters in SSANOVA models, and a scalable algorithm for estimating multiple smoothing parameters in SSANOVAs. To validate our approach, we present two simulation studies comparing our reparameterization and algorithm to implementations of SSANOVAs and GAMs that are currently available in R. Our simulation results demonstrate that (a) our scalable SSANOVA algorithm outperforms the currently used SSANOVA algorithm, and (b) SSANOVAs can be a fast and reliable alternative to GAMs. We also provide an example with oceanographic data that demonstrates the practical advantage of our SSANOVA framework. Supplementary materials that are available online can be used to replicate the analyses in this article. [ABSTRACT FROM PUBLISHER]