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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]

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We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size |$n$|⁠ , the smoothing spline estimator can be expressed as a linear combination of |$n$| basis functions, requiring |$O(n^3)$| computational time when the number |$d$| of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on |$q$| randomly selected basis functions, resulting in a computational cost of |$O(nq^2)$|⁠. It is known that these two estimators converge at the same rate when |$q$| is of order |$O\{n^{2/(pr+1)}\}$|⁠ , where |$p\in [1,2]$| depends on the true function and |$r > 1$| depends on the type of spline. Such a |$q$| is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease |$q$| to around |$O\{n^{1/(pr+1)}\}$| when |$d\leq pr+1$|⁠. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods. [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]

Cyclic biomechanical data are commonplace in orthopedic, rehabilitation, and sports research, where the goal is to understand and compare biomechanical differences between experimental conditions and/or subject populations. A common approach to analyzing cyclic biomechanical data involves averaging the biomechanical signals across cycle replications, and then comparing mean differences at specific points of the cycle. This pointwise analysis approach ignores the functional nature of the data, which can hinder one׳s ability to find subtle differences between experimental conditions and/or subject populations. To overcome this limitation, we propose using mixed-effects smoothing spline analysis of variance (SSANOVA) to analyze differences in cyclic biomechanical data. The SSANOVA framework makes it possible to decompose the estimated function into the portion that is common across groups (i.e., the average cycle, AC) and the portion that differs across groups (i.e., the contrast cycle, CC). By partitioning the signal in such a manner, we can obtain estimates of the CC differences (CCDs), which are the functions directly describing group differences in the cyclic biomechanical data. Using both simulated and experimental data, we illustrate the benefits of using SSANOVA models to analyze differences in noisy biomechanical (gait) signals collected from multiple locations (joints) of subjects participating in different experimental conditions. Using Bayesian confidence intervals, the SSANOVA results can be used in clinical and research settings to reliably quantify biomechanical differences between experimental conditions and/or subject populations. [ABSTRACT FROM AUTHOR]