## Adaptive smoothing spline estimator for the function-on-function linear regression model

### Description

The adaptive smoothing spline (AdaSS) estimator for the function-on-function linear regression proposed in Centofanti et al., 2020.

### Usage

adass.fr(
Y_fd,
X_fd,
basis_s,
basis_t,
beta_ders = NULL,
beta_dert = NULL,
grid_eval_ders = NULL,
grid_eval_dert = NULL,
tun_par = c(lambda_s = 10^4, lambda_t = 10^4, delta_s = 0, gamma_s = 1, delta_t = 0,
delta_t = 1),
CV = FALSE,
K = 10,
X_fd_test = NULL,
Y_fd_test = NULL
)


### Arguments

 Y_fd An object of class fd corresponding to the response functions. X_fd An object of class fd corresponding to the covariate functions. basis_s B-splines basis along the s-direction of class basisfd. basis_t B-splines basis along the t-direction of class basisfd. beta_ders Initial estimate of the partial derivative of the coefficient function along the s-direction. Either a matrix or a class basisfd object. If NULL no adaptive penalty is used along the s-direction. beta_dert Initial estimate of the partial derivative of the coefficient function along the t-direction. Either a matrix or a class basisfd object. If NULL no adaptive penalty is used along the t-direction. grid_eval_ders Grid of evaluation of the partial derivatives along the s-direction. grid_eval_dert Grid of evaluation of the partial derivatives along the t-direction. tun_par Vector of tuning parameters. CV If TRUE the K-fold cross-validation prediction error is calculated. Default is FALSE. If X_fd_test and Y_fd_test are both provided the prediction error on the test set is calculated in place of the cross-validation prediction error when CV is TRUE. K Number of folds. Default is 10. X_fd_test Test set covariate functions. Default is NULL. Y_fd_test Test set response functions. Default is NULL.

### Value

A list containing the following arguments:

• B: The basis coefficients matrix estimate of the coefficient function.

• Beta_hat_fd: The coefficient function estimate of class bifd.

• alpha: The intercept function estimate.

• tun_par: Vector of tuning parameters.

• CV: Estimated prediction error.

• CV_sd: Standard error of the estimated prediction error.

• Y_fd: The response functions.

• X_fd: The covariate functions.

### References

Centofanti, F., Lepore, A., Menafoglio, A., Palumbo, B., Vantini, S. (2020). Adaptive Smoothing Spline Estimator for the Function-on-Function Linear Regression Model. arXiv preprint arXiv:2011.12036.

adass.fr_eaass

### Examples

library(adass)
data<-simulate_data("Scenario HAT",n_obs=100)
X_fd=data$X_fd Y_fd=data$Y_fd
basis_s <- fda::create.bspline.basis(c(0,1),nbasis = 10,norder = 4)
basis_t <- fda::create.bspline.basis(c(0,1),nbasis = 10,norder = 4)
mod_smooth <-adass.fr(Y_fd,X_fd,basis_s = basis_s,basis_t = basis_t,tun_par=c(10^-6,10^-6,0,0,0,0))
grid_s<-seq(0,1,length.out = 10)
grid_t<-seq(0,1,length.out = 10)
beta_der_eval_s<-fda::eval.bifd(grid_s,grid_t,mod_smooth$Beta_hat_fd,sLfdobj = 2) beta_der_eval_t<-fda::eval.bifd(grid_s,grid_t,mod_smooth$Beta_hat_fd,tLfdobj = 2)