Package 'slasso'

Smooth LASSO Estimator for the Function-on-Function Linear Regression Model

Description

Implements the Smooth LASSO Estimator for the Function-on-Function Linear Regression Model described in Centofanti et al. (2020) <arXiv:2007.00529>.

Details

Package:	slasso
Type:	Package
Version:	1.0.0
Date:	2021-10-13
License:	GPL (>= 3)

Author(s)

Fabio Centofanti, Matteo Fontana, Antonio Lepore, Simone Vantini

References

Centofanti, F., Fontana, M., Lepore, A., & Vantini, S. (2020). Smooth LASSO Estimator for the Function-on-Function Linear Regression Model. arXiv preprint arXiv:2007.00529.

See Also

slasso.fr, slasso.fr_cv

Examples

library(slasso)
data<-simulate_data("Scenario II",n_obs=150)
X_fd=data$X_fd
Y_fd=data$Y_fd
domain=c(0,1)
n_basis_s<-30
n_basis_t<-30
breaks_s<-seq(0,1,length.out = (n_basis_s-2))
breaks_t<-seq(0,1,length.out = (n_basis_t-2))
basis_s <- fda::create.bspline.basis(domain,breaks=breaks_s)
basis_t <- fda::create.bspline.basis(domain,breaks=breaks_t)

mod_slasso_cv<-slasso.fr_cv(Y_fd = Y_fd,X_fd=X_fd,basis_s=basis_s,basis_t=basis_t,
lambda_L_vec = 10^seq(0,1,by=1),lambda_s_vec = 10^-9,lambda_t_vec = 10^-7,
B0=NULL,max_iterations=10,K=2,invisible=1,ncores=1)
mod_slasso<-slasso.fr(Y_fd = Y_fd,X_fd=X_fd,basis_s=basis_s,basis_t=basis_t,
lambda_L = 10^0.7,lambda_s =10^-5,lambda_t = 10^-6,B0 =NULL,invisible=1,max_iterations=10)

plot(mod_slasso_cv)
plot(mod_slasso)

---

Plot the results of the S-LASSO method

Description

This function provides plots of the S-LASSO coefficient function estimate when applied to the output of slasso.fr, whereas provides the cross-validation plots when applied to the output of slasso.fr_cv. In the latter case the first plot displays the CV values as a function of lambda_L, lambda_s and lambda_t, and the second plot displays the CV values as a function of lambda_L with lambda_s and lambda_t fixed at their optimal values.

Usage

## S3 method for class 'slasso_cv'
plot(x, ...)

## S3 method for class 'slasso'
plot(x, ...)

Arguments

x	The output of either slasso.fr_cv or slasso.fr.
...	No additional parameters, called for side effects.

Value

No return value, called for side effects.

Examples

library(slasso)
data<-simulate_data("Scenario II",n_obs=150)
X_fd=data$X_fd
Y_fd=data$Y_fd
domain=c(0,1)
n_basis_s<-30
n_basis_t<-30
breaks_s<-seq(0,1,length.out = (n_basis_s-2))
breaks_t<-seq(0,1,length.out = (n_basis_t-2))
basis_s <- fda::create.bspline.basis(domain,breaks=breaks_s)
basis_t <- fda::create.bspline.basis(domain,breaks=breaks_t)
mod_slasso<-slasso.fr(Y_fd = Y_fd,X_fd=X_fd,basis_s=basis_s,basis_t=basis_t,
lambda_L = -1.5,lambda_s =-8,lambda_t = -7,B0 =NULL,invisible=1,max_iterations=10)
plot(mod_slasso)

---

Simulate data through the function-on-function linear regression model

Description

Generate synthetic data as in the simulation study of Centofanti et al. (2020).

Usage

simulate_data(scenario, n_obs = 3000, type_x = "Bspline")

Arguments

scenario	A character strings indicating the scenario considered. It could be "Scenario I", "Scenario II", "Scenario III", and "Scenario IV".
n_obs	Number of observations.
type_x	Covariate generating mechanism, either Bspline or Brownian.

Value

A list containing the following arguments:

X: Covariate matrix, where the rows correspond to argument values and columns to replications.

Y: Response matrix, where the rows correspond to argument values and columns to replications.

X_fd: Coavariate functions.

Y_fd: Response functions.

clus: True cluster membership vector.

References

Centofanti, F., Fontana, M., Lepore, A., & Vantini, S. (2020). Smooth LASSO Estimator for the Function-on-Function Linear Regression Model. arXiv preprint arXiv:2007.00529.

Examples

library(slasso)
data<-simulate_data("Scenario II",n_obs=150)

---

Smooth LASSO estimator for the function-on-function linear regression model

Description

The smooth LASSO (S-LASSO) method for the function-on-function linear regression model provides interpretable coefficient function estimates that are both locally sparse and smooth (Centofanti et al., 2020).

Usage

slasso.fr(
  Y_fd,
  X_fd,
  basis_s,
  basis_t,
  lambda_L,
  lambda_s,
  lambda_t,
  B0 = 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.
lambda_L	Regularization parameter of the functional LASSO penalty.
lambda_s	Regularization parameter of the smoothness penalty along the s-direction.
lambda_t	Regularization parameter of the smoothness penalty along the t-direction.
B0	Initial estimator of the basis coefficients matrix of the coefficient function. Should have dimensions in accordance with the basis dimensions of basis_s and basis_t.
...	Other arguments to be passed to the Orthant-Wise Limited-memory Quasi-Newton optimization function. See the lbfgs help page of the package lbfgs.

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.

• lambdas_L: Regularization parameter of the functional LASSO penalty.

• lambda_s: Regularization parameter of the smoothness penalty along the s-direction.

• lambda_t: Regularization parameter of the smoothness penalty along the t-direction.

• Y_fd: The response functions.

• X_fd: The covariate functions.

• per_0: The fraction of domain where the coefficient function is zero.

• type: The output type.

References

Centofanti, F., Fontana, M., Lepore, A., & Vantini, S. (2020). Smooth LASSO Estimator for the Function-on-Function Linear Regression Model. arXiv preprint arXiv:2007.00529.

See Also

slasso.fr_cv

Examples

library(slasso)
data<-simulate_data("Scenario II",n_obs=150)
X_fd=data$X_fd
Y_fd=data$Y_fd
domain=c(0,1)
n_basis_s<-30
n_basis_t<-30
breaks_s<-seq(0,1,length.out = (n_basis_s-2))
breaks_t<-seq(0,1,length.out = (n_basis_t-2))
basis_s <- fda::create.bspline.basis(domain,breaks=breaks_s)
basis_t <- fda::create.bspline.basis(domain,breaks=breaks_t)
mod_slasso<-slasso.fr(Y_fd = Y_fd,X_fd=X_fd,basis_s=basis_s,basis_t=basis_t,
lambda_L = -1.5,lambda_s =-8,lambda_t = -7,B0 =NULL,invisible=1,max_iterations=10)

---

Cross-validation for the S-LASSO estimator

Description

K-fold cross-validation procedure to choose the tuning parameters for the S-LASSO estimator (Centofanti et al., 2020).

Usage

slasso.fr_cv(
  Y_fd,
  X_fd,
  basis_s,
  basis_t,
  K = 10,
  kss_rule_par = 0.5,
  lambda_L_vec = NULL,
  lambda_s_vec = NULL,
  lambda_t_vec = NULL,
  B0 = NULL,
  ncores = 1,
  ...
)

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.
K	Number of folds. Default is 10.
kss_rule_par	Parameter of the k-standard error rule. If kss_rule_par=0 the tuning parameters that minimize the estimated prediction error are chosen. Default is 0.5.
lambda_L_vec	Vector of regularization parameters of the functional LASSO penalty.
lambda_s_vec	Vector of regularization parameters of the smoothness penalty along the s-direction.
lambda_t_vec	Vector of regularization parameters of the smoothness penalty along the t-direction.
B0	Initial estimator of the basis coefficients matrix of the coefficient function. Should have dimensions in accordance with the basis dimensions of basis_s and basis_t.
ncores	If ncores>1, then parallel computing is used, with ncores cores. Default is 1.
...	Other arguments to be passed to the Orthant-Wise Limited-memory Quasi-Newton optimization function. See the lbfgs help page of the package lbfgs.

Value

A list containing the following arguments:

• lambda_opt_vec: Vector of optimal tuning parameters.

• CV: Estimated prediction errors.

• CV_sd: Standard errors of the estimated prediction errors.

• per_0: The fractions of domain where the coefficient function is zero for all the tuning parameters combinations.

• comb_list: The combinations of lambda_L,lambda_s and lambda_t explored.

• Y_fd: The response functions.

• X_fd: The covariate functions.

References

Centofanti, F., Fontana, M., Lepore, A., & Vantini, S. (2020). Smooth LASSO Estimator for the Function-on-Function Linear Regression Model. arXiv preprint arXiv:2007.00529.

See Also

slasso.fr

Examples

library(slasso)
data<-simulate_data("Scenario II",n_obs=150)
X_fd=data$X_fd
Y_fd=data$Y_fd
domain=c(0,1)
n_basis_s<-60
n_basis_t<-60
breaks_s<-seq(0,1,length.out = (n_basis_s-2))
breaks_t<-seq(0,1,length.out = (n_basis_t-2))
basis_s <- fda::create.bspline.basis(domain,breaks=breaks_s)
basis_t <- fda::create.bspline.basis(domain,breaks=breaks_t)
mod_slasso_cv<-slasso.fr_cv(Y_fd = Y_fd,X_fd=X_fd,basis_s=basis_s,basis_t=basis_t,
lambda_L_vec=seq(0,1,by=1),lambda_s_vec=c(-9),lambda_t_vec=-7,B0=NULL,
max_iterations=10,K=2,invisible=1,ncores=1)

---