f2sSP: Overlap Group Least Absolute Shrinkage and Selection Operator...
In fdaSP: Sparse Functional Data Analysis Methods
f2sSP R Documentation
Overlap Group Least Absolute Shrinkage and Selection Operator for scalar-on-function regression model
Description
Overlap Group-LASSO for scalar-on-function regression model solves the following optimization problem
\textrm{min}_{\psi,\gamma} ~ \frac{1}{2} \sum_{i=1}^n \left( y_i - \int x_i(t) \psi(t) dt-z_i^\intercal\gamma \right)^2 + \lambda \sum_{g=1}^{G} \Vert S_{g}T\psi \Vert_2
to obtain a sparse coefficient vector \psi\in\mathbb{R}^{M} for the functional penalized predictor x(t) and a coefficient vector \gamma\in\mathbb{R}^q for the unpenalized scalar predictors z_1,\dots,z_q. The regression function is \psi(t)=\varphi(t)^\intercal\psi
where \varphi(t) is a B-spline basis of order d and dimension M. For each group g, each row of
the matrix S_g\in\mathbb{R}^{d\times M} has non-zero entries only for those bases belonging
to that group. These values are provided by the arguments groups and group_weights (see below).
Each basis function belongs to more than one group. The diagonal matrix T\in\mathbb{R}^{M\times M} contains
the basis-specific weights. These values are provided by the argument var_weights (see below).
The regularization path is computed for the overlap group-LASSO penalty at a grid of values for the regularization
parameter \lambda using the alternating direction method of multipliers (ADMM). See Boyd et al. (2011) and Lin et al. (2022)
for details on the ADMM method.
Usage
f2sSP(
vY,
mX,
mZ = NULL,
M,
group_weights = NULL,
var_weights = NULL,
standardize.data = TRUE,
splOrd = 4,
lambda = NULL,
nlambda = 30,
lambda.min.ratio = NULL,
intercept = FALSE,
overall.group = FALSE,
control = list()
)
Arguments
vY
a length-n vector of observations of the scalar response variable.
mX
a (n\times r) matrix of observations of the functional covariate.
mZ
an (n\times q) full column rank matrix of scalar predictors that are not penalized.
M
number of elements of the B-spline basis vector \varphi(t).
group_weights
a vector of length G containing group-specific weights. The default is square root of the group cardinality, see Bernardi et al. (2022).
var_weights
a vector of length M containing basis-specific weights. The default is a vector where
each entry is the reciprocal of the number of groups including that basis. See Bernardi et al. (2022) for details.
standardize.data
logical. Should data be standardized?
splOrd
the order d of the spline basis.
lambda
either a regularization parameter or a vector of regularization parameters.
In this latter case the routine computes the whole path. If it is NULL values for lambda are provided by the routine.
nlambda
the number of lambda values - default is 30.
lambda.min.ratio
smallest value for lambda, as a fraction of the maximum lambda value. If n>M, the default is 0.0001, and if n<M, the default is 0.01.
intercept
logical. If it is TRUE, a column of ones is added to the design matrix.
overall.group
logical. If it is TRUE, an overall group including all penalized covariates is added.
control
a list of control parameters for the ADMM algorithm. See ‘Details’.
Value
A named list containing
- sp.coefficients
a length-M solution vector for the parameters \psi, which corresponds to the minimum in-sample MSE.
- sp.coef.path
an (n_\lambda\times M) matrix of estimated \psi coefficients for each lambda.
- sp.fun
a length-r vector providing the estimated functional coefficient for \psi(t).
- sp.fun.path
an (n_\lambda\times r) matrix providing the estimated functional coefficients for \psi(t) for each lambda.
- coefficients
a length-q solution vector for the parameters \gamma, which corresponds to the minimum in-sample MSE.
It is provided only when either the matrix Z in input is not NULL or the intercept is set to TRUE.
- coef.path
an (n_\lambda\times q) matrix of estimated \gamma coefficients for each lambda.
It is provided only when either the matrix Z in input is not NULL or the intercept is set to TRUE.
- lambda
sequence of lambda.
- lambda.min
value of lambda that attains the minimum in-sample MSE.
- mse
in-sample mean squared error.
- min.mse
minimum value of the in-sample MSE for the sequence of lambda.
- convergence
logical. 1 denotes achieved convergence.
- elapsedTime
elapsed time in seconds.
- iternum
number of iterations.
When you run the algorithm, output returns not only the solution, but also the iteration history recording
following fields over iterates,
- objval
objective function value.
- r_norm
norm of primal residual.
- s_norm
norm of dual residual.
- eps_pri
feasibility tolerance for primal feasibility condition.
- eps_dual
feasibility tolerance for dual feasibility condition.
Iteration stops when both r_norm and s_norm values
become smaller than eps_pri and eps_dual, respectively.
Details
The control argument is a list that can supply any of the following components:
- adaptation
logical. If it is TRUE, ADMM with adaptation is performed. The default value is TRUE. See Boyd et al. (2011) for details.
- rho
an augmented Lagrangian parameter. The default value is 1.
- tau.ada
an adaptation parameter greater than one. Only needed if adaptation = TRUE. The default value is 2. See Boyd et al. (2011) and Lin et al. (2022) for details.
- mu.ada
an adaptation parameter greater than one. Only needed if adaptation = TRUE. The default value is 10. See Boyd et al. (2011) and Lin et al. (2022) for details.
- abstol
absolute tolerance stopping criterion. The default value is sqrt(sqrt(.Machine$double.eps)).
- reltol
relative tolerance stopping criterion. The default value is sqrt(.Machine$double.eps).
- maxit
maximum number of iterations. The default value is 100.
- print.out
logical. If it is TRUE, a message about the procedure is printed. The default value is TRUE.
References
\insertRefbernardi_etal.2022fdaSP
\insertRefboyd_etal.2011fdaSP
\insertRefjenatton_etal.2011fdaSP
\insertReflin_etal.2022fdaSP
Examples
## generate sample data
set.seed(1)
n <- 40
p <- 18 # number of basis to GENERATE beta
r <- 100
s <- seq(0, 1, length.out = r)
beta_basis <- splines::bs(s, df = p, intercept = TRUE) # basis
coef_data <- matrix(rnorm(n*floor(p/2)), n, floor(p/2))
fun_data <- coef_data %*% t(splines::bs(s, df = floor(p/2), intercept = TRUE))
x_0 <- apply(matrix(rnorm(p, sd=1),p,1), 1, fdaSP::softhresh, 1) # regression coefficients
x_fun <- beta_basis %*% x_0
b <- fun_data %*% x_fun + rnorm(n, sd = sqrt(crossprod(fun_data %*% x_fun ))/10)
l <- 10^seq(2, -4, length.out = 30)
maxit <- 1000
## set the hyper-parameters
maxit <- 1000
rho_adaptation <- TRUE
rho <- 1
reltol <- 1e-5
abstol <- 1e-5
mod <- f2sSP(vY = b, mX = fun_data, M = p,
group_weights = NULL, var_weights = NULL, standardize.data = FALSE, splOrd = 4,
lambda = NULL, nlambda = 30, lambda.min = NULL, overall.group = FALSE,
control = list("abstol" = abstol,
"reltol" = reltol,
"adaptation" = rho_adaptation,
"rho" = rho,
"print.out" = FALSE))
# plot coefficiente path
matplot(log(mod$lambda), mod$sp.coef.path, type = "l",
xlab = latex2exp::TeX("$\\log(\\lambda)$"), ylab = "", bty = "n", lwd = 1.2)
fdaSP documentation built on Oct. 6, 2023, 1:07 a.m.
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| f2sSP | R Documentation |
Overlap Group Least Absolute Shrinkage and Selection Operator for scalar-on-function regression model
Description
Overlap Group-LASSO for scalar-on-function regression model solves the following optimization problem
\textrm{min}_{\psi,\gamma} ~ \frac{1}{2} \sum_{i=1}^n \left( y_i - \int x_i(t) \psi(t) dt-z_i^\intercal\gamma \right)^2 + \lambda \sum_{g=1}^{G} \Vert S_{g}T\psi \Vert_2
to obtain a sparse coefficient vector \psi\in\mathbb{R}^{M} for the functional penalized predictor x(t) and a coefficient vector \gamma\in\mathbb{R}^q for the unpenalized scalar predictors z_1,\dots,z_q. The regression function is \psi(t)=\varphi(t)^\intercal\psi
where \varphi(t) is a B-spline basis of order d and dimension M. For each group g, each row of
the matrix S_g\in\mathbb{R}^{d\times M} has non-zero entries only for those bases belonging
to that group. These values are provided by the arguments groups and group_weights (see below).
Each basis function belongs to more than one group. The diagonal matrix T\in\mathbb{R}^{M\times M} contains
the basis-specific weights. These values are provided by the argument var_weights (see below).
The regularization path is computed for the overlap group-LASSO penalty at a grid of values for the regularization
parameter \lambda using the alternating direction method of multipliers (ADMM). See Boyd et al. (2011) and Lin et al. (2022)
for details on the ADMM method.
Usage
f2sSP(
vY,
mX,
mZ = NULL,
M,
group_weights = NULL,
var_weights = NULL,
standardize.data = TRUE,
splOrd = 4,
lambda = NULL,
nlambda = 30,
lambda.min.ratio = NULL,
intercept = FALSE,
overall.group = FALSE,
control = list()
)
Arguments
vY |
a length- |
mX |
a |
mZ |
an |
M |
number of elements of the B-spline basis vector |
group_weights |
a vector of length |
var_weights |
a vector of length |
standardize.data |
logical. Should data be standardized? |
splOrd |
the order |
lambda |
either a regularization parameter or a vector of regularization parameters. In this latter case the routine computes the whole path. If it is NULL values for lambda are provided by the routine. |
nlambda |
the number of lambda values - default is 30. |
lambda.min.ratio |
smallest value for lambda, as a fraction of the maximum lambda value. If |
intercept |
logical. If it is TRUE, a column of ones is added to the design matrix. |
overall.group |
logical. If it is TRUE, an overall group including all penalized covariates is added. |
control |
a list of control parameters for the ADMM algorithm. See ‘Details’. |
Value
A named list containing
- sp.coefficients
a length-
Msolution vector for the parameters\psi, which corresponds to the minimum in-sample MSE.- sp.coef.path
an
(n_\lambda\times M)matrix of estimated\psicoefficients for each lambda.- sp.fun
a length-
rvector providing the estimated functional coefficient for\psi(t).- sp.fun.path
an
(n_\lambda\times r)matrix providing the estimated functional coefficients for\psi(t)for each lambda.- coefficients
a length-
qsolution vector for the parameters\gamma, which corresponds to the minimum in-sample MSE. It is provided only when either the matrixZin input is not NULL or the intercept is set to TRUE.- coef.path
an
(n_\lambda\times q)matrix of estimated\gammacoefficients for each lambda. It is provided only when either the matrixZin input is not NULL or the intercept is set to TRUE.- lambda
sequence of lambda.
- lambda.min
value of lambda that attains the minimum in-sample MSE.
- mse
in-sample mean squared error.
- min.mse
minimum value of the in-sample MSE for the sequence of lambda.
- convergence
logical. 1 denotes achieved convergence.
- elapsedTime
elapsed time in seconds.
- iternum
number of iterations.
When you run the algorithm, output returns not only the solution, but also the iteration history recording following fields over iterates,
- objval
objective function value.
- r_norm
norm of primal residual.
- s_norm
norm of dual residual.
- eps_pri
feasibility tolerance for primal feasibility condition.
- eps_dual
feasibility tolerance for dual feasibility condition.
Iteration stops when both r_norm and s_norm values
become smaller than eps_pri and eps_dual, respectively.
Details
The control argument is a list that can supply any of the following components:
- adaptation
logical. If it is TRUE, ADMM with adaptation is performed. The default value is TRUE. See Boyd et al. (2011) for details.
- rho
an augmented Lagrangian parameter. The default value is 1.
- tau.ada
an adaptation parameter greater than one. Only needed if adaptation = TRUE. The default value is 2. See Boyd et al. (2011) and Lin et al. (2022) for details.
- mu.ada
an adaptation parameter greater than one. Only needed if adaptation = TRUE. The default value is 10. See Boyd et al. (2011) and Lin et al. (2022) for details.
- abstol
absolute tolerance stopping criterion. The default value is sqrt(sqrt(.Machine$double.eps)).
- reltol
relative tolerance stopping criterion. The default value is sqrt(.Machine$double.eps).
- maxit
maximum number of iterations. The default value is 100.
- print.out
logical. If it is TRUE, a message about the procedure is printed. The default value is TRUE.
References
\insertRefbernardi_etal.2022fdaSP
\insertRefboyd_etal.2011fdaSP
\insertRefjenatton_etal.2011fdaSP
\insertReflin_etal.2022fdaSP
Examples
## generate sample data
set.seed(1)
n <- 40
p <- 18 # number of basis to GENERATE beta
r <- 100
s <- seq(0, 1, length.out = r)
beta_basis <- splines::bs(s, df = p, intercept = TRUE) # basis
coef_data <- matrix(rnorm(n*floor(p/2)), n, floor(p/2))
fun_data <- coef_data %*% t(splines::bs(s, df = floor(p/2), intercept = TRUE))
x_0 <- apply(matrix(rnorm(p, sd=1),p,1), 1, fdaSP::softhresh, 1) # regression coefficients
x_fun <- beta_basis %*% x_0
b <- fun_data %*% x_fun + rnorm(n, sd = sqrt(crossprod(fun_data %*% x_fun ))/10)
l <- 10^seq(2, -4, length.out = 30)
maxit <- 1000
## set the hyper-parameters
maxit <- 1000
rho_adaptation <- TRUE
rho <- 1
reltol <- 1e-5
abstol <- 1e-5
mod <- f2sSP(vY = b, mX = fun_data, M = p,
group_weights = NULL, var_weights = NULL, standardize.data = FALSE, splOrd = 4,
lambda = NULL, nlambda = 30, lambda.min = NULL, overall.group = FALSE,
control = list("abstol" = abstol,
"reltol" = reltol,
"adaptation" = rho_adaptation,
"rho" = rho,
"print.out" = FALSE))
# plot coefficiente path
matplot(log(mod$lambda), mod$sp.coef.path, type = "l",
xlab = latex2exp::TeX("$\\log(\\lambda)$"), ylab = "", bty = "n", lwd = 1.2)
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