f2sSP_cv: Cross-validation for Overlap Group Least Absolute Shrinkage and Selection Operator on 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_cv(
  vY,
  mX,
  mZ = NULL,
  M,
  group_weights = NULL,
  var_weights = NULL,
  standardize.data = FALSE,
  splOrd = 4,
  lambda = NULL,
  lambda.min.ratio = NULL,
  nlambda = NULL,
  cv.fold = 5,
  intercept = FALSE,
  overall.group = FALSE,
  control = list()
)

Value

A named list containing

sp.coefficients: a length-$M$ solution vector solution vector for the parameters $\psi$, which corresponds to the minimum cross-validated MSE.
sp.fun: a length-$r$ vector providing the estimated functional coefficient for $\psi(t)$ corresponding to the minimum cross-validated MSE.
coefficients: a length-$q$ solution vector for the parameters $\gamma$, which corresponds to the minimum cross-validated MSE. 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 cross-validated MSE.
mse: cross-validated mean squared error.
min.mse: minimum value of the cross-validated MSE for the sequence of lambda.
convergence: logical. 1 denotes achieved convergence.
elapsedTime: elapsed time in seconds.
iternum: number of iterations.

Iteration stops when both r_norm and s_norm values become smaller than eps_pri and eps_dual, respectively.

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.
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.
nlambda: the number of lambda values - default is 30.
cv.fold: the number of folds - default is 5.
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’.

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

bernardi_etal.2022fdaSP

boyd_etal.2011fdaSP

jenatton_etal.2011fdaSP

lin_etal.2022fdaSP

Examples

Run this code


## generate sample data and functional coefficients
set.seed(1)
n     <- 40
p     <- 18                                 
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)  
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

## run cross-validation
mod_cv <- f2sSP_cv(vY = b, mX = fun_data, M = p,
                   group_weights = NULL, var_weights = NULL, standardize.data = FALSE, splOrd = 4,
                   lambda = NULL, lambda.min = 1e-5, nlambda = 30, cv.fold = 5, intercept = FALSE, 
                   control = list("abstol" = abstol, 
                                  "reltol" = reltol, 
                                  "adaptation" = rho_adaptation,
                                  "rho" = rho, 
                                  "print.out" = FALSE))
                                          
### graphical presentation
plot(log(mod_cv$lambda), mod_cv$mse, type = "l", col = "blue", lwd = 2, bty = "n", 
     xlab = latex2exp::TeX("$\\log(\\lambda)$"), ylab = "Prediction Error", 
     ylim = range(mod_cv$mse - mod_cv$mse.sd, mod_cv$mse + mod_cv$mse.sd),
     main = "Cross-validated Prediction Error")
fdaSP::confband(xV = log(mod_cv$lambda), yVmin = mod_cv$mse - mod_cv$mse.sd, 
                yVmax = mod_cv$mse + mod_cv$mse.sd)       
abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), 
       col = "red", lwd = 1.0)

### comparison with oracle error
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 = 1e-5, intercept = FALSE,
             control = list("abstol" = abstol, 
                            "reltol" = reltol, 
                            "adaptation" = rho_adaptation, 
                            "rho" = rho, 
                            "print.out" = FALSE))
                                    
err_mod <- apply(mod$sp.coef.path, 1, function(x) sum((x - x_0)^2))
plot(log(mod$lambda), err_mod, type = "l", col = "blue", 
     lwd = 2, xlab = latex2exp::TeX("$\\log(\\lambda)$"), 
     ylab = "Estimation Error", main = "True Estimation Error", bty = "n")
abline(v = log(mod$lambda[which(err_mod == min(err_mod))]), col = "red", lwd = 1.0)
abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), 
       col = "red", lwd = 1.0, lty = 2)

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