f2fSP_cv: Cross-validation for Overlap Group Least Absolute Shrinkage and Selection Operator for function-on-function regression model

Description

Overlap Group-LASSO for function-on-function regression model solves the following optimization problem $$\textrm{min}_{\psi} ~ \frac{1}{2} \sum_{i=1}^n \int \left( y_i(s) - \int x_i(t) \psi(t,s) dt \right)^2 ds + \lambda \sum_{g=1}^{G} \Vert S_{g}T\psi \Vert_2$$ to obtain a sparse coefficient vector $\psi=\mathsf{vec}(\Psi)\in\mathbb{R}^{ML}$ for the functional penalized predictor $x(t)$, where the coefficient matrix $\Psi\in\mathbb{R}^{M\times L}$, the regression function $\psi(t,s)=\varphi(t)^\intercal\Psi\theta(s)$, $\varphi(t)$ and $\theta(s)$ are two B-splines bases of order $d$ and dimension $M$ and $L$, respectively. For each group $g$, each row of the matrix $S_g\in\mathbb{R}^{d\times ML}$ 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 $T\in\mathbb{R}^{ML\times ML}$ 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

f2fSP_cv(
  mY,
  mX,
  L,
  M,
  group_weights = NULL,
  var_weights = NULL,
  standardize.data = FALSE,
  splOrd = 4,
  lambda = NULL,
  lambda.min.ratio = NULL,
  nlambda = NULL,
  cv.fold = 5,
  overall.group = FALSE,
  control = list()
)

Value

A named list containing

sp.coefficients: an $(M\times L)$ solution matrix for the parameters $\Psi$, which corresponds to the minimum cross-validated MSE.
sp.fun: an $(r_x\times r_y)$ matrix providing the estimated functional coefficient for $\psi(t,s)$ corresponding to the minimum cross-validated MSE.
lambda: sequence of lambda.
lambda.min: value of lambda that attains the cross-validated minimum mean squared error.
indi.min.mse: index of the lambda sequence corresponding to lambda.min.
mse: cross-validated mean squared error.
min.mse: minimum value of the cross-validated MSE for the sequence of lambda.
mse.sd: standard deviation of the cross-validated mean squared error.
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

mY: an $(n\times r_y)$ matrix of observations of the functional response variable.
mX: an $(n\times r_x)$ matrix of observations of the functional covariate.
L: number of elements of the B-spline basis vector $\theta(s)$.
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 $ML$ 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 $nr_y>LM$, the default is 0.0001, and if $nr_y<LM$, the default is 0.01.
nlambda: the number of lambda values - default is 30.
cv.fold: the number of folds - default is 5.
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
set.seed(4321)
s  <- seq(0, 1, length.out = 100)
t  <- seq(0, 1, length.out = 100)
p1 <- 5
p2 <- 6
r  <- 10
n  <- 50

beta_basis1 <- splines::bs(s, df = p1, intercept = TRUE)    # first basis for beta
beta_basis2 <- splines::bs(s, df = p2, intercept = TRUE)    # second basis for beta

data_basis <- splines::bs(s, df = r, intercept = TRUE)    # basis for X

x_0   <- apply(matrix(rnorm(p1 * p2, sd = 1), p1, p2), 1, 
               fdaSP::softhresh, 1.5)  # regression coefficients 
x_fun <- beta_basis2 %*% x_0 %*%  t(beta_basis1)  

fun_data <- matrix(rnorm(n*r), n, r) %*% t(data_basis)
b        <- fun_data %*% x_fun + rnorm(n * 100, sd = sd(fun_data %*% x_fun )/3)

## set the hyper-parameters
maxit          <- 1000
rho_adaptation <- FALSE
rho            <- 0.01
reltol         <- 1e-5
abstol         <- 1e-5

## fit functional regression model
mod_cv <- f2fSP_cv(mY = b, mX = fun_data, L = p1, M = p2,
                   group_weights = NULL, var_weights = NULL, 
                   standardize.data = FALSE, splOrd = 4,
                   lambda = NULL, nlambda = 30, cv.fold = 5, 
                   lambda.min.ratio = NULL,
                   control = list("abstol" = abstol, 
                                  "reltol" = reltol, 
                                  "maxit" = maxit, 
                                  "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 <- f2fSP(mY = b, mX = fun_data, L = p1, M = p2,
             group_weights = NULL, var_weights = NULL, 
             standardize.data = FALSE, splOrd = 4,
             lambda = NULL, nlambda = 30, lambda.min.ratio = NULL, 
             control = list("abstol" = abstol, 
                            "reltol" = reltol, 
                            "maxit" = maxit,
                            "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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