Select the ARFC penalty family and resolution by validation loss

arfc_select() fits the four prespecified candidates ASR-MCP, ASR-SCAD, AMR-MCP, and AMR-SCAD to the same training data, pilot, tuning grids, and preprocessing. It selects the smallest response MSE on an explicitly supplied validation sample. The validation responses are never used to fit a candidate. Numerical ties follow the documented order above.

Usage

arfc_select(
  y,
  x,
  argvals,
  validation,
  lambda,
  kappa,
  levels = c(8L, 16L, 32L),
  level_weights = NULL,
  adaptive = TRUE,
  degree = 3L,
  penalty_order = 2L,
  mcp_gamma = 2.7,
  scad_a = 3.7,
  integration = c("simpson", "trapezoid"),
  control = arfc_control()
)

Arguments

y

Finite numeric response vector.

x

Finite numeric matrix with subjects in rows and functional grid points in columns.

argvals

Strictly increasing functional argument values.

validation

Named list with finite y and numeric matrix x on the same functional grid as the training data. A split is never created silently.

lambda

Nonempty nonnegative folded-concave tuning grid.

kappa

Nonempty nonnegative roughness grid for the common smooth pilot. The BIC-selected value is used for the ARFC path.

levels

Strictly increasing nested region counts. Every value must divide the largest value. The default reproduces the study specification.

level_weights

Positive weights, one per levels, summing to one. NULL gives equal weights. They are not silently normalized.

adaptive

Logical. Use smooth-pilot adaptive regional weights. Setting this to FALSE gives the corresponding nonadaptive regional fit.

degree

B-spline polynomial degree.

penalty_order

Derivative order for roughness penalization.

mcp_gamma

MCP shape parameter greater than one.

scad_a

SCAD shape parameter greater than two. The defaults mcp_gamma = 2.7 and scad_a = 3.7 have equal scalar maximum concavity.

integration

Numerical integration rule for the observed predictor.

control

Numerical and adaptive-weight settings from arfc_control().

Value

An object of class arfc_select. It contains one selected_fit, all four candidate fits, validation losses, tie diagnostics, and an explicit no-leakage record. The candidates are not averaged or ensembled. A final combined-data refit is not performed automatically.

Examples

set.seed(2027)
argvals <- seq(0, 1, length.out = 21)
x <- matrix(rnorm(48 * length(argvals)), 48)
beta <- ifelse(argvals >= 0.35 & argvals <= 0.65, 0,
               sin(2 * pi * argvals))
w <- c(0.025, rep(0.05, 19), 0.025)
y <- as.numeric(x %*% (w * beta)) + rnorm(48, sd = 0.2)
selected <- arfc_select(
  y[1:32], x[1:32, , drop = FALSE], argvals,
  validation = list(y = y[33:48], x = x[33:48, , drop = FALSE]),
  lambda = c(0.002, 0.02),
  kappa = c(1e-5, 1e-3),
  levels = c(4L, 8L),
  control = arfc_control(max_iter = 40L)
)
selected$selection