Fit adaptive regional folded-concave functional regression

arfc() fits a scalar-on-function linear model with pilot-adaptive regional MCP or SCAD penalties. A single-resolution fit uses the finest partition; a multi-resolution fit uses a nested collection of partitions in the same B-spline coefficient space. Regional multipliers are normalized by domain mass, and level weights must sum to one.

Usage

arfc(
  y,
  x,
  argvals,
  lambda,
  kappa,
  resolution = c("single", "multi"),
  penalty = c("MCP", "SCAD"),
  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.

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.

resolution

Either "single" (ASR) or "multi" (AMR).

penalty

Either "MCP" or "SCAD".

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 containing the training-BIC-tuned fit for the one requested penalty-resolution specification: its coefficient function, regional norms and weights, complete tuning diagnostics, smooth pilot, fitted values, and prediction metadata. This function does not choose between ASR and AMR or between MCP and SCAD; use arfc_select() for that four-candidate comparison.

Details

The roughness-only pilot chooses one kappa value by strict BIC. Its regional norms define level-normalized and clipped adaptive weights. The folded-concave path is then evaluated over lambda at that common smoothness value. No response or predictor standardization is performed; centering estimates the intercept.

References

Fan, J., and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348-1360.

Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38(2), 894-942.

Lin, Z., Cao, J., Wang, L., and Wang, H. (2017). Locally sparse estimator for functional linear regression models. Journal of Computational and Graphical Statistics, 26(2), 306-318.

Examples

set.seed(2026)
n <- 36
argvals <- seq(0, 1, length.out = 21)
x <- matrix(rnorm(n * length(argvals)), n)
beta <- ifelse(argvals < 0.35, 1.5 * sin(2 * pi * argvals), 0)
w <- c(0.025, rep(0.05, 19), 0.025)
y <- as.numeric(x %*% (w * beta)) + rnorm(n, sd = 0.2)
fit <- arfc(
  y, x, argvals,
  lambda = c(0.002, 0.02),
  kappa = c(1e-5, 1e-3),
  resolution = "single",
  penalty = "MCP",
  levels = c(4L, 8L),
  control = arfc_control(max_iter = 40L)
)
coef(fit, type = "basis")
predict(fit, x[1:2, , drop = FALSE])