build_vcmm_design: Build the design matrix and penalty for a VCMM

In cevcmm: Communication-Efficient Varying Coefficient Mixed-Effects Models

Build the design matrix and penalty for a VCMM

Description

Constructs the fixed-effects design matrix and matching penalty for a VCMM with one or more varying coefficients. Given covariates X (n by K) and a time/index vector t (length n), builds the cubic-by-default B-spline basis B at t and assembles

• X_design = cbind(1, B * X[,1], B * X[,2], ..., B * X[,K]), dimension n by (1 + K * n_basis); the first column is the intercept, then each covariate gets its own n_basis-column block.

• penalty: a (1 + K * n_basis) by (1 + K * n_basis) block-diagonal second-order difference penalty matrix with the intercept unpenalised.

This is the natural multi-covariate generalisation of the single-covariate design used in the simulation code, matching the paper's general VCMM specification.

Usage

build_vcmm_design(
  X,
  t,
  n_basis = NULL,
  degree = 3L,
  lambda = 1,
  normalize_t = TRUE
)

Arguments

X	Numeric n by K matrix (or length-n vector if K = 1) of covariates that get varying coefficients in t.
t	Numeric vector of length n. The variable in which the coefficients vary smoothly (time, index, location, etc.).
n_basis	Integer (\geq degree + 1) or NULL. Number of B-spline basis columns per varying coefficient. If NULL (default), chosen as max(floor(n^(1/3)) + 4, 10) matching the simulation code's rule of thumb.
degree	Integer. B-spline degree (default 3 = cubic).
lambda	Non-negative numeric. Smoothing parameter for the penalty (default 1).
normalize_t	Logical. If TRUE (default), t is linearly mapped to [0, 1] before building the basis. Set FALSE only if t is already in [0, 1].

Value

A list with elements:

• X_design: numeric n by (1 + K * n_basis) design matrix.

• penalty: numeric (1 + K * n_basis) by (1 + K * n_basis) penalty matrix.

• B_spline: numeric n by n_basis basis matrix at t.

• internal_knots, boundary_knots: spline knot locations, recorded so predictions at new t can use the same basis.

• degree, n_basis, K, lambda: scalars recording how the design was built.

• normalize_t, t_min, t_max: how to remap new t values at prediction time.

References

Jalili, L. and Lin, L.-H. (2025). Scalable and Communication-Efficient Varying Coefficient Mixed-Effects Models.

Examples

set.seed(1)
n <- 200
t <- runif(n)
x1 <- runif(n); x2 <- runif(n)

# Single varying coefficient
d1 <- build_vcmm_design(X = x1, t = t, n_basis = 8)
dim(d1$X_design)   # 200 x 9   (1 + 1*8)

# Two varying coefficients
d2 <- build_vcmm_design(X = cbind(x1, x2), t = t, n_basis = 8)
dim(d2$X_design)   # 200 x 17  (1 + 2*8)

cevcmm documentation built on July 24, 2026, 5:07 p.m.