build_penalty_matrix: Build a B-spline second-order difference penalty matrix

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

Build a B-spline second-order difference penalty matrix

Description

Constructs a symmetric p \times p penalty matrix \mathbf{P}_\lambda for use with penalised B-spline varying coefficients. For a single varying coefficient (n_blocks = 1, default), p = n\_basis + 1; the intercept (row/column 1) is unpenalised, and the remaining block is \lambda \mathbf{D}_2^\top \mathbf{D}_2 where \mathbf{D}_2 is the second-order difference operator on n_basis coefficients.

Usage

build_penalty_matrix(n_basis, lambda, n_blocks = 1L)

Arguments

n_basis	Integer (\geq 3). Number of B-spline basis columns per varying coefficient.
lambda	Non-negative numeric. Smoothing parameter \lambda.
n_blocks	Integer (\geq 1). Number of varying coefficients (covariates) sharing the same penalty structure. Default 1.

Details

For n_blocks > 1 (multiple covariates each with their own varying coefficient), the result is block-diagonal: one zero entry for the intercept, followed by n_blocks copies of the same n_basis x n_basis penalty block. The total dimension is 1 + n_blocks * n_basis.

A small ridge is added if any block is not positive-definite, to guarantee numerical stability in downstream solves.

Value

A symmetric (1 + n_blocks * n_basis) x (1 + n_blocks * n_basis) numeric matrix. Row/column 1 is zero (intercept is unpenalised).

References

Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89–121.

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

Examples

# Single varying coefficient
P1 <- build_penalty_matrix(n_basis = 10, lambda = 1)
dim(P1)            # 11 x 11
P1[1, 1]           # 0 -- intercept is not penalised

# Three varying coefficients sharing the same penalty
P3 <- build_penalty_matrix(n_basis = 10, lambda = 1, n_blocks = 3)
dim(P3)            # 31 x 31  (1 + 3*10)
isSymmetric(P3)

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