Build a B-spline second-order difference penalty matrix

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.

Value

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

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.

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
#> [1] 11 11
P1[1, 1]           # 0 -- intercept is not penalised
#> [1] 0

# 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)
#> [1] 31 31
isSymmetric(P3)
#> [1] TRUE