Pspline

Pspline, or P-spline, is a method for smoothing and nonparametric regression that combines a B-spline basis with a roughness penalty on the coefficients. Introduced by Eilers and Marx in 1996, P-splines provide a flexible and computationally efficient approach to estimate smooth functions from noisy data. The model expresses the unknown function f(x) as f(x) = sum_j B_j(x) α_j, where B_j are B-spline basis functions and α_j are coefficients. The roughness penalty typically takes the form λ ∑ (Δ^d α_j)^2, a discrete approximation to the integral of the squared d-th derivative of f. The smoothing parameter λ controls the trade-off between fidelity to the data and smoothness.

Estimation is done by solving a penalized least squares problem: (X'X + λP) α = X'y, with P the penalty matrix derived from the chosen difference order. The same framework supports generalized responses (via a link function) in generalized additive models and can handle multiple covariates using tensor-product bases. Selection of λ is usually via generalized cross-validation, AIC, REML, or cross-validation.

Compared with classical smoothing splines, P-splines use a fixed B-spline basis with a relatively small number of knots, which improves computational efficiency and numerical stability, especially for large datasets. They are widely used in smoothing, density estimation, and nonparametric regression, and have many extensions, including monotone and constrained P-splines, multivariate smoothing, and Bayesian formulations.