Regression spline

A regression spline fits a regression function in a finite-dimensional spline approximation space with selected knots. Evaluating its basis at predictor values gives a design matrix, reducing fitting to regression on basis coefficients.

A cubic regression spline represents a mean function in a finite-dimensional space of piecewise cubic functions, with continuity of the function and its first two derivatives at chosen knots. A penalized fit minimizes a residual criterion plus $\lambda \int (f^{\prime \prime }{)}^2$, using ${\Omega }_{jk}=\int B_j^{\prime \prime }{B}_k^{\prime \prime }$. Unlike a full cubic smoothing spline with knots at every distinct observation, its knot set can be much smaller. The mgcv basis bs="cr" uses a penalized natural cubic regression-spline basis with linear tails.

A penalized regression spline estimates basis coefficients by minimizing a residual or likelihood criterion plus a curvature penalty. Basis size limits the available complexity; the penalty controls how much of that complexity is used.