Cubic regression spline ($f(x)={\sum }_j{c}_j{B}_j(x)$)

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.