Scalar-on-function linear model

A scalar-on-function linear model has $Y=\alpha +⟨X,\beta ⟩+\epsilon$, where the response is scalar and the predictor and slope are functions. Basis truncation turns it into a finite-dimensional linear regression.

For basis functions $B_1,\dots ,B_K$, the second-derivative roughness penalty has matrix entries ${\Omega }_{jk}=\int B_j^{\prime \prime }(t)B_k^{\prime \prime }(t)dt$. It is positive semidefinite and satisfies $c^{\mathsf{T}}\Omega c=\Vert ({\sum }_k{c}_k{B}_k{)}^{\prime \prime }{\Vert }_2^2$.