Functional linear model

A functional linear model relates functional or scalar responses to functional predictors through a linear operator.

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 $c^{\mathsf{T}}\Omega c=\Vert ({\sum }_k{c}_k{B}_k{)}^{\prime \prime }{\Vert }_2^2$.

A function-on-function linear model has $Y(t)=\int \beta (t,s)X(s)ds+\epsilon (t)$. If the centered error is independent of $X$, then $\mathbb{E}[Y\mid X]$ is the regression operator applied to $X$.

For centered Hilbert-space random variables $X$ and $Y$, the cross-covariance operator is $C_{YX}h=\mathbb{E}[⟨X,h⟩Y]$. In the functional linear model $Y=BX+\epsilon$ with independent centered error, $C_{YX}=B{C}_X$.