Ex. 5.9

Ex. 5.9

Derive the Reinsch form \(\bb{S}_\lambda = (\bb{I} + \lambda \bb{K})^{-1}\) for the smoothing spline.

Soln. 5.9

We have

\[\begin{eqnarray} \bb{S} &=& \bb{N}(\bb{N}^T\bb{N} + \lambda\bm{\Omega}_N)^{-1}\bb{N}^T\non\\ &=&\bb{N}(\bb{N}^T(\bb{I} + \lambda (\bb{N}^T)^{-1}\bm{\Omega}_N\bb{N}^{-1})\bb{N})^{-1}\bb{N}^T\non\\ &=&(\bb{I} + \lambda \bb{K})^{-1}\non \end{eqnarray}\]

where \(\bb{K} = (\bb{N}^T)^{-1}\bm{\Omega}_N\bb{N}^{-1}\).