The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy

Abstract

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares ${\sum }_j(Y_j-\mu (t_j){)}^2+\lambda {\int }_a^b[\mu "(t){]}^{2}dt$, where the data are $t_j,Y_j$, $j=1,...,n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function $\mu$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, ${\int }_a^b[\mu "(t){]}^{2}dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to penalties based on linear differential operators. In this case, one can sometimes easily calculate the minimizer explicitly, using Green's functions.