The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy
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The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares ∑_j(Y_j - μ(t_j))^2 + λ∫_a^b [μ"(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 μ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, ∫_a^b [μ"(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.
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