= Cubic smoothing spline
{title2=$\widehat g_\lambda=N(I+\lambda\Gamma)^{-1}\mathbf Y$}
= Natural cubic smoothing spline
{synonym}
For $\lambda>0$ and at least two distinct design points, the unique $C^2[a,b]$ minimizer of squared residuals plus $\lambda J(g)$ is $N(I+\lambda\Gamma)^{-1}\mathbf Y$. The <minimum roughness property of the natural cubic spline interpolant> reduces the optimization to the value vector. The <matrix> $I+\lambda\Gamma$ is a <positive-definite matrix>, so the normal equations have a unique solution; the interpolation equality case gives uniqueness as a function. Linear tails are part of the natural boundary convention.
Back to article page
