A penalized function estimate balancing a residual loss against an integrated derivative penalty. For squared-error loss and the second derivative roughness penalty, the estimate is a cubic smoothing spline. Higher derivative orders give other spline degrees.
For and at least two distinct design points, the unique minimizer of squared residuals plus is . The minimum roughness property of the natural cubic spline interpolant reduces the optimization to the value vector. The matrix 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.
Articles by others on the same topic
A smoothing spline is a type of statistical tool used for analyzing and fitting data. Specifically, it is a form of spline, which is a piecewise-defined polynomial function that is used to create a smooth curve through a given set of data points. The primary objective of using a smoothing spline is to find a curve that balances fidelity to the data (i.e., minimizing the error in fitting the data) with smoothness (i.e., avoiding overfitting the data).