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).
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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.