SmoothingParameter

A smoothing parameter is a tuning parameter used to control the degree of smoothness of an estimated function or signal. In smoothing methods, it balances fidelity to observed data against the roughness of the resulting fit. Larger values produce smoother estimates with lower variance but potentially higher bias; smaller values fit the data more closely and may capture noise as structure.

Common contexts include kernel smoothing, smoothing splines, and LOESS. In kernel methods such as kernel regression or kernel density estimation, the bandwidth parameter determines the kernel width; increasing the bandwidth yields smoother curves and densities, while decreasing it makes the estimate more variable. In smoothing splines, the smoothing parameter (often denoted lambda) weights a penalty on roughness, such as the integral of squared second derivatives. As lambda increases, the fit becomes smoother, and in the limit may resemble a low-degree polynomial or linear function; as lambda decreases, the fit approaches an interpolating curve through the data.

LOESS or LOWESS uses a span parameter that controls the fraction of data used in local fits, effectively adjusting smoothness.

Selecting an appropriate smoothing parameter is a model-choice problem and is typically guided by cross-validation, generalized cross-validation, information criteria, or visual inspection, sometimes aided by L-curve analysis or Bayesian methods.

It is important to distinguish the smoothing parameter from inherent data noise; it is a tuning parameter rather than a fixed property of the underlying process.