LOESSsmoothing

LOESS smoothing, or locally estimated scatterplot smoothing, is a nonparametric regression technique for estimating a smooth relationship between a predictor and a response from scattered data. It fits simple models to localized subsets of the data and combines the results to form a smooth curve, without assuming a single global functional form.

Procedure: For each x_i, select a neighborhood containing a fraction α of the data (the span). Assign weights to observations in the neighborhood using a kernel, typically the tri-cube kernel, so observations closer to x_i receive greater weight. Fit a weighted least squares regression of degree p (commonly p = 1 or p = 2) to the neighborhood, and evaluate the fitted model at x_i to obtain ŷ_i. The collection of ŷ_i over all x_i traces the LOESS curve. To downweight outliers, a robust version performs several iterations, reweighting observations by the residuals and recomputing the local fits (the robust locally weighted regression, often referred to as LOWESS).

Parameters and choices: The span α controls smoothness (smaller α yields more wiggly curves; larger α yields smoother curves). The local polynomial degree p determines the local fit's flexibility. Boundary handling and computational cost are practical considerations; LOESS is data-driven and does not extrapolate beyond the observed range.

History and use: LOESS was introduced by William S. Cleveland and colleagues in 1979 as a flexible method for scatterplot smoothing. It is implemented in many statistical software packages and is widely used for exploratory data analysis, trend extraction in time series, and easing nonlinear relationships in regression diagnostics.