LOESSLOWESS

LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing) are nonparametric regression techniques for fitting smooth curves to scatterplot data without assuming a single global model. The approach builds simple local models for neighborhoods of the predictor variable and stitches them into a smooth curve.

For each target x0, select a neighborhood containing a fraction 'span' of the data, usually determined by a smoothing parameter between 0.2 and 0.5. Within this neighborhood, fit a low-degree polynomial (commonly linear or quadratic) to the (x,y) pairs using weighted least squares, with weights decreasing with distance from x0 (for example, a tricube kernel). The fitted value at x0 is taken as the estimate yhat0. This process is repeated for a grid of x0 values to produce the smooth curve.

To mitigate outliers, LOESS can be made robust by an iterative reweighting step: after an initial fit, compute residuals, assign robustness weights, and refit; repeat a few times. The final curve then preserves structure while resisting outliers.

The method is flexible and requires only mild assumptions; it handles nonlinear relationships well and provides confidence intervals via resampling or asymptotic theory. However, it can be sensitive to the span choice, suffers from boundary bias, does not extrapolate well beyond the data range, and has higher computational cost relative to global fits (O(n^2) for naïve implementations).

LOESS/LOWESS was developed by William S. Cleveland and colleagues in the late 1970s as a practical smoothing technique. It is widely implemented in statistical software (for example, R's loess, Python's statsmodels lowess/loess) and is commonly used for exploratory data analysis, time series smoothing, and plotting.