Lowess

LOWESS, short for locally weighted scatterplot smoothing, is a nonparametric regression technique used to reveal smooth trends in scatterplot data. The term LOESS, locally estimated scatterplot smoothing, is often used interchangeably. Both methods aim to fit simple models locally to the data rather than assume a single global function.

The core idea is to estimate the value of y at each x by fitting a weighted regression to nearby points. For a target x0, a neighborhood comprising a fraction f of the data is chosen. We assign weights to observations based on their distance from x0, typically using a kernel such as the tricube function. A weighted least squares regression is then performed on the local subset, often with a local linear (degree 1) or quadratic (degree 2) model. The fitted value at x0 becomes the smooth estimate ŷ(x0). This process is repeated for all x to produce a smooth curve.

Robustness is commonly added by iterating the procedure with robust weights that reduce the influence of outliers. After an initial fit, residuals are scaled to form weights (e.g., bisquare weights) and the local fitting is repeated a few times. The result is a smoother that resists uneven outlier effects.

Differences and usage: LOWESS typically refers to the linear local fit, while LOESS often implies higher-order local polynomials and may include robust fitting. In practice, many software implementations provide both options. The method is popular for exploratory data analysis due to its flexibility and minimal assumptions, but it can be computationally intensive, sensitive to the smoothing parameter, and not intended for extrapolation beyond observed data. It is implemented in various stats packages, including R and Python libraries.