The LOESS Procedure

Local Weighting

The size of the local neighborhoods that PROC LOESS uses in performing local fitting is determined by the smoothing parameter value s. When s < 1, the local neighborhood used at a point contains the s fraction of the data points closest to the point . When $s \geq 1$ , all data points are used.

Suppose q denotes the number of points in the local neighborhoods and $d_1,d_2,\dots ,d_ q$ denote the distances in increasing order of the q points closest to . The point at distance from is given a weight in the local regression that decreases as the distance from increases. PROC LOESS uses a tricube weight function to define

$w_ i= \frac{32}{5} \left(1-(\frac{d_ i}{d_ q} )^3 \right)^3$

If , then is replaced by $d_ q s^{1/p}$ in the previous formula, where p is the number of predictors in the model.

Finally, note that if a weight variable has been specified using a WEIGHT statement, then is multiplied by the corresponding value of the specified weight variable.