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 $x_ i$ contains the s fraction of the data points closest to the point $x_ i$. 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 $x_ i$. The point at distance $d_ i$ from $x_ i$ is given a weight $w_ i$ in the local regression that decreases as the distance from $x_ i$ increases. PROC LOESS uses a tricube weight function to define

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

If $s>1$, then $d_ q$ 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 $w_ i$ is multiplied by the corresponding value of the specified weight variable.