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 $\text{[math]}$ . When $\text{[math]}$ , the local neighborhood used at a point $\text{[math]}$ contains the $\text{[math]}$ fraction of the data points closest to the point $\text{[math]}$ . When $\text{[math]}$ , all data points are used.

Suppose $\text{[math]}$ denotes the number of points in the local neighborhoods and $\text{[math]}$ denote the distances in increasing order of the $\text{[math]}$ points closest to $\text{[math]}$ . The point at distance $\text{[math]}$ from $\text{[math]}$ is given a weight $\text{[math]}$ in the local regression that decreases as the distance from $\text{[math]}$ increases. PROC LOESS uses a tricube weight function to define

$\text{[math]}$

If $\text{[math]}$ , then $\text{[math]}$ is replaced by $\text{[math]}$ in the previous formula, where $\text{[math]}$ is the number of predictors in the model.

Finally, note that if a weight variable has been specified using a WEIGHT statement, then $\text{[math]}$ is multiplied by the corresponding value of the specified weight variable.