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 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 s. When s < 1, the local neighborhood used at a point x contains the s fraction of the data points closest to the point x. When , all data points are used.

Suppose q denotes the number of points in the local neighborhoods and d1,d2, ... ,dq denote the distances in increasing order of the q points closest to x. The point at distance di from x is given a weight wi in the local regression that decreases as the distance from x increases. PROC LOESS uses a tricube weight function to define

wi = [32/5] (1-([(di)/(dq)])3 )3

If s>1, then dq is replaced by dq s1/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 wi is multiplied by the corresponding value of the specified weight variable.

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