## Local Regression and the Loess Method

Assume that for *i*=1 to *n*,
the *i*th measurement *y*_{i} of the response *y* and the corresponding
measurement *x*_{i} of the vector *x* of *p* predictors are related by

where *g* is the regression function and is a random error.
The idea of local regression is that at a predictor *x*, the regression function *g*(*x*)
can be locally approximated by the value of a function in some specified
parametric class. Such a local approximation is
obtained by fitting a regression surface to the data points within a
chosen neighborhood of the point *x*.
In the loess method, weighted least squares is used to
fit linear or quadratic functions of the predictors at the centers of
neighborhoods. The radius of each neighborhood is chosen so that the neighborhood
contains a specified percentage of the data points. The fraction of the data,
called the *smoothing parameter,* in each local neighborhood
controls the smoothness of the estimated surface. Data points in a given local
neighborhood are weighted by a smooth decreasing function of their distance
from the center of the neighborhood.

In a direct implementation,
such fitting is done at each point at which the regression surface is to be estimated.
A much faster computational procedure is to perform such local fitting at a selected
sample of points in predictor space and then to blend these local
polynomials to obtain a regression surface.

You can use the LOESS procedure to perform statistical inference provided the error
distribution satisfies some basic assumptions. In particular, such analysis is
appropriate when the are i.i.d. normal random variables
with mean 0. By using the iterative reweighting, the LOESS procedure can also
provide statistical inference when the error distribution is symmetric
but not necessarily normal. Furthermore, by doing iterative reweighting, you can use
the LOESS procedure to perform robust fitting in the presence of outliers in the data.

While all output of the LOESS procedure can be optionally displayed, most often the
LOESS procedure is used to produce output data sets that will be viewed and
manipulated by other SAS procedures. PROC LOESS uses the Output
Delivery System (ODS) to place results in output data sets. This is a departure from
older SAS procedures that provide OUTPUT statements to create SAS data sets from
analysis results.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.