The LOESS Procedure
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Overview
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Getting Started
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Syntax
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Details
Missing Values Output Data Sets Data Scaling Direct versus Interpolated Fitting kd Trees and Blending Local Weighting Iterative Reweighting Specifying the Local Polynomials Smoothing Matrix Model Degrees of Freedom Statistical Inference and Lookup Degrees of Freedom Automatic Smoothing Parameter Selection Sparse and Approximate Degrees of Freedom Computation Scoring Data Sets ODS Table Names ODS Graphics
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Examples
- References
| Local Regression and the Loess Method |
Assume that for 
to 
, the 
th measurement 
of the response 
and the corresponding measurement 
of the vector 
of 
predictors are related by
 |
where 
is the regression function and 
is a random error. The idea of local regression is that at a predictor , the regression function 
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 .
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 that 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.