Details

Missing Values

PROC LOESS deletes any observation with missing values for any variable specified in the MODEL statement. This enables the procedure to reuse the kd tree for all the dependent variables that appear in the MODEL statement. If you have multiple dependent variables with different missing value structures for the same set of independent variables , you may want to use separate PROC LOESS steps for each dependent variable.

Output Data Sets

PROC LOESS assigns a name to each table it creates. You can use the ODS OUTPUT statement to place one or more of these tables in output data sets. See the section ODS Table Names on page 2248 for a list of the table names created by PROC LOESS. For detailed information on ODS, see Chapter 14, Using the Output Delivery System.

For example, the following statements create an output data set named MyOutStats containing the OutputStatistics table and an output data set named MySummary containing the FitSummary table.

  proc loess data=Melanoma;   model Incidences=Year;   ods output OutputStatistics = MyOutStats   FitSummary       = MySummary;   run;  

Often, a single MODEL statement describes more than one model. For example, the following statements fit eight different models (4 smoothing parameter values for each dependent variable).

  proc loess data=notReal;   model y1 y2 = x1 x2 x3/smooth =0.1 to 0.7 by 0.2;   ods output OutputStatistics = MyOutStats;   run;  

The eight OutputStatistics tables for these models are stacked in a single data set called MyOutStats. The data set contains a column named DepVarName and a column named SmoothingParameter that distinguish each model (see Figure 41.4 on page 2224 for an example). If you want the OutputStatistics table for each model to be in its own data set, you can do so by using the MATCH_ ALL option in the ODS OUTPUT statement. The following statements create eight data sets named MyOutStats, MyOutStats1, , MyOutStats7.

  proc loess data=notReal;   model y1 y2 = x1 x2 x3/smooth =0.1 to 0.7 by 0.2;   ods output OutputStatistics(match_all) = MyOutStats;   run;  

For further options available in the ODS OUTPUT statement, see Chapter 14, Using the Output Delivery System.

Only the ScaleDetails and FitSummary tables are displayed by default. The other tables are optionally displayed by using the DETAILS option in the MODEL statement and the PRINT option in the SCORE statement. Note that it is not necessary to display a table in order for that table to be used in an ODS OUTPUT statement. For example, the following statements display the OutputStatistics and kdTree tables but place the OutputStatistics and PredAtVertices tables in output data sets.

  proc loess data=Melanoma;   model Incidences=Year/details(OutputStatistics kdTree);   ods output OutputStatistics = MyOutStats   PredAtVertices   = MyVerticesOut;   run;  

Using the DETAILS option alone causes all tables to be displayed.

The MODEL statement options CLM, RESIDUAL , STD, SCALEDINDEP, and T control which optional columns are added to the OutputStatistics table. For example, to obtain an OutputStatistics output data set containing residuals and confidence limits in addition to the model variables and predicted value, you need to specify the RESIDUAL and CLM options in the MODEL statement as in the following example:

  proc loess data=Melanoma;   model Incidences=Year/residual clm;   ods output OutputStatistics = MyOutStats;   run;  

Finally, note that the ALL option in the MODEL statement includes all optional columns in the output. Also, ID columns can be added to the OutputStatistics table by using the ID statement.

Data Scaling

The loess algorithm to obtain a predicted value at a given point in the predictor space proceeds by doing a least squares fit using all data points that are close to the given point. Thus the algorithm depends critically on the metric used to define closeness. This has the consequence that if you have more than one predictor variable and these predictor variables have significantly different scales , then closeness depends almost entirely on the variable with the largest scaling. It also means that merely changing the units of one of your predictors can significantly change the loess model fit.

To circumvent this problem, it is necessary to standardize the scale of the independent variables in the loess model. The SCALE= option in the MODEL statement is provided for this purpose. PROC LOESS uses a symmetrically trimmed standard deviation as the scale estimate for each independent variable of the loess model. This is a robust scale estimator in that extreme values of a variable are discarded before estimating the data scaling. For example, to compute a 10% trimmed standard deviation of a sample, you discard the smallest and largest 5% of the data and compute the standard deviation of the remaining 90% of the data points. In this case, the trimming fraction is 0 . 1.

For example, the following statements specify that the variables Temperature and Catalyst are scaled before performing the loess fitting. In this case, because the trimming fraction is 0 . 1, the scale estimate used for each of these variables is a 10% trimmed standard deviation.

  model Yield=Temperature Catalyst / scale = SD(0.1);  

The default trimming fraction used by PROC LOESS is 0 . 1 and need not be specified by the SCALE= option. Thus the following MODEL statement is equivalent to the previous MODEL statement.

  model Yield=Temperature Catalyst / scale = SD;  

If the SCALE= option is not specified, no scaling of the independent variables is done. This is appropriate when there is only a single independent variable or when all the independent variables are a priori scaled similarly.

When the SCALE= option is specified, the scaling details for each independent variable are added to the ScaleDetails table (see Output 41.3.2 on page 2265 for an example). By default, this table contains only the minimum and maximum values of each independent variable in the model. Finally, note that when the SCALE= option is used, specifying the SCALEDINDEP option in the MODEL statement adds the scaled values of the independent variables to the OutputStatistics and PredAtVertices tables. If the SCALEDINDEP option is specified in the SCORE statement then scaled values of the independent variables are included in the ScoreResults table. By default, only the unscaled values are placed in these tables.

Direct versus Interpolated Fitting

Local regression to obtain a predicted value at a given point in the predictor space is done by doing a least squares fit using all data points in a local neighborhood of the given point. This method is computationally expensive because a local neighborhood must be determined and a least squares problem solved for each point at which a fitted value is required. A faster method is to obtain such fits at a representative sample of points in the predictor space and to obtain fitted values at all other points by interpolation.

PROC LOESS can fit models using either of these two paradigms . By default, PROC LOESS uses fitting at a sample of points and interpolation. The method fitting a local model at every data point is selected by specifying the DIRECT option in the MODEL statement.

kd Trees and Blending

PROC LOESS uses a kd tree to divide the box (also called the initial cell or bucket ) enclosing all the predictor data points into rectangular cells. The vertices of these cells are the points at which local least squares fitting is done.

Starting from the initial cell, the direction of the longest cell edge is selected as the split direction. The median of this coordinate of the data in the cell is the split value. The data in the starting cell are partitioned into two child cells. The left child consists of all data from the parent cell whose coordinate in the split direction is less than the split value. The above procedure is repeated for each child cell that has more than a prespecified number of points, called the bucket size of the kd tree.

You can specify the bucket size with the BUCKET= option in the MODEL statement. If you do not specify the BUCKET= option, the default value used is the largest integer less than or equal to ns/ 5, where n is the number of observations and s is the value of the smoothing parameter. Note that if fitting is being done for a range of smoothing parameter values, the bucket size may change for each value.

The set of vertices of all the cells of the kd tree are the points at which PROC LOESS performs its local fitting. The fitted value at an original data point (or at any other point within the original data cell) is obtained by blending the fitted values at the vertices of the kd tree cell that contains that data point.

The univariate blending methods available in PROC LOESS are linear and cubic polynomial interpolation, with linear interpolation being the default. You can request cubic interpolation by specifying the INTERP=CUBIC option in the MODEL statement. In this case, PROC LOESS uses the unique cubic polynomial whose values and first derivatives match those of the fitted local polynomials evaluated at the two endpoints of the kd tree cell edge.

In the multivariate case, such univariate interpolating polynomials are computed on each edge of the kd-tree cells, and are combined using blending functions (Gordon 1971). In the case of two regressors, if you specify INTERP=CUBIC in the MODEL statement, PROC LOESS uses Hermite cubic polynomials as blending functions. If you do not specify INTERP=CUBIC, or if you specify a model with more than two regressors, then PROC LOESS uses linear polynomials as blending functions. In these cases, the blending method reduces to tensor product interpolation from the 2 ^p vertices of each kd tree cell, where p is the number of regressors.

While the details of the kd tree and the fitted values at the vertices of the kd tree are implementation details that seldom need to be examined, PROC LOESS does provide options for their display. Each kd tree subdivision of the data used by PROC LOESS is placed in the kdTree table. The predicted values at the vertices of each kd tree are placed in the PredAtVertices table. You can request these tables using the DETAILS option in the MODEL statement.

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 contains the s fraction of the data points closest to the point x . When s ‰ 1, all data points are used.

Suppose q denotes the number of points in the local neighborhoods and d ₁ ,d ₂ , ,d _q denote the distances in increasing order of the q points closest to x . The point at distance d _i from x is given a weight w _i in the local regression that decreases as the distance from x increases . PROC LOESS uses a tricube weight function to define

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.

Iterative Reweighting

PROC LOESS can do iterative reweighting to improve the robustness of the fitinthe presence of outliers in the data. Iterative reweighting is also appropriate when statistical inference is requested and the error distribution is symmetric but not Gaussian.

The number of iterations is specified by the ITERATIONS= option in the MODEL statement. The default is ITERATIONS=1, which corresponds to no reweighting.

At iterations beyond the first iteration, the local weights w _i of the previous section are replaced by r _i w _i where r _i is a weight that decreases as the residual of the fitted value at the previous iteration at the point corresponding to d _i increases. Refer to Cleveland and Grosse (1991) and Cleveland, Grosse, and Shyu (1992) for details.

Specifying the Local Polynomials

PROC LOESS uses linear or quadratic polynomials in doing the local least squares fitting. The option DEGREE = in the MODEL statement is used to specify the degree of the local polynomials used by PROC LOESS, with DEGREE = 1 being the default. In addition, when DEGREE = 2 is specified, the MODEL statement DROPSQUARE= option can be used to exclude specific monomials during the least squares fitting.

For example, the following statements use the monomials 1, x1 , x2 , x1*x2 ,and x2*x2 for the local least squares fitting.

  proc loess data=notReal;   model y= x1 x2/ degree=2 dropsquare=(x1);   run;  

Statistical Inference

If you denote the i th measurement of the response by y _i and the corresponding measurement of predictors by x _i , then

where g is the regression function and ˆˆ _i are independent random errors with mean zero. If the errors are normally distributed with constant variance, then you can obtain confidence intervals for the predictions from PROC LOESS. You can also obtain confidence limits in the case where ˆˆ _i is heteroscedastic but a _i ˆˆ _i has constant variance and a _i are a priori weights that are specified using the WEIGHT statement of PROC LOESS. You can do inference in the case in which the error distribution is symmetric by using iterative reweighting.

Formulae for doing statistical inference under the preceding conditions can be found in Cleveland and Grosse (1991) and Cleveland, Grosse, and Shyu (1992). The main result of their analysis is that a standardized residual for a loess model follows a t distribution with degrees of freedom, where is called the lookup degrees of freedom. is a function of the smoothing matrix L , which defines the linear relationship between the fitted and observed dependent variable values of a loess model.

The determination of is computationally expensive and is not done by default. It is computed if you specify the DFMETHOD=EXACT or DFMETHOD=APPROX option in the MODEL statement. It is also computed if you specify any of the options CLM, STD, or T in the MODEL statement.

If you specify the CLM option in the MODEL statement, confidence limits are added to the OutputStatistics table. By default, 95% limits are computed, but you can change this by using the ALPHA= option in the MODEL statement.