Getting Started: TPSPLINE Procedure

The following example demonstrates how you can use the TPSPLINE procedure to fit a semiparametric model.

Suppose that y is a continuous variable and x1 and x2 are two explanatory variables of interest. To fit a bivariate thin-plate spline model, you can use a MODEL statement similar to that used in many regression procedures in the SAS System:

   proc tpspline;
       model y = (x1 x2);
   run;

The TPSPLINE procedure can fit semiparametric models; the parentheses in the preceding MODEL statement separate the smoothing variables from the regression variables. The following statements illustrate this syntax:

   proc tpspline;
       model y = z1 (x1 x2);
   run;

This model assumes a linear relation with z1 and an unknown functional relation with x1 and x2.

If you want to fit several responses by using the same explanatory variables, you can save computation time by using the multiple responses feature in the MODEL statement. For example, if y1 and y2 are two response variables, the following MODEL statement can be used to fit two models. Separate analyses are then performed for each response variable.

   proc tpspline;
       model y1 y2 = (x1 x2);
   run;

The following example illustrates the use of PROC TPSPLINE. The data are from Bates et al. (1987).

   data Measure;
       input x1 x2 y @@;
   datalines;
   -1.0 -1.0   15.54483570   -1.0 -1.0    15.76312613
    -.5 -1.0   18.67397826    -.5 -1.0    18.49722167
     .0 -1.0   19.66086310     .0 -1.0    19.80231311
     .5 -1.0   18.59838649     .5 -1.0    18.51904737
    1.0 -1.0   15.86842815    1.0 -1.0    16.03913832
   -1.0  -.5   10.92383867   -1.0  -.5    11.14066546
    -.5  -.5   14.81392847    -.5  -.5    14.82830425
     .0  -.5   16.56449698     .0  -.5    16.44307297
     .5  -.5   14.90792284     .5  -.5    15.05653924
    1.0  -.5   10.91956264    1.0  -.5    10.94227538
   -1.0   .0    9.61492010   -1.0   .0     9.64648093
    -.5   .0   14.03133439    -.5   .0    14.03122345
     .0   .0   15.77400253     .0   .0    16.00412514
     .5   .0   13.99627680     .5   .0    14.02826553
    1.0   .0    9.55700164    1.0   .0     9.58467047
   -1.0   .5   11.20625177   -1.0   .5    11.08651907
    -.5   .5   14.83723493    -.5   .5    14.99369172
     .0   .5   16.55494349     .0   .5    16.51294369
     .5   .5   14.98448603     .5   .5    14.71816070
    1.0   .5   11.14575565    1.0   .5    11.17168689
   -1.0  1.0   15.82595514   -1.0  1.0    15.96022497
    -.5  1.0   18.64014953    -.5  1.0    18.56095997
     .0  1.0   19.54375504     .0  1.0    19.80902641
     .5  1.0   18.56884576     .5  1.0    18.61010439
    1.0  1.0   15.86586951    1.0  1.0    15.90136745
   ;

The data set Measure contains three variables x1, x2, and y. Suppose that you want to fit a surface by using the variables x1 and x2 to model the response y. The variables x1 and x2 are spaced evenly on a square, and the response y is generated by adding a random error to a function . The raw data are plotted by using the G3D procedure. In order to visualize those replicates, half of the data are shifted a little bit by adding a small value () to x1 values, as in the following statements:

   data Measure1;
      set Measure;
   run;
    
   proc sort data=Measure1;
      by x2 x1;
   run;
    
   data Measure1;
      set Measure1;
      if mod(_N_, 2) = 0 then x1=x1+0.001;
   run;

   proc g3d data=Measure1;
      scatter x2*x1=y /size=.5
                       zmin=9 zmax=21
                       zticknum=4;
      title "Raw Data";
   run;

Figure 89.1 displays the raw data.

Figure 89.1 Plot of Data Set MEASURE

The following statements invoke the TPSPLINE procedure, by using the Measure data set as input. In the MODEL statement, the x1 and x2 variables are listed as smoothing variables. The LOGNLAMBDA=  option returns a list of GCV values with ranging from to . The OUTPUT statement creates the data set estimate to contain the predicted values and the upper and lower confidence limits.

   proc tpspline data=Measure;
      model y=(x1 x2) /lognlambda=(-4 to -2 by 0.1);
      output out=estimate pred uclm lclm;
   run;
    
   proc print data=estimate;
   run;

The results of this analysis are displayed in the following figures. Figure 89.2 shows that the data set Measure contains observations with unique design points. The final model contains no parametric regression terms and smoothing variables. The order derivative in the penalty is by default, and the dimension of polynomial space is . See the section Computational Formulas for definitions.

The GCV values are listed along with the of . The value of that minimizes the GCV function is around . The final thin-plate smoothing spline estimate is based on LOGNLAMBDA = . The residual sum of squares is , and the degrees of freedom are . The standard deviation, defined as , is . The predictions and confidence limits are displayed in Figure 89.3.