TPSPLNEV Call

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TPSPLNEV Call

evaluates the thin-plate smoothing spline at new data points

It can be used only after the TPSPLINE call.

CALL TPSPLNEV( pred, xpred, x, coeff);

The TPSPLNEV subroutine returns the following value:

pred: is an m ×1 vector of the predicated values of the TPSS fit evaluated at m new data points.

The inputs to the TPSPLNEV subroutine are as follows:

xpred: is an m ×k matrix of data points at which the $f_{\lambda}$ is evaluated, where m is the number of new data points and k is the number of variables in the spline model.
x: is an n ×k matrix of design points which is used as an input of TPSPLINE call.
coeff: is the coefficient vector returned from the TPSPLINE call.

See the previous section on the TPSPLINE call for details about the TSPLNEV subroutine.

As an example, consider the following data set, which consists of two independent variables. The plot of the raw data can be found in the first panel of Figure 17.11.

   proc iml;
   x={ -1.0 -1.0,    -1.0 -1.0,     -.5 -1.0,     -.5 -1.0,
         .0 -1.0,      .0 -1.0,      .5 -1.0,      .5 -1.0,
        1.0 -1.0,     1.0 -1.0,     -1.0 -.5,     -1.0 -.5,
         -.5 -.5,      -.5 -.5,       .0 -.5,       .0 -.5,
          .5 -.5,       .5 -.5,      1.0 -.5,      1.0 -.5,
         -1.0 .0,      -1.0 .0,       -.5 .0,       -.5 .0,
           .0 .0,        .0 .0,        .5 .0,        .5 .0,
          1.0 .0,       1.0 .0,      -1.0 .5,      -1.0 .5,
          -.5 .5,       -.5 .5,        .0 .5,        .0 .5,
           .5 .5,        .5 .5,       1.0 .5,       1.0 .5,
        -1.0 1.0,     -1.0 1.0,      -.5 1.0,      -.5 1.0,
          .0 1.0,       .0 1.0,       .5 1.0,       .5 1.0,
         1.0 1.0,      1.0 1.0 };

   y={15.54483570,  15.76312613,   18.67397826,   18.49722167,
      19.66086310,  19.80231311,   18.59838649,   18.51904737,
      15.86842815,  16.03913832,   10.92383867,   11.14066546,
      14.81392847,  14.82830425,   16.56449698,   16.44307297,
      14.90792284,  15.05653924,   10.91956264,   10.94227538,
      9.614920104,  9.646480938,   14.03133439,   14.03122345,
      15.77400253,  16.00412514,   13.99627680,   14.02826553,
      9.557001644,  9.584670472,   11.20625177,   11.08651907,
      14.83723493,  14.99369172,   16.55494349,   16.51294369,
      14.98448603,  14.71816070,   11.14575565,   11.17168689,
      15.82595514,  15.96022497,   18.64014953,   18.56095997,
      19.54375504,  19.80902641,   18.56884576,   18.61010439,
      15.86586951,  15.90136745 };

Now generate a sequence of $\lambda$ from -3.8 to -3.3 so that we can study the GCV function within this range.

   do j=-3.8 to -3.3 by 0.1;
     lambda=lambda||j;
   end;
   lambda=t(lambda);

Use the following IML statement to do the thin-plate smoothing spline fit and returning the fitted values on those design points.

   call tpspline(fit,coef,adiag,gcv, x, y,lambda);

The output from this call is

Output 17. 0.1: Thin-plate Smoothing Spline Fitted Values

                   SUMMARY OF TPSPLINE CALL

        Number of observations                    50
        Number of unique design points            25
        Dimension of polynomial Space              3
        Number of Parameters                      28



        GCV Estimate of Lambda              0.00000668
        Smoothing Penalty                   2558.14323
        Residual Sum of Squares                0.24611
        Trace of (I-A)                        25.40680
        Sigma^2 estimate                       0.00969
        Sum of Squares for Replication         0.24223

After this TPSPLINE call, you obtained the fitted value. The fitted surface is plotted in the second panel of Figure 17.11. Also in Figure 17.11, panel 4, you plot the GCV function values against lambda. From panel 2, you see that because of the spare design points, the fitted surface is a little bit rough. In order to study the TPSS fit $f_\lambda(x)$ more closely, you use the following IML statements to generate a more dense grid on [-1,1] ×[-1,1].

   do i1=-1 to 1 by 0.1;
      do i2=-1 to 1 by 0.1;
         x1=x1||i1;
         x2=x2||i2;
      end;
   end;
   x1=t(x1);
   x2=t(x2);
   xpred=x1||x2;

Now you can use the function TPSPLNEV to evaluate $f_\lambda(x)$ on this dense grid.

   call tpsplnev(pred,  xpred, x, coef);

The final fitted surface is plotted in Figure 17.11, panel 3.

Figure 17.11: Plots of Fitted Surface

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