Example | SAS/STAT 9.1, Users Guide, Volume 3 (volume 3 ONLY)

Example 37.1. Investigating the Effect of Model Specification on Prediction

In the Getting Started section of the chapter on the VARIOGRAM procedure, a particular variogram is chosen for the coal seam thickness data. The chosen variogram is Gaussian with a scale (sill) of c = 7 . 5, and a range of a = 30. This choice of the variogram is based on a visual fit ”a comparison of the plots of the regular and robust sample variograms and the Gaussian variogram for various scale (sill) and range values.

Another possible choice of model is the spherical variogram with the same scale (sill) of c = 7 . 5 but with a range of a = 60. This choice of range is again based on a visual fit; while not as good as the Gaussian model, the fit is reasonable.

It is generally held that spatial prediction is robust against model specification, while the standard error computation is not so robust.

This example investigates the effect of using these different models on the prediction and associated standard errors.

  data thick;   input east north thick @@;   datalines;   0.7  59.6  34.1   2.1  82.7  42.2   4.7  75.1  39.5   4.8  52.8  34.3   5.9  67.1  37.0   6.0  35.7  35.9   6.4  33.7  36.4   7.0  46.7  34.6   8.2  40.1  35.4   13.3   0.6  44.7  13.3  68.2  37.8  13.4  31.3  37.8   17.8   6.9  43.9  20.1  66.3  37.7  22.7  87.6  42.8   23.0  93.9  43.6  24.3  73.0  39.3  24.8  15.1  42.3   24.8  26.3  39.7  26.4  58.0  36.9  26.9  65.0  37.8   27.7  83.3  41.8  27.9  90.8  43.3  29.1  47.9  36.7   29.5  89.4  43.0  30.1   6.1  43.6  30.8  12.1  42.8   32.7  40.2  37.5  34.8   8.1  43.3  35.3  32.0  38.8   37.0  70.3  39.2  38.2  77.9  40.7  38.9  23.3  40.5   39.4  82.5  41.4  43.0   4.7  43.3  43.7   7.6  43.1   46.4  84.1  41.5  46.7  10.6  42.6  49.9  22.1  40.7   51.0  88.8  42.0  52.8  68.9  39.3  52.9  32.7  39.2   55.5  92.9  42.2  56.0   1.6  42.7  60.6  75.2  40.1   62.1  26.6  40.1  63.0  12.7  41.8  69.0  75.6  40.1   70.5  83.7  40.9  70.9  11.0  41.7  71.5  29.5  39.8   78.1  45.5  38.7  78.2   9.1  41.7  78.4  20.0  40.8   80.5  55.9  38.7  81.1  51.0  38.6  83.8   7.9  41.6   84.5  11.0  41.5  85.2  67.3  39.4  85.5  73.0  39.8   86.7  70.4  39.6  87.2  55.7  38.8  88.1   0.0  41.6   88.4  12.1  41.3  88.4  99.6  41.2  88.8  82.9  40.5   88.9   6.2  41.5  90.6   7.0  41.5  90.7  49.6  38.9   91.5  55.4  39.0  92.9  46.8  39.1  93.4  70.9  39.7   94.8  71.5  39.7  96.2  84.3  40.3  98.2  58.2  39.5   ;   /*- Run KRIGE2D on original Gaussian model ------------ */   proc krige2d data=thick outest=est1;   pred var=thick r=60;   model scale=7.5 range=30 form=gauss;   coord xc=east yc=north;   grid x=0 to 100 by 10 y=0 to 100 by 10;   run;   /*- Run KRIGE2D using Spherical Model, modified range  -*/   proc krige2d data=thick outest=est2;   pred var=thick r=60;   model scale=7.5 range=60 form=spherical;   coord xc=east yc=north;   grid x=0 to 100 by 10 y=0 to 100 by 10;   run;   data compare ;   merge est1(rename=(estimate=g_est stderr=g_std))   est2(rename=(estimate=s_est stderr=s_std));   est_dif=g_est-s_est;   std_dif=g_std-s_std;   run;   proc print data=compare;   title 'Comparison of Gaussian and Spherical Models';   title2 'Differences of Estimates and Standard Errors';   var gxc gyc npoints g_est s_est est_dif g_std s_std   std_dif;   run;

Output 37.1.1: Comparison of Gaussian and Spherical Models

  Comparison of Gaussian and Spherical Models   Differences of Estimates and Standard Errors: First 50 Observations   Obs  GXC  GYC  NPOINTS   g_est    s_est    est_dif   g_std    s_std     std_dif   1    0    0     23    43.9408  42.6700   1.27087  0.68260  2.05947   1.37687   2    0   10     28    41.6828  41.6780   0.00483  0.55909  2.03464   1.47554   3    0   20     31    38.9601  39.7285   0.76843  0.30185  1.93478   1.63293   4    0   30     32    36.1701  37.3275   1.15739  0.12705  1.54844   1.42139   5    0   40     39    33.8376  35.4320   1.59440  0.04872  1.37821   1.32949   6    0   50     38    32.8375  34.3930   1.55550  0.02983  1.22584   1.19602   7    0   60     35    33.9576  34.3155   0.35785  0.00195  0.54122   0.53927   8    0   70     30    36.9502  37.6669   0.71673  0.04006  1.20451   1.16444   9    0   80     31    41.1097  41.1016   0.00812  0.04705  0.99544   0.94839   10    0   90     28    43.6671  42.5216   1.14546  0.10236  1.57357   1.47121   11    0  100     23    41.9443  42.6511   0.70681  0.53646  2.20792   1.67146   12   10    0     25    44.6795  44.1959   0.48355  0.07833  1.09743   1.01910   13   10   10     31    42.8397  42.7496   0.09008  0.10982  1.46686   1.35703   14   10   20     34    40.3120  40.3634   0.05140  0.05315  1.54889   1.49574   15   10   30     39    37.7593  37.7648   0.00544  0.00889  0.94136   0.93247   16   10   40     44    35.6365  35.5471   0.08940  0.00595  0.75920   0.75325   17   10   50     44    35.0603  34.7042   0.35612  0.01564  1.05033   1.03469   18   10   60     41    36.0716  35.4737   0.59794  0.01321  1.18277   1.16957   19   10   70     36    38.1196  38.1040   0.01565  0.00315  0.89157   0.88842   20   10   80     33    41.2799  41.0734   0.20644  0.02446  1.22772   1.20326   21   10   90     30    43.2193  42.8904   0.32890  0.05988  1.49438   1.43450   22   10  100     26    41.0358  43.1350   2.09918  0.19050  1.93434   1.74384   23   20    0     29    44.4890  44.4359   0.05317  0.06179  1.23618   1.17439   24   20   10     35    43.3391  43.2938   0.04531  0.00526  0.95512   0.94986   25   20   20     39    41.1293  40.9885   0.14079  0.00675  1.18544   1.17870   26   20   30     43    38.6060  38.5300   0.07598  0.00898  1.08973   1.08075   27   20   40     49    36.5013  36.5275   0.02623  0.03037  1.33620   1.30583   28   20   50     49    36.1158  35.7959   0.31990  0.02535  1.31986   1.29451   29   20   60     49    36.8115  36.5397   0.27182  0.00835  1.11490   1.10656   30   20   70     39    38.4308  38.5182   0.08746  0.00257  0.89419   0.89162   31   20   80     36    41.0601  41.0449   0.01511  0.00766  1.18548   1.17781   32   20   90     33    43.1788  43.1073   0.07144  0.00613  0.94924   0.94311   33   20  100     27    42.7757  43.4689   0.69313  0.06770  1.52094   1.45324   34   30    0     35    43.3601  43.9579   0.59779  0.04662  1.32306   1.27644   35   30   10     39    43.1539  43.1448   0.00912  0.00245  0.72413   0.72167   36   30   20     44    41.2400  41.2166   0.02336  0.00528  1.10234   1.09706   37   30   30     52    38.9296  39.0178   0.08816  0.00489  1.04501   1.04012   38   30   40     57    37.2813  37.3412   0.05992  0.00804  0.89242   0.88438   39   30   50     57    36.7198  36.7558   0.03597  0.00652  0.83517   0.82865   40   30   60     55    37.2047  37.3407   0.13597  0.00682  1.00330   0.99648   41   30   70     48    38.8856  38.8919   0.00628  0.00285  1.01430   1.01145   42   30   80     43    41.0627  41.0663   0.00359  0.00260  0.97336   0.97077   43   30   90     36    43.0969  43.0465   0.05038  0.00194  0.51312   0.51118   44   30  100     29    44.5840  43.3474   1.23663  0.13593  1.57267   1.43674   45   40    0     36    42.8186  43.5157   0.69706  0.01976  1.25689   1.23713   46   40   10     40    42.8970  42.9168   0.01984  0.00301  0.95163   0.94862   47   40   20     52    41.1025  41.1824   0.07989  0.00193  0.96204   0.96012   48   40   30     60    39.3288  39.2992   0.02960  0.00451  1.05561   1.05111   49   40   40     67    38.2096  37.9680   0.24161  0.01791  1.29139   1.27349   50   40   50     68    37.3139  37.5055   0.19150  0.04039  1.51095   1.47056

The predicted values at each of the grid locations do not differ greatly for the two variogram models. However, the standard error of prediction for the spherical model is substantially larger than the Gaussian model.