Example 46.2 Data Quality and Prediction with Missing Values

Kriging methods depend primarily on your data. The quantity and quality of your observations are important factors in minimizing prediction errors and increasing accuracy in your prediction analysis.

A typical aspect of data quality is measurement accuracy. In principle, the accuracy level of your data is not a parameter in kriging prediction; kriging assumes by definition that your data are perfectly accurate (hard) measurements. Whether you accept this assumption depends on your application. For example, an instrumentation error of 1% in the data values might be regarded as considerable in one case, whereas the same level of uncertainty might be trivial within a different framework. Your experience and judgment are crucial when you consider whether observations in a data set might be too noisy for kriging predictions to be useful.

A second aspect of data quality involves the spatial arrangement of your observations. You need to have a sufficient number of observations in order to perform spatial prediction. Also, a key element in minimizing prediction errors is an adequate sampling density. Interpretation of the expressions "sufficient number" and "adequate sampling" is again case-specific. In any event, you want enough measurements so that you can deduce the underlying spatial correlation in the working domain; see also the discussion in the section Choosing the Size of Classes in the VARIOGRAM procedure.

This example focuses on the effects of different sampling densities on the prediction analysis. The demonstration is a slight variation of the example in the section Getting Started: KRIGE2D Procedure. Specifically, you use the same correlation structure and prediction grid. However, the thick data set, is modified as follows: three values in the central area of the grid are assumed missing, namely the observation values at locations , , and . These locations have been selected so that an extended area without observations is created in the domain. The following DATA step is the input for the modified thick data set:

   data thick;
      input East North Thick @@;
      label Thick='Coal Seam Thickness';
      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     .  52.9  32.7     .
      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
      55.8  50.5     .  96.2  84.3  40.3  98.2  58.2  39.5
      ;

   title 'Coal Seam Thickness Kriging';
   ods graphics on;

Note that here you assume prior knowledge of the correlation structure model, because its parameters are based on the complete thick data set. A covariance model extracted from the incomplete set with the missing values would be a covariance model coming from a different data set; hence, it is likely to have different parameters.

After you define the modified data set, you run PROC KRIGE2D and request the OBSERVATIONS plot with the SHOWMISSING suboption. You also request two instances of the PREDICTION plot: one that displays the prediction surface and contours, and another that plots the kriging standard error surface and contours. In both of these PREDICTION plots you specify that the observations be shown as gradient markers with outlines. The following statements compute the kriged predictions and produce the requested graphics:

   proc krige2d data=thick outest=predictions
                plots(only)=(observ(showmissing)
                             pred(fill=pred line=pred obs=linegrad)
                             pred(fill=se line=se obs=linegrad));
      coordinates xc=East yc=North;
      predict var=Thick r=60;
      model scale=7.2881 range=30.6239 form=gauss;
      grid x=0 to 100 by 2.5 y=0 to 100 by 2.5;
   run;
   
   ods graphics off;

The number of observations table indicates the three missing values in Output 46.2.1.