Example 24.3: Using an Initial Design to Search an Optimal Design

Details of the OPTEX Procedure

Example 24.3: Using an Initial Design to Search an Optimal Design

See OPTEX4 in the SAS/QC Sample Library

This example is a continuation of Example 24.2.

You can customize the runs used to initialize the search in the OPTEX procedure. For example, you can use the INITDESIGN=SEQUENTIAL option to use an initial design chosen by the sequential search. Or you can place specific points in a data set and use the INITDESIGN=SAS-data-set option. In both cases, the search time can be significantly reduced, since the search only has to be done once. This example illustrates both of these options.

The previous example compared the results of the DETMAX and sequential search algorithms. You can use the design chosen by the sequential search as the starting point for the DETMAX algorithm. The following statements specify the DETMAX search method, replacing the default initialization method with the sequential search:

   proc optex data=a seed=33805;
      model af|egr|sa@2 af*af egr*egr sa*sa;
      generate n=50 method=detmax initdesign=sequential;
   run;

The results, which are displayed in Output 24.3.1, show an improvement over the sequential design itself (Output 24.2.2) but not over the DETMAX algorithm with the default initialization method (Output 24.2.1). Evidently the sequential design represents a local optimum that is not the global optimum, which is a common phenomenon in combinatorial optimization problems such as this one.

Output 24.3.1: Initializing with a Sequential Design

The OPTEX Procedure

Design Number	D-Efficiency	A-Efficiency	G-Efficiency	Average Prediction Standard Error
1	46.4333	25.0321	95.1371	0.4199

Prior knowledge of the design problem at hand may also provide a specific set of factor combinations to use as the initial design. For example, many D-optimal designs are composed of replications of the optimal saturated design -that is, the optimal design with exactly as many points as there are parameters to be estimated. In this case, there are 10 parameters in the model. Thus, you can find the optimal saturated design in 10 points, replicate it five times, and use the resulting design as an initial design, as follows:

   proc optex data=a seed=33805;      
       model af|egr|sa@@2       
             af*af egr*egr sa*sa;
       generate n=saturated
                method=detmax;
       output out=b;

   data c; set b; drop i;            
      do i=1 to 5; output; end;       

   proc optex data=a seed=33805;   
       model af|egr|sa@@2           
             af*af egr*egr sa*sa;
       generate n=50                 
                method=detmax      
                initdesign=c;
   run;

The results are displayed in Output 24.3.2 and Output 24.3.3. The resulting design is 99.9% D-efficient and 98.3% A-efficient relative to the best design found by the straight-forward approach (Output 24.2.1), and it takes considerably less time to produce.

Output 24.3.2: Efficiencies for the Unreplicated Saturated Design

The OPTEX Procedure

Design Number	D-Efficiency	A-Efficiency	G-Efficiency	Average Prediction Standard Error
1	41.6990	24.8480	67.6907	0.9508
2	41.4931	22.2840	70.8532	0.9841
3	40.9248	20.7672	62.2177	1.0247
4	40.7447	21.6253	52.7537	1.0503
5	39.9563	20.1557	46.4244	1.0868
6	39.9287	19.5856	45.9023	1.0841
7	39.9287	19.5856	45.9023	1.0841
8	38.9078	13.5976	37.7964	1.2559
9	38.9078	13.5976	37.7964	1.2559
10	37.6832	12.5540	45.3315	1.3036

Output 24.3.3: Initializing with a Data Set

The OPTEX Procedure

Design Number	D-Efficiency	A-Efficiency	G-Efficiency	Average Prediction Standard Error
1	46.4388	24.4951	96.0717	0.4242

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