Previous Page

|

Next Page

The SIMNORMAL Procedure

Getting Started: SIMNORMAL Procedure

The following example illustrates the use of PROC SIMNORMAL to generate variable values conditioned on a set of related or correlated variables.

Suppose you are given a sample of size 50 from ten normally distributed, correlated random variables, . The first five variables represent input variables for a chemical manufacturing process, and the last five are output variables.

First, the data are input and the correlation structure is determined by using PROC CORR, as in the following statements. The results are shown in Figure 80.1.

   data a ;
      input in1-in5 out1-out5 ;
      datalines ;
    9.3500    10.0964     7.3177    10.3617    10.3444    9.4612  
   10.7443     9.9026     9.0144    11.7968
   
   ... more lines ...   

    8.9174     9.9623     9.5742     9.9713
   run ;

   proc corr data=a cov nocorr outp=outcov ;
     var in1-in5 out1-out5 ;
   run ;

Figure 80.1 Correlation of Chemical Process Variables

The CORR Procedure

10 Variables:	in1 in2 in3 in4 in5 out1 out2 out3 out4 out5

Covariance Matrix, DF = 49
	in1	in2	in3	in4	in5	out1	out2	out3	out4	out5
in1	1.019198331	0.128086799	0.291646382	0.327014916	0.417546732	0.097650713	0.206698403	0.516271121	0.118726106	0.261770905
in2	0.128086799	1.056460818	0.143581799	0.095937707	0.104117743	0.056612934	-0.121700731	0.266581451	0.092288067	-0.020971411
in3	0.291646382	0.143581799	1.384051249	0.058853960	0.326107730	0.093498839	0.078294087	0.481576554	0.057816322	0.259053423
in4	0.327014916	0.095937707	0.058853960	1.023128678	0.347916864	0.022915645	0.125961491	0.179627237	0.075028230	0.078147576
in5	0.417546732	0.104117743	0.326107730	0.347916864	1.606858140	0.360270318	0.297046593	0.749212945	0.220196337	0.349618466
out1	0.097650713	0.056612934	0.093498839	0.022915645	0.360270318	0.807007554	0.217285879	0.064816340	-0.053931448	0.037758721
out2	0.206698403	-0.121700731	0.078294087	0.125961491	0.297046593	0.217285879	0.929455806	0.206825664	0.138551008	0.054039499
out3	0.516271121	0.266581451	0.481576554	0.179627237	0.749212945	0.064816340	0.206825664	1.837505268	0.292963975	0.165910481
out4	0.118726106	0.092288067	0.057816322	0.075028230	0.220196337	-0.053931448	0.138551008	0.292963975	0.832831377	-0.067396486
out5	0.261770905	-0.020971411	0.259053423	0.078147576	0.349618466	0.037758721	0.054039499	0.165910481	-0.067396486	0.697717191

Simple Statistics
Variable	N	Mean	Std Dev	Sum	Minimum	Maximum
in1	50	10.18988	1.00955	509.49400	7.63500	12.58860
in2	50	10.10673	1.02784	505.33640	8.12580	13.78310
in3	50	10.14888	1.17646	507.44420	7.31770	12.40080
in4	50	10.03884	1.01150	501.94200	7.40490	11.99060
in5	50	10.22587	1.26762	511.29340	7.23350	12.93360
out1	50	9.85347	0.89834	492.67340	8.01220	12.24660
out2	50	9.96857	0.96408	498.42840	7.76420	12.09450
out3	50	10.29588	1.35555	514.79410	7.29660	13.74200
out4	50	10.15856	0.91260	507.92780	8.43090	12.45230
out5	50	10.26023	0.83529	513.01130	7.86060	11.96000

After the mean and correlation structure are determined, any subset of these variables can be simulated. Suppose you are interested in a particular function of the output variables for two sets of values of the input variables for the process. In particular, you are interested in the mean and variability of the following function over 500 runs of the process conditioned on each set of input values:

Although the distribution of these quantities could be determined theoretically, it is simpler to perform a conditional simulation by using PROC SIMNORMAL.

To do this, you first append a _TYPE_=’COND’ observation to the covariance data set produced by PROC CORR for each group of input values:

   data cond1 ;
         _TYPE_='COND' ;
         in1 = 8    ;
         in2 = 10.5 ;
         in3 = 12   ;
         in4 = 13.5 ;
         in5 = 14.4 ;
         output ;
   run ;
   
   data cond2 ;
         _TYPE_='COND' ;
         in1 = 15.4 ;
         in2 = 13.7 ;
         in3 = 11 ;
         in4 = 7.9 ;
         in5 = 5.5 ;
         output ;
   run ;

Next, each of these conditioning observations is appended to a copy of the OUTP=OUTCOV data from the CORR procedure, as in the following statements. A new variable, INPUT, is added to distinguish the sets of input values. This variable is used as a BY variable in subseqent steps.

   data outcov1 ;
      input=1 ;
      set outcov cond1 ;
   run ;
   
   data outcov2 ;
      input=2 ;
      set outcov cond2 ;
   run ;

Finally, these two data sets are concatenated:

   data outcov ;
     set outcov1 outcov2 ;
   run ;
   proc print data=outcov ;
   where (_type_ ne 'COV') ;
   run ;

Figure 80.2 shows the added observations.

The specification of PROC SIMNORMAL now follows from the problem description. The condition variables are IN1–IN5, the analysis variables are OUT1–OUT5, and 500 realizations are required. A seed value can be chosen arbitrarily, or the system clock can be used. Note that in the following statements, the simulation is done for each of the values of the BY variable INPUT:

   proc simnormal data=outcov(type=cov) 
         out = osim
         numreal = 500
         seed = 33179 
         ;
      by input ;
      var out1-out5 ;
      cond in1-in5 ;
      run;
   
   data b; 
      set osim ;
      denom = sum(of out1-out5) ;
      if abs(denom) < 1e-8 then ff = . ;
      else ff = (out1-out3)/denom ;
   run ;

The DATA step that follows the simulation computes the function ; in the following statements the UNIVARIATE procedure computes the simple statistics for this function for each set of conditioning input values. This is shown in Figure 80.3, and Figure 80.4 shows the distribution of the function values for each set of input values by using the SGPANEL procedure.

   proc univariate data=b ;
      by input ;
      var ff ;
   run ;      
   title ;
   proc sgpanel data=b ;
      panelby input ;
      REFLINE 0 / axis= x ;
      density ff ;
   run ;

Figure 80.4 Frequency Plot for ff for Each Set of Input Values

Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.