1. Wicklin, Rick. Simulating data with SAS. SAS Institute, 2013.

2. Wicklin, Rick. Statistical programming with SAS/IML software
Wicklin, Rick. Statistical programming with SAS/IML software. SAS Institute, 2010.

3. SIMULATION
Real World
Traffic Flow
Arrivals
Climate
Epidemics …

4. SIMULATION Statistics Examine distributions
Examine sampling distributions
Examine theoretical results
Compare models …

5. Simulating a sample from Univariate Distributions in SAS The Data Step

6. Everything starts with random uniform(0,1).
Pseudo random number generators
In SAS, the RAND function generates realizations from multiple distributions

7. The RAND function %let obs=10; %let seed=54321; data tmp;
do i=1 to &obs;
x=rand("uniform");
output;
end;
run;
proc print data=tmp;run;

8. From SAS help
The RAND function uses the Mersenne-Twister random number generator (RNG) that was developed by Matsumoto and Nishimura (1998). The random number generator has a very long period ( – 1) and very good statistical properties. The period is a Mersenne prime.

9. Desired properties of random number generators.
Repeatability -- the same sequence should be produced with the same initial values (seeds).
Randomness -- Produce independent uniformly distributed
random variables (pass statistical tests for randomness.)
Long period --a pseudo-random number sequence uses finite
Precision arithmetic, so the sequence must repeat itself with a finite period.
This should be much longer than the amount of random numbers needed for the
simulation.
Period and randomness properties should not depend on the initial seeds.

10. Randomness -- Produce independent uniformly distributed random variables (pass statistical tests for randomness.)
data uniforms(drop=i);
call streaminit(45321);
do i=1 to ;
x=ceil(10*rand("uniform"));
output;
end;
run;
proc freq data=uniforms ;
tables x/chisq plots=none;

11. Randomness -- Produce independent uniformly distributed random variables (pass statistical tests for randomness.)
data uniforms2;
set uniforms;
x1=lag(x);
run;
proc corr data=uniforms2;

12. From SAS Help
The Mersenne-Twister RNG algorithm has an extremely long period, but this does not imply that large random samples are devoid of duplicate values. The RAND function returns at most 232 distinct values (4,294,967,296). In a random uniform sample of size 105, the chance of drawing at least one duplicate is greater than 50%. The expected number of duplicates in a random uniform sample of size M is approximately M2/233 when M is much less than 232. For example, you should expect about 115 duplicates in a random uniform sample of size M=106. These results are consequences of the famous “birthday matching problem” in probability theory.

13. How many duplicates? data rand (drop=i); call streaminit(54321);
do i=1 to ;
x=rand("Uniform");
output;
end;
run;
proc sort data=rand out=tmp nodupkeys;
by x;

14. How Long before repeat? %let seed=54321;/*i=372,972,159*/
data _null_;
call streaminit(&seed);
x0=rand("Uniform");
do until (x=x0);
i+1;
x=rand("Uniform");
end;
put "i=" i ;
run;

15. Univariate Distributions Available with the RAND function.
Bernoulli Distribution
Beta Distribution
Binomial Distribution
Cauchy Distribution
Chi-Square Distribution
Erlang Distribution
Exponential Distribution
F Distribution
Gamma Distribution
Geometric Distribution
Hypergeometric Distribution
Lognormal Distribution
Negative Binomial Distribution
Normal Distribution
Poisson Distribution
T Distribution
Tabled Distribution
Triangular Distribution
Uniform Distribution
Weibull Distribution
Univariate Distributions Available with the RAND function.

16. Some Continuous Distributions

17. Generating a random sample from a univariate distribution in the data step – The standard normal
%let seed=754313;
%let numobs=10000;
data normals;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("normal");
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=normals;
var x;
histogram x/normal;

18. Generating a random sample from a univariate distribution in the data step – chi-square (4 df)
%let seed=754313;
%let numobs=10000;
data chisquares;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("chisq",4);
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=chisquares;
var x;
histogram x/gamma;

19. Generating a random sample from a univariate distribution in the data step – chi-square (4 df)
%let seed=754313;
%let numobs=10000;
data chisquares;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("chisq",4);
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=chisquares;
var x;
histogram x/gamma(alpha=2,sigma=2);

20. Generating a random sample from a univariate distribution in the data step – chi-square (4 df)
%let seed=754313;
%let numobs=10000;
data chisquares;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("chisq",4);
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=chisquares;
var x;
histogram x/gamma(alpha=2,sigma=2) kernel(color=red);

21. Exponential %let seed=754313; %let numobs=10000; data exponentials;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("exponential");
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=exponentials;
var x;
histogram x/exponential kernel(color=red);

22. Exponential %let seed=754313; %let numobs=10000;
data exponentials(drop=i);
call streaminit(&seed);
do i=1 to &numobs;
x=5*rand("exponential");
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=exponentials;
var x;
histogram x/exponential kernel(color=red);
proc means data=exponentials n std mean; run;

23. Weibull %let seed=754313; %let numobs=10000; data exponentials;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("weibull",1,.2);
output;
end;
run;
ods select histogram goodnessoffit;
proc univariate data=exponentials;
var x;
histogram x/exponential kernel(color=red);

24. Discrete Distributions

25. Bernoulli %let seed=754313; %let numobs=10000; data bernoullis;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("bernoulli",.25);
output;
end;
run;
proc freq data=bernoullis;
tables x;

26. Binomial %let seed=754313; %let numobs=10000; data binomials;
call streaminit(&seed);
do i=1 to &numobs;
x=rand("binomial",.25,12);
output;
end;
run;
proc freq data=binomials;
tables x;
proc means data=binomials;
var x;

27. Poisson %let seed=754313; %let numobs=10000; data poissons(drop=i);
call streaminit(&seed);
do i=1 to &numobs;
x=rand("poisson",10);
output;
end;
run;
proc means data=poissons;

28. Tabled distribution %let seed=754313; %let numobs=10000;
data tabular(drop=i);
call streaminit(&seed);
do i=1 to &numobs;
x=rand("table",.2,.3,.1,.4);
output;
end;
run;
proc freq data=tabular;