1. Latin Hypercube Sampling Example Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

2. Example Z=XY X and Y follow exponential distributions x =1 y =1/2 Uncertainty Analysis for Engineers2

3. Step 1 Divide cdfs into even intervals (vertical axis) Uncertainty Analysis for Engineers3

4. Step 2 Now sample a number in each section For example, pick a random number between 0.8 and 1.0 and use it to get a random value for x xx=expinv(rx,mux); Uncertainty Analysis for Engineers4

5. Step 3 Sort the values (shuffle) nux=xx(randperm(n)); Uncertainty Analysis for Engineers5

6. An Alternative Instead of using expinv, we can generate the inverse ourselves Just take the CDF and solve for x xx=-mux*log(1-rx); Uncertainty Analysis for Engineers6

7. First Script n=10000000; mux=1; muy=2; x=exprnd(mux,n,1); y=exprnd(muy,n,1); mz=mean(x.*y); error=abs(mz-mux*muy)/mux/muy d=linspace(0,1,n+1); rx=unifrnd(d,d+1/n); ry=unifrnd(d,d+1/n); xx=expinv(rx,mux); yy=expinv(ry,muy); nux=xx(randperm(n)); nuy=yy(randperm(n)); mz=mean(nux.*nuy); error=abs(mz-mux*muy)/mux/muy Uncertainty Analysis for Engineers7

8. Alternative n=10000000; mux=1; muy=2; x=exprnd(mux,n,1); y=exprnd(muy,n,1); mz=mean(x.*y); error=abs(mz-mux*muy)/mux/muy d=linspace(0,1,n+1); rx=unifrnd(d,d+1/n); ry=unifrnd(d,d+1/n); xx=-mux*log(1-rx); yy=-muy*log(1-ry); nux=xx(randperm(n)); nuy=yy(randperm(n)); mz=mean(nux.*nuy); error=abs(mz-mux*muy)/mux/muy Uncertainty Analysis for Engineers8

9. Test Z=XY X and Y are beta with mean of 1 and 2, respectively Use simple Monte Carlo Then use LHS without sorting Then use LHS with sorting N=100,000 For each case, find mean 100 times and then take standard deviation of results Uncertainty Analysis for Engineers9

10. Case 1 n=100000; ntrials=100; mz=zeros(ntrials,1); for i=1:ntrials x=exprnd(mux,n,1); y=exprnd(muy,n,1); mz(i)=mean(x.*y); end std(mz) Uncertainty Analysis for Engineers10

11. Case 2 d=linspace(0,1,n+1); for i=1:ntrials rx=unifrnd(d,d+1/n); rx=rx(1:end-1); ry=unifrnd(d,d+1/n); ry=ry(1:end-1); x=expinv(rx,mux); y=expinv(ry,muy); mz(i)=mean(x.*y); end std(mz) Uncertainty Analysis for Engineers11

12. Case 3 d=linspace(0,1,n+1); for i=1:ntrials rx=unifrnd(d,d+1/n); rx=rx(1:end-1); ry=unifrnd(d,d+1/n); ry=ry(1:end-1); x=expinv(rx,mux); y=expinv(ry,muy); nux=x(randperm(n)); nuy=y(randperm(n)); mz(i)=mean(nux.*nuy); end std(mz) Uncertainty Analysis for Engineers12

13. Results Mean(mz)Std(mz) Case 11.99990.0098 Case 24.00000.00029 Case 31.99980.0064 Uncertainty Analysis for Engineers13