Download Probability Density Functions

Latin Hypercube Sampling
Example
Jake Blanchard
Spring 2010
Uncertainty Analysis for Engineers
1
Example
Z=XY
 X and Y follow exponential distributions
 x=1
 x
 y=1/2
pdf f   e

cdf
F  1 e
mean  
 x
1

Uncertainty Analysis for Engineers
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Step 1
Divide cdfs into even intervals (vertical
axis)
1
0.9
0.8
CDF -  =1

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
x
Uncertainty Analysis for Engineers
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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 Engineers
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Step 3
Sort the values (shuffle)
 nux=xx(randperm(n));

25
20
20
yy
15
15
nyy
10
5
0
0
1
2
3
4
5
xx
6
7
8
9
10
10
5
0
0
2
4
6
8
nux
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5
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 Engineers
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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 Engineers
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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 Engineers
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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 Engineers
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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 Engineers
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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 Engineers
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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)
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Results
Mean(mz)
Std(mz)
Case 1
1.9999
0.0098
Case 2
4.0000
0.00029
Case 3
1.9998
0.0064
Uncertainty Analysis for Engineers
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