Slide 2 How do we generate the statistics of a function of a
random variable? Why is the method called Monte Carlo? How do we
use the uniform random number generator to generate other
distributions? Are other distributions directly available in
matlab? How do we accelerate the brute force approach? Probability
distributions and moments Web links:
http://www.riskglossary.com/link/monte_carlo_method.htm
http://physics.gac.edu/~huber/envision/instruct/montecar.htm Monte
Carlo Simulation SOURCE:
http://pics.hoobly.com/full/AA7G6VQPPN2A.jpg Slide 3 Basic Monte
Carlo Given a random variable X and a function h(X): sample X: [x
1,x 2,,x n ]; Calculate [h(x 1 ),h(x 2 ),,h(x n )]; use to
approximate statistics of h. Example: X is U[0,1]. Use MCS to find
mean of X 2 x=rand(10); y=x.^2; %generates 10x10 random matrix
mean=sum(y)/10 x =0.4017 0.5279 0.1367 0.3501 0.3072 0.3362 0.3855
0.3646 0.5033 0.2666 mean=0.3580 What is the true mean SOURCE:
http://schools.sd68.bc.ca/ed611/akerley/question.jpg SOURCE:
http://www.sz-wholesale.com/uploadFiles/041022104413s.jpg Slide 4
Obtaining distributions Histogram: y=randn(100,1); hist(y) Slide 5
Cumulative density function Cdfplot(y) [f,x]=ecdf(y); Slide 6
Histogram of average x=rand(100); y=sum(x)/100; hist(y) Slide 7
Histogram of average x=rand(1000); y=sum(x)/1000; hist(y ) What is
the law of large numbers? Slide 8 Distribution of x 2
x=rand(10000,1); x2=x.^2; hist(x2,20) Slide 9 Other distributions
Other distributions available in matlab For example, Weibull
distribution r=wblrnd(1,1,1000); hist(r,20) Slide 10 Correlated
Variables For normal distribution can use Matlabs mvnrnd R =
MVNRND(MU,SIGMA,N) returns a N-by- D matrix R of random vectors
chosen from the multivariate normal distribution with 1-by-D mean
vector MU, and D-by-D covariance matrix SIGMA. Slide 11 Example mu
= [2 3]; sigma = [1 1.5; 1.5 3]; r = mvnrnd(mu,sigma,20);
plot(r(:,1),r(:,2),'+') What is the correlation coefficient? Slide
12 Problems Monte Carlo Use Monte Carlo simulation to estimate the
mean and standard deviation of x 2, when X follows a Weibull
distribution with a=b=1. Calculate by Monte Carlo simulation and
check by integration the correlation coefficient between x and x 2,
when x is uniformly distributed in [0,1] Slide 13 Latin hypercube
sampling X = lhsnorm(mu,SIGMA,n) generates a latin hypercube sample
X of size n from the multivariate normal distribution with mean
vector mu and covariance matrix SIGMA. X is similar to a random
sample from the multivariate normal distribution, but the marginal
distribution of each column is adjusted so that its sample marginal
distribution is close to its theoretical normal distribution. Slide
14 Comparing MCS to LHS mu = [2 2]; sigma = [1 0; 0 3]; r =
lhsnorm(mu,sigma,20); sum(r)/20 ans = 1.9732 2.0259 r =
mvnrnd(mu,sigma,20); sum(r)/20 ans =2.3327 2.2184 Slide 15
Evaluating probabilities of failure Failure is defined in terms of
a limit state function that must satisfy g(r)>0, where r is a
vector of random variables. Probability of failure is estimated as
the ratio of number of negative gs, m, to total MC sample size, N
The accuracy of the estimate is poor unless N is much larger than
1/P f For small P f Slide 16 problems probability of failure
1.Derive formula for the standard deviation of estimate of P f 2.If
x is uniformly distributed in [0,1], use MCS to estimate the
probability that x2>0.95 and estimate the accuracy of your
estimate from the formula. 3. Calculate the exact value of the
answer to Problem 2 (that is without MCS). Source: Smithsonian
Institution Number: 2004-57325 Slide 17 Separable Monte Carlo
Usually limit state function is written in terms of response vs.
capacity g=C(r)-R(r)>0 Failure typically corresponds to
structures with extremely low capacity or extremely high response
but not both Can take advantage of that in separable MC Slide 18
Reading assignment Ravishankar, Bharani, Smarslok B.P., Haftka
R.T., Sankar B.V. (2010)Error Estimation and Error Reduction in
Separable Monte Carlo Method AIAA Journal,Vol 48(11), 22252230.
Source: www.library.veryhelpful.co.uk/ Page11.htm
