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How do we generate the statistics of a function of a random

Monte Carlo Simulation

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How do we generate the statistics of a function of a random variable?– Why is the method called “Monte Carlo?” S

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How do we use the uniform random number generator to generate other distributions?

A th di t ib ti di tl il bl i tl b?– Are other distributions directly available in matlab?

How do we accelerate the brute force approach?Probability distributions and moments– Probability distributions and moments

Web links: htt // i k l /li k/ t l th d hthttp://www.riskglossary.com/link/monte_carlo_method.htmhttp://physics.gac.edu/~huber/envision/instruct/montecar.htm

Basic Monte Carlo

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• Given a random variable X and a function h(X):

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sample X: [x1,x2,…,xn];Calculate [h(x1),h(x2),…,h(xn)]; use to approximate statistics of h.

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• Example: X is U[0,1]. Use MCS to find mean of X2

x=rand(10); y=x.^2; %generates 10x10 random matrix( ) y gsumy=sum(y)/10sumy =0.4017 0.5279 0.1367 0.3501 0.3072 0.3362 0.3855 0.3646 0.5033 0.2666sum(sumy)/10 ans =0 3580sum(sumy)/10 ans =0.3580

• What is the true mean

SOURCE: http://schools.sd68.bc.ca/ed611/akerley/question.jpg

Basic Monte Carlo• Given a random variable X and a function h(X): sample

X: [x1,x2,…,xn]; Calculate [h(x1),h(x2),…,h(xn)]; use to [ 1, 2, , n]; [ ( 1), ( 2), , ( n)];approximate statistics of h.

• Example: X is U[0,1]. Use MCS to find mean of X2

x=rand(10); y=x.^2; %generates 10x10 random matrixsumy=sum(y)/10sumy =0 4017 0 5279 0 1367 0 3501 0 3072sumy =0.4017 0.5279 0.1367 0.3501 0.3072

0.3362 0.3855 0.3646 0.5033 0.2666sum(sumy)/10 ans =0.3580( y)• What is the true mean SOURCE:

http://schools.sd68.bc.ca/ed611/akerley/question.jpg

Obtaining distributionsObtaining distributions

• Histogram: y=randn(100 1); hist(y)Histogram: y=randn(100,1); hist(y)25

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Cumulative density functionCumulative density function• Cdfplot(y)p (y)

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Experimental CDFExperimental CDF

• [f x]=ecdf(y); f’[f,x]=ecdf(y); fColumns 1 through 8

0 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700Columns 9 through 16Columns 9 through 16

0.0800 0.0900 0.1000 0.1100 0.1200 0.1300 0.1400 0.1500

• x‘Columns 1 through 8

-2.2529 -2.2529 -2.1763 -2.1746 -1.9539 -1.9068 -1.6354 -1.5729Columns 9 through 16

1 4127 1 2568 1 2203 1 2158 1 1957 1 1684 1 1645 1 0954-1.4127 -1.2568 -1.2203 -1.2158 -1.1957 -1.1684 -1.1645 -1.0954

Columns 89 through 960.9940 1.0437 1.0945 1.1654 1.2020 1.2750 1.5081 1.5338

Columns 97 through 1011.5908 1.7909 1.8918 2.4124 2.7335

Histogram of averageg g• x=rand(100); y=sum(x)/100; hist(y)

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0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.580

Histogram of average• x=rand(1000); y=sum(x)/1000; hist(y)

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0 46 0 47 0 48 0 49 0 5 0 51 0 52 0 530

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What is the law of large numbers?

Distribution of x2Distribution of x

x=rand(10000,1);( , );x2=x.^2;hist(x2,20)

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Other distributions• Other distributions available in matlab• For example, Weibul distributionFor example, Weibul distribution

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Correlated Variables

For normal distrib tion can se Matlab’s m nrnd• For normal distribution can use Matlab’s mvnrnd• R = MVNRND(MU,SIGMA) returns an n-by-d

matrix R of random vectors chosen from thematrix R of random vectors chosen from the multivariate normal distribution with mean vector MU, and covariance matrix SIGMA. MU is anMU, and covariance matrix SIGMA. MU is an n-by-d matrix, and MVNRND generates each row of R using the corresponding row of MU. SIGMA is a d-by-d symmetric positive semi-definite matrix

Examplepmu = [2 3];sigma = [1 1 5; 1 5 3];sigma = [1 1.5; 1.5 3];r = mvnrnd(mu,sigma,20);plot(r(:,1),r(:,2),'+') 4.5

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Latin hypercube samplingyp p g

X = lhsnorm(mu,SIGMA,n) generates a latinX 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

lti i t l di t ib ti b t thmultivariate normal distribution, but the marginal distribution of each column is adjusted so that its sample marginaladjusted so that its sample marginal distribution is close to its theoretical normal distributionnormal distribution.

Comparing MCS to LHS

mu = [2 2];sigma = [1 0; 0 3];sigma [1 0; 0 3];r = lhsnorm(mu,sigma,20);sum(r)/20

r = mvnrnd(mu,sigma,20);sum(r)/20

2 3327 2 2184( )

ans = 1.9732 2.0259

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Evaluating probabilities of failureEvaluating probabilities of failure

• Failure is defined in terms of a limit stateFailure is defined in terms of a limit state function that must satisfy g(r)>0, where r is a vector of random variables. /fP m N

• Probability of failure is estimated as the ratio of number of negative g’s, m, to total MC sample

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i h l h 1/Pis much larger than 1/Pf

• For small Pf

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Separable Monte CarloSeparable Monte Carlo

• Usually limit state function is written inUsually limit state function is written in terms of response vs. capacity g=C(r)-R(r)>0g C(r) R(r)>0

• Failure typically corresponds to structures with extremely low capacity or extremelywith extremely low capacity or extremely high response but not bothC t k d t f th t i bl• Can take advantage of that in separable MC

Reading assignmentg gSDM paper by Ben Smarslok

Source: www.library.veryhelpful.co.uk/ Page11.htm

problems1. Derive formula for the standard

deviation of estimate of Pfdeviation of estimate of Pf

2. Use MCS to calculate the probability of failure when astructure is designed with 50% safety factor

Source: Smithsonian InstitutionNumber: 2004-57325

structure is designed with 50% safety factor and both stress and failure stress have a normal distribution with 10% COV.normal distribution with 10% COV.

3. Calculate the exact value of the answer to Problem 2 (that is without MCS).