Transcript Estimating probability of failure - UF-MAE

Estimating probability of failure
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Probability of failure.
The reliability index.
Monte Carlo simulation.
Uncertainty in Monte Carlo simulations.
The uncertainty in the number of rare
occurrence is the square root of the observed
number.
Limit state function
• Simulation to predict the behavior of a physical system
often try to answer what is the probability of a catastrophic
event.
• We will use the term “probability of failure.”
• Following the textbook by Choi, Grandhi, and Canfield
(CGC) we will denote the behavioral quantity that defines
system failure by S(X), where X is a random vector of
uncertainties in the simulation.
• The limit on S, or the resistance of the system is denoted as
R(X). The limit state function is g=R-S.
• Failure occurs when g=R-S<0.
• Unfortunately an alternate definition is system response is
denoted as R and resistance by C for capacity.
Probability of failure and reliability
index
• The probability of failure
is
Pf  P[ g ( X )  0]
• Another way to measure
safety is by the number of
standard deviations the
mean of g is away from
the failure boundary.
• This is the reliability index 
SOURCE:
http://www.sz-wholesale.com/uploadFiles/041022104413s.jpg
Monte Carlo Simulation
• Given a random variable X and
a limit state function g(X):
sample X: [x1,x2,…,xn];
Calculate [g(x1),g(x2),…,g(xn)]; use to approximate statistics of
g.
• Example: X is U[0,1]. Use MCS to find mean of X2
x=rand(10); y=x.^2; %generates 10x10 random matrix
sumy=sum(y)/10
sumy =0.4017 0.5279 0.1367 0.3501 0.3072 0.3362 0.3855
0.3646 0.5033 0.2666
sum(sumy)/10 ans =0.3580
• What is the true mean
SOURCE: http://schools.sd68.bc.ca/ed611/akerley/question.jpg
Evaluating probabilities of failure
• Failure is defined in terms of a limit state function
where failure occurs when g(X)<0, where X is a
vector of random variables.
Pf  m / N
• Probability of failure is estimated as the ratio of
number of negative g’s, m, to total MC sample size,
N
• The accuracy of the estimate is poor unless N is
Pf (1  Pf )
much larger than 1/Pf
  Pf  
N
• For small Pf
  Pf 
Pf
1

m
Example
• Estimate the probability that x=N(0,1)>1
x=randn(1,1000); x1=0.5*(sign(x-1)+1); pf=sum(x1)/1000.; pf =0.1550
• Repeating the process obtained: 0.136, 0.159, 0.160, 0.172, 0.154,
0.166.
• Exact value 0.1587.
  Pf  
Pf (1  Pf )
N

0.1587  0.8413
 0.0116
1, 000
With pf from MC =0.136   Pf   0.108
  Pf 
Pf

1
1

 0.086
m
136
• In general, for 10% accuracy of probability you need 100 failed
samples.
Top Hat question
• Sampling a distribution with 10,000 points, the
mean of the sample was 6, the standard
deviation of the sample was 2, and 100 points
were negative. Estimate the noise (standard
deviation) in the mean and number of negative
points over repeated 10,000 samples.
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0.02, 10
0.2,1
0.02,1
0.2,10