Transcript Session 14a Monte Carlo Simulation

Delivering Integrated, Sustainable,
Water Resources Solutions
Monte Carlo Simulation
Robert C. Patev
North Atlantic Division – Regional Technical
Specialist
(978) 318-8394
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Water Resources Solutions
– “Monte Carlo” is the method (code name) for
simulations relating to development of atomic bomb
during WWII
• Traditional – static not dynamic (not involve time), U(0,1)
• Non-Traditional – multi-integral problems, dynamic (time)
– Applied to wide variety of complex problems involving
random behavior
– Procedure that generates values of a random variable
based on one or more probability distributions
– Not simulation method per se – just a name!
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• Analytical solution
– For many problems analytical solution difficult or may not exist
– An approximate solution
• Monte Carlo Simulation
• Monte Carlo process
– Frequently used in risk analysis to generate sample of realizations of
the model
– Not always necessary
• Can enumerate all the end-points
• Model collapsed into single period
– Full enumeration may not be feasible
– Simulation produces no new information about contributing variables
• The analyst must provide the distributions for the contributing variables
– Monte Carlo simulation is a numerical technique for estimating a
distribution
• Resulting statistics are reflection of sample
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Simple Example:
Determine the expected value and distribution of
the sum of two die
Each face of each die has equal probability = 1/6
Multiple ways of getting the same sum
Analytical Solution:
Enumerate all possible combinations
EX.
Pr(3) = Pr(1 and 2) + Pr(2 and 1)
= 1/36+1/36 = 0.05556
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Two Die Toss Distribution
Outcome Probability
2
0.02778
3
0.05556
4
0.08333
5
0.11111
6
0.13889
7
0.16667
8
0.13889
9
0.11111
10
0.08333
11
0.05556
12
0.02778
0.180
0.160
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
2
3
4
5
6
7
8
9
10
11
12
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Requirements for Monte Carlo Simulation
(1)Model describing in quantitative terms the variable(s) of
interest and the relationship among them
(2)Estimate distributions of the contributing variables
(3)Generate random numbers
(4)Transform random numbers into useful values using a
specific probability distribution
(5)Criteria for determining sample size
(6)Perform a statistical analysis on the random sample to
determine
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Results from Monte Carlo Simulation
• Means often stabilize quickly--few hundred
• Estimating probabilities of outcomes take MANY more
• Defining tails of output distribution takes MANY MANY
more iterations
• If extreme events are important it make take MANY
MANY MANY more
– Convergence is key to a believable result from a
simulation. Documentation is critical.
• Depends on the degree of accuracy desired
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Criteria for Determining Sample Size
Number of iterations required increases with
– increases in variance and skew
– reductions in probability
– the number of variables simulated
• Many rules of thumb in literature…..
• Really need to set and examine convergence criteria
• Convergence
– Sample mean
– Sample variance
– Extremes--maximum and minimum
– Percentiles
– Confidence intervals
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Running Average
500 0
450 0
400 0
350 0
300 0
250 0
200 0
0
100 0
200 0
300 0
Life-cycles Simulated
400 0
500 0
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Stopping Rules
• Some commercial Monte Carlo simulation software rely
on convergence
– User specified percentage change in sample statistics
– Careful…..THIS MAY NOT BE TRUE
CONVERGENCE OF THE LIMIT STATE!
– Careful….CONVERGENCE ON TIME-DEPENDENT
PROBLEMS NEEDS TO BE EXAMINED ON EACH
TIME INCREMENT!