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Monte Carlo Process
Charles Yoe, Ph.D.
College of Notre Dame of Maryland
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Simulation
Numerical technique used to estimate analytical solutionsto a
problem
Not an optimization technique, answers what-if questionsResults
are not analytical solutions
Analytical solutions are preferred
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Suppose . . .
You have a variable that varies between 10 and 50
All you know is theoretical maximum and minimum, any
number between is equally likely
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Monte Carlo Process
Is a process that can generate numbers within that range
According to the rules you specify
In this case a min and a max
Any number as likely as any other number
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Monte Carlo Process
Two steps
Generate a simple random number
Transform it into a useful value using a specific
probabilitydistribution
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Random Number Generation
Pseudorandom Numbers [0,1]
Seed = 6721 (any number)
Mid-square Method (John von Neumann)
(6721) 2 = 45171841; r1= 0.1718
(1718) 2 = 29515240; r2= 0.5152
(5152) 2 = 26543204; r3= 0.5432etc.
More sophisticated methods now used
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Transformation (1)
Assume Uniform Distribution, U(a,b) where a = 10 and b = 50
To obtain a value, x, we use x = a + (b - a)uIn this case, x =
10 + 40u
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Transformation (2)
Generate U~U(0,1), say u = 0.1718 then
x = 10 +(50 - 10)0.1718 = 16.9
x = 10 +(50 - 10)0.5152 = 30.6
x = 10 +(50 - 10)0.5432 = 31.7, etc.
Other distributions are similar but more
complextransformations
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Some Language
Iteration--one recalculation of the model during asimulation.
Uncertain variables are sampled once duringeach iteration according
to their probability distributions
Simulation--technique for calculating a model output valuemany
times with different input values. Purpose is to getcomplete range
of all possible scenarios
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Monte Carlo Simulation
Simulation model that uses the Monte Carlo process
Deterministic values in models replaced by distributions
Values randomly generated for each probabilistic variablein
model and calculations are completed
Process repeated desired # times
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Monte Carlo Simulation
0.00
0.08
0 10 20 30 40 0.0
0.4
5.0 8.8 12.5 16.3 20.0
X =
0.00
0.02
0 100 200 300 400
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How Many Iterations?
Means often stabilize quickly--few hundred
Estimating probabilities of outcomes takes more
Defining tails of output distribution takes many
moreiterations
If extreme events are important it make take many manymore
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Some Examples
The Monte Carlo process is used for several riskassessments
linked to the Clearinghouse
Salmonellahttp://www.fsis.usda.gov/ophs/risk/index.htm
Antimicrobial Resistant
Campylobacterhttp://www.fda.gov/cvm/fda/mappgs/ra/risk.html
http://www.fsis.usda.gov/ophs/risk/index.htmhttp://www.fda.gov/cvm/fda/mappgs/ra/risk.htmlhttp://www.fda.gov/cvm/fda/mappgs/ra/risk.htmlhttp://www.fsis.usda.gov/ophs/risk/index.htm
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The End
