Introduction to Monte Carlo
Simulation
Robert L. Harrison
University of Washington Medical Center
Seattle, Washington, USA
Supported in part by PHS grants CA42593 and CA126593
What is Monte Carlo?
• Monte Carlo
methods are a class
of algorithms that
rely on repeated
random sampling to
compute their
results. (Wikipedia)
An example: estimating an
integral
1
• Randomly generate
points in square.
100
400
1000points.
points.
•• Count
the points
• 278
69
under
curve.
curve.
717under
under
the curve.
Estimate
0.69.
0.695.
0.717.
•• Divide
the=number
under curve by total
number of points.
0
0
0
0
1
1
A little history…
• Buffon’s needle.
• Student / William Gosset.
• Manhattan project.
Buffon’s needle
• Georges-Louis Leclerc,
Comte de Buffon (1707 –
1788).
• A party game using a
lined piece of paper and
a pin.
• Estimates the value of 
 = 2*drops/hits.
http://www.shodor.org/interactivate/activities/buffon
William Gosset (1876-1937)
• Supervised the experimental brewery at
Guiness.
• Developed methods to draw
conclusions from small sample size
experiments.
• Published under the penname A.
Student.
• Student t-test.
Manhattan project
• Development of first nuclear weapons.
• Needed to extrapolate knowledge of a
single neutron’s behavior to the
behavior of a system of neutrons.
• Experimentation was to dangerous,
time-consuming, costly.
Manhattan project…
• John von Neumann (1903-1957) and
Stanislaus Ulam (1909-1984).
• Developed particle tracking Monte Carlo
simulation.
Why do people simulate?
Why do people simulate?
• Low cost way to experiment.
• Easy to repeat/modify.
• Study systems to complex to solve
analytically.
• Examine hidden quantities.
Disadvantages
•
•
•
•
Time consuming.
Doesn’t give exact solution.
Is the model correct?
Bugs, bugs, bugs.
Alternatives
• Analytic solution.
• Experiment.
Mechanics of simulation
•
•
•
•
General method.
Random number generator.
Sampling from a distribution.
Accelerating a simulation.
General Monte Carlo
Method
• Model a physical system as a (series of)
probability density function(s).
• Repeatedly sample from the pdf(s).
• Tally the statistics of interest.
• (There are many Monte Carlo
simulations that don’t fit the above
exactly.)
Random numbers
• Generally we sample from a uniform
distribution of integers over [0,n].
• Convert to a uniform distribution on the
interval (0,0), [0,1) or [0,1].
– (e.g., divide integer value by n or n+1.)
• Lists of truly random numbers.
– E.g., time between radioactive decays.
• Pseudo-random number generators.
– Calculate a randoms number!?
Pseudo-random number
generators
• Usually called random number
generators (RNG).
• (Early - bad) Middle squares.
• (Still around - bad) Linear congruential
generator.
• (Good) Mersenne twister.
Middle square
• Von Neumann, 1946.
To generate a sequence of 10 digit integers,
start with one, and square it and then take the
middle 10 digits from the result as the next
number in the sequence
2
• e.g. 5772156649 = 33317792380594909291
the next number is
• Not a recommended method!
Linear congruential
generator
• Lehmer, 1948.
In+1 = (aIn + c) mod m
• I0 = Starting value (seed)
• a, c, and m are specially chosen
constants.
Linear congruential
generator…
• Values used by
some built-in
random functions
are very poor.
• Good values for a,
c, and m on-line.
• Even with good
values, there can be
problems.
Three dimensional sampling from
a linear congruential generator
with poorly chosen values,
RANDU (Wikipedia).
Mersenne twister
• Used in MatLab, Maple.
• Does very well on most tests of
randomness.
• Ridiculously long period.
• Reasonably fast.
Mersenne twister
• Used in MatLab, Maple.
• Does very well on most tests of
randomness.
• Ridiculously long period.
• Reasonably fast.
• Quite complicated to
implement/describe.
Sampling from a distribution
• We have a uniform distribution, how do
we sample from
– An exponential distribution?
– Other continuous distributions?
– A Poisson distribution?
Exponential distribution
• Time between
radioactive decays.
• Distance a photon
travels before an
interaction.
• Mean 1/, variance
1/
Probability density function (pdf)
(Wikipedia)
Cumulative distribution function (cdf)
Sampling from the
exponential distribution
• Sample u from the
uniform distribution
on [0,1).
Sampling from the
exponential distribution
• Sample u from [0,1).
• Locate u on the yaxis of the
u
exponential
distribution cdf.
Sampling from the
exponential distribution
• Sample u from [0,1).
• Locate u on y-axis.
• Find the x value that
u
corresponds to u.
x
Sampling from the
exponential distribution
• Sample u from [0,1).
• Locate u on y-axis.
• Find x = F-1(u).
u
x
Sampling from an invertible
continuous distribution
• Sample u from the unit interval.
• Find x = F-1(u).
Other sampling methods for
continuous distributions
• Approximate F-1(u), e.g., as a piecewise
linear function.
– Table lookups are fast!
– Approximate => bias.
• Acceptance-rejection method.
– AKA rejection method, accept-reject
method, rejection sampling.
Acceptance-rejection
•
To sample from a
pdf f(x):
1. Choose pdf g(x)
and constant c such
that
c*g(x) ≥ f(x) for all x
www.mathworks.com
Acceptance-rejection
•
To sample from f(x):
1. Choose g(x) and c:
c*g(x) ≥ f(x) for all x.
2. Generate a random
number v from g(x).
3. Generate a random
number u from the
uniform distribution
on (0,1).
v
Acceptance-rejection
•
To sample from f(x):
1. Choose g(x) and c:
c*g(x) ≥ f(x) for all x.
2. Generate v from g(x).
3. Generate u from
uniform distribution.
4. If c*u ≤ f(v)/g(v)
accept v
5. else reject v and go
to 2.
Acceptance-rejection
•
To sample from f(x):
1. Choose g(x) and c:
c*g(x) ≥ f(x) for all x.
2. Generate v from g(x).
3. Generate uniform u.
4. If c*u ≤ f(v)/g(v)
accept v
5. else reject, go to 2.
•
The average number
of iterations is c.
Acceptance-rejection
• Sampling from a normal distribution.
– While the cdf is invertible, the resulting formula is
computationally inefficient.
– Generally sampling is done using the Box-Muller
or Ziggurat algorithms.
– Use a library function!
• Acceptance-rejection also be used to remove
the bias from an approximation/table lookup
sampling method.
Sampling from a normal
distribution
• Gaussian, bell curve.
• Everything is normal.
– Central limit theorem:
the sum of a large
number of independent
random variables is
approximately normally
distributed.
pdf
• Generally sampled
using the acceptancerejection method.
• Use a library function!
cdf
(Wikipedia)
Poisson distribution
• Discrete probability
distribution.
• {0,1,2…} (nonnegative integer)
result.
Probability mass function
(Wikipedia)
• Mean = variance = 
Cumulative density function
Poisson distribution
• Distribution for: how
many light bulbs will
burn out in a month?
– Large building.
– Maintenance
department immediately
replaces burned out
bulb.
– Choose  = the average
number that burn out in
a month
How many
mathematicians
does it take to
change a light bulb?
(www.clipartof.com)
In earlier work,
Haynor[1] has
shown that one
mathematician can
change a light bulb.
Poisson distribution
• Other examples:
– Number of radioactive
decays.
– Number of bugs
discovered in a
program.
– Number of calls to a
call center.
– Number of scintillation
photons.
How many
mathematicians
does it take to
change a light bulb?
None. It's left to the
reader as an
(www.clipartof.com)
exercise.
Sampling from Poisson
distributions
•
•
Use a library function!
Use the exponential
distribution.
1. Let T be the time period,
set t = 0, k = 0.
2. Sample tk from the
exponential distribution.
3. t = t + tk, k = k + 1;
4. If t < T, go to 2.
5. Return (k - 1).
•
For  large approximate
using a normal
distribution.
How many quantum
physicists does it
take to change a
light bulb?
(www.clipartof.com)
They can't. If they
know where the
socket is, they
cannot locate the
new bulb.
Keeping track of results
• Tally (score, bin, histogram) the
quantities we are interested in, e.g.:
– Number of light bulbs replaced by month;
– Time to failure for a product.
Building a simulation
• Often a single ‘event’ in a simulation will
require sampling from many different
pdfs, some many times, and be tallied in
many ways.
Building a simulation
• For example, a nuclear physics simulation
might model
–
–
–
–
the time between decays
the decay process
the paths of photons and particles
the energy deposited during tracking and any
secondary particles created thereby
– the detection electronics…
• and tally
– energy, position, and type of detected particles.
Take away
• Simulation a key tool for analytically
intractable problems.
• Currently the Mersenne Twister is a good
choice of random number generator.
• Random sampling for functions:
–
–
–
–
cdfs are analytically invertible;
use library functions if available;
acceptance-rejection method;
piecewise approximation.
• Four key distributions: uniform, exponential,
normal, Poisson.
References
J.T. Bushberg, The essential physics of medical imaging, Lippincott Williams & Wilkins, 2002.
K.P. George et al, Brain Imaging in Neurocommunicative Disorders, in Medical speech-language pathology: a
practitioner's guide, ed. A.F. Johnson, Thieme, 1998.
D.E. Heron et al, FDG-PET and PET/CT in Radiation Therapy Simulation and Management of Patients Who Have
Primary and Recurrent Breast Cancer, PET Clin, 1:39–49, 2006.
E.G.A. Aird and J. Conway, CT simulation for radiotherapy treatment planning, British J Radiology, 75:937-949, 2002.
R. McGarry and A.T. Turrisi, Lung Cancer, in Handbook of Radiation Oncology: Basic Principles and Clinical
Protocols, ed. B.G. Haffty and L.D. Wilson, Jones & Bartlett Publishers, 2008.
R. Schmitz et al, The Physics of PET/CT Scanners, in PET and PET/CT: a clinical guide, ed. E. Lin and A. Alavi,
Thieme, 2005.
W.P. Segars and B.M.W. Tsui, Study of the efficacy of respiratory gating in myocardial SPECT using the new 4-D
NCAT phantom, IEEE Transactions on Nuclear Science, 49(3):675-679, 2002.
V.W. Stieber et al, Central Nervous System Tumors, in Technical Basis of Radiation Therapy: Practical Clinical
Applications, ed. S.H. Levitt et al, Springer, 2008.
P. Suetens, Fundamentals of medical imaging, Cambridge University Press, 2002.
www.wofford.org/ecs/ScientificProgramming/MonteCarlo/index.htm
www.shodor.org/interactivate/activities/buffon
www.impactscan.org/slides/impactcourse/introduction_to_ct_in_radiotherapy