9. Monte Carlo Methods
Monte Carlo (MC) methods use repeated random sampling to solve
computational problems when there is no affordable deterministic algorithm.
Most often used in
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Physics
Chemistry
Finance and risk analysis
Engineering
MC is typically used in high dimensional problems where a lower dimensional
approximation is inaccurate.
Example: n year mortgages paid once a month. The risk analysis is in 12n
dimensions. For a 30 year mortgage we have a 360 dimensional problem.
Integration (quadrature rules) above 8 dimensions is impractical.
Main drawback is the addition of statistical errors to the systematic errors. A
balance between the two error types has to be made intelligently, which is not
always easy nor obvious.
Short history: MC methods have been used since 1777 when the Compte de
Buffon and Laplace each solved problems. In the 1930’s Enrico Fermi used MC
to estimate what lab experiments would show for neutron transport in fissile
materials. Metropolis and Ulam first called the method MC in the 1940’s. In
1950’s MC was expanded to use an probability distribution, not just Gaussian.
In the 1960’s and 1970’s, quantum MC and variational MC methods were
developed.
MC Simulations: The problem being solved is stochastic and the MC method
mimics the stochastic properties well. Example: neutron transport an decay in a
nuclear reactor.
MC Calculations: The problem is not stochastic, but is solved using a stochastic
MC method. Example: high dimension integration.
Quick review of probability
Event B is a set of possible outcomes that has probability Pr(B). The set of all
events is denoted by W and particular outcomes are w . Hence, BÎW .
Suppose B,C ÎW . Then BÇC represents events in both B and C. Similarly,
BÈC represents events that are in B or C.
Some axioms of probability are
1. Pr(B)Î[0,1]
2. Pr(AÈB)= Pr(A)+Pr(B) if AÇB¹Æ
3. BÍC ÞPr(B)£Pr(C)
4. Pr(W)=1
The conditional probability that a C outcome is also a B outcome is given by
Bayes formula,
Pr(B|C)= Pr(BÇC) .
Pr(C)
Frequently, we already know both Pr(B|C) and Pr(C) and use Bayes formula to
calculate Pr(BÇC).
Events B and C are independent if Pr(BÇC)= Pr(B)Pr(C)ÞPr(B)= Pr(B|C).
If W is either finite or countable, we call W discrete. In this case we can specify
all probabilities of possible outcomes as
f k = Pr(w = w k )
and an event B has probability
Pr(B)=
å
w k ÎB
Pr(w k ) =
å
w k ÎB
fk .
A discrete random variable is a number X(w ) that depends on the random
outcome w . As an example, in coin tossing, X(w ) could represent how many
heads or tails came up. For xk = X(w k ), define the expected value by
E[X]=
å X(w )Pr(w ) = å xk fk .
w ÎW
w ÎW
The probability distribution of a continuous random variable is described using a
probability density function (PDF) f (x). If X Î n and BÍ n , then
Pr(B)=
ò
xÎB
f (x)dx and E[X]=
òn xf (x)dx.
The variance in 1D is given by
s 2 = var(X)= E[(X - E[X])2 ]= ò (x- E[x])2 f (x)dx .
The notation is identical for discrete and continuous random variables.
For 2 or higher dimensions, there is a symmetric n´n variance/covariance
matrix given by
C = E[(X - E[X])(X - E[X])T ],
where the matrix elements C jk are given by
C jk = E[(X j - E[X j ])(Xk - E[Xk ])]= cov[X j ,Xk ].
The covariance matrix is positive semidefinite.
Common random variables
The standard uniform random variable U has a probability density of
ì
ï
í
ï
î
f (u)= 1 if 0 £u £1
0 otherwise.
We can create a random variable in [a,b] by Y =(b-a)U +a . The PDF for Y is
ì
ï
ï
í
ï
ïî
1 if a £ y £b
g(y)= b- a
0 otherwise.
The exponential random variable T with rate constant l >0 has a PDF
ì
ï
í
ï
î
- lt
if 0 £t
f (t)= l e
0 otherwise.
The standard normal is denoted by Z and has a PDF
f (z)= 1 e-z2 /2 .
2p
The general normal with mean m and variance s 2 is given by X = s Z + m and
has PDF
f (x)=
1
2ps 2
e-(x-m)2 /2s 2 .
We write X ~ N(m,s 2 ) in this case. A standard distribution has X ~ N(0,1).
If an n component random variable X is a multivariate normal with mean m and
covariance C, then it has a probability density
n
-(x-m )T C-1 x-m /2
1
æ
ö
f (x)= e
, where Z = çè 2p ÷ø det(C) .
Z
æ
ç
è
ö
÷
ø
Multivariate normal possess a linear transformation property: suppose L is an
m´n matrix with rank m, so L: n ® m and onto. If X Î n and Y Î m are
multivariate normal, then the covariance matrix for Y is
CY = LCX LT assuming that m = 0 .
Finally, there are two probability laws/theorems that are crucial to believing that
MC is relevant to any problem:
1. Law of large numbers
2. Central limit theorem
Law of large numbers: Suppose A= E[X] and ìí Xk |k =1,2,
î
approximation of A is
ü
ý
þ
Í X . The
n
1
Ân = n å Xk ® A as n®¥.
k=1
All estimators satisfying the law of large numbers are denoted consistent.
Central limit theorem: If s 2 = var[X], then Rn = Ân - A» N(0,s 2 /n).
Hence, recalling that A is not random, that
Var(Rn )=Var(Ân )= 1
n Var(X).
The law of large numbers makes the estimator unbiased. The central limit
theorem follows from the independence of the Xk . When n is large enough, Rn
is approximately normal, independent of the distribution of X as long as
E[X]<¥.
Random number generators
Beware simple random number generators. For example, never, ever use the
UNIX/Linux function rand. It repeats much too quickly. The function random
repeats less frequently, but is not useful for parallel computing. Matlab has a
very good random number generator that is operating system independent.
Look for digital based codes developed 20 years ago by Michael Mascagni for
good parallel random number generators. These are the state of the art even
today.
However, the best ones are analog: they measure the deviations in the electrical
line over time and normalize them to the interval [0,1]. Some CPUs do this as a
hardware instruction for sampling the deviation. These are the only true random
number generators available on computers. The Itanium2 CPU line has this built
in. Some other chips have this, too, but finding operating systems that will
sample this instruction is hard to find.
Sampling
A simple sampler produces and independent sample of X each time it is called.
The simple sampler turns standard uniforms into samples of some other random
variable.
MC codes spend almost all of their time in the sampler. Optimizing the sampler
code to reduce its execution time can have a profound effect on the overall run
time of the MC computation.
In the discussion below rng() is a good random number generator.
Bernoulli coin tossing
A Bernoulli random variable with parameter p is a random variable X with
Pr(X =1)= p and Pr(X = 0)=1- p .
If U is a standard uniform, then p = Pr(U £ p). So we can sample X using the
code fragment
if ( rng() <= p ) X = 1; else X = 0;
For a random variable with a finite number of values
Pr(X = xk )= pk with å pk =1,
we sample it using the unit interval and dividing it into subintervals of length pk.
This works well with Markov chains.
Exponential
If U is a standard uniform, then
T = -l -1 ln(U)
is an exponential with rate parameter l with units 1/Time. Since 0<U<1,
ln(U)<0 and T>0. We can sample T with the code fragment
T = -(1/lambda)*log(rng());
The PDF of the random variable T is given by f (t)= le-lt for some t>0.
Cumulative density function (CDF)
Suppose X is a one component random variable with PDF f (x). Then the CDF
F(x)= Pr(X £ x)= ò f (x')dx' . We know that 0 £ F(x)£1, "x and any u Î[0,1]
x'£x
there is an x such that F(x)=u .
The simple sampler can be coded with
1. Choose U = rng()
2. Find X such that F(X)=U
Note that step 2 can be quite difficult and time consuming. Good programming
reduces the time.
There is no elementary formula for the cumulative normal N(z). However there
is software available to compute it to approximately double precision. The
inverse cumulative normal z = N -1(u) can also be approximated.
The Box Muller method
We can generate two independent standard normal from two independent
standard uniforms using the formulas
R
Q
Z1
Z2
= -2ln(U1)
= 2pU2
= Rcos(Q)
= Rsin(Q)
We can make N independent standard normal by making N standard uniforms
and then using them in pairs to make N/2 pairs of independent standard normal.
Multivariate normals
Let X Î n be a multivariate normal random variable with mean 0 and
covariance matrix C. We sample X using the Cholesky factorization of C = LT L ,
where L is lower triangular. Let Z Î n be a vector of n independent standard
normal generated by the Box Muller method (or similarly). Then cov[Z]= I . If
X = LZ , then X is multivariate normal and has cov[X]= LILT =C .
There are many more methods that can be studied, e.g., Rejection.
Testing samplers
All scientific software should be presumed wrong until demonstrated to be
correct. Simple 1D samplers are testing using tables and histograms.
Errors
Estimating the error in a MC calculation is straightforward. Normally a result
with an error estimate is given when using a MC method.
n
Suppose X is a scalar random variable. Approximate A= E[X] by Ân = 1
n åk=1 Xk .
The central limit theorem states that Rn = Ân - A» s nZ, where s n is the standard
deviation of Ân and Z ~ N(0,1) . It can be shown that
s n = 1 , where s 2 = var[X]= E[(X - A)2 ],
s2
which we estimate using
ŝ 2 = 1
n
n
2 then take ŝ = 1 ŝ 2 .
(X
Â
)
n
å k
n
n k=1
n n
Since Z is of order 1, Rn is of order ŝ n .
We typically report the MC data as A = Ân ±ŝ n . We can plot circles with a line
for the diameter called the (standard deviation) error bar. We can think of k
é
ù
êë
úû
standard deviation error bars ê Ân - kŝ n, Ân + kŝ n ú , which are confidence levels.
The central limit theorem can be used to show that
æé
çê
çê
èë
Pr Ân - ŝ n, Ân + ŝ
ùö
÷
n úú÷
ûø
æé
çê
çê
èë
» 66% and Pr Ân -2ŝ n, Ân +2ŝ
ùö
÷
n úú÷
ûø
» 95%.
It is common in MC to report one standard deviation error bar. To interpret the
data correctly, knowledge that the real data is outside of the circle one-third of
the time has to be understood.
Integration (quadrature)
We want to approximate a d dimensional integral to an accuracy of e >0 .
Assume we can do this using N quadrature points. Consider Simpson’s rule. For
a function f (x): n ® , e µN -d/4 . MC integration can be done so that
e µN -1/2 independent of d as long as the variance of the integrand is finite.
MC integration
Let V be the domain of integration. Define I( f )= ò f (x)dx and for uniform xi ÎV
V
let
N
N
< f >= 1 åi=1 f (xi ) and < f 2 >= 1 åi=1 f 2(xi ).
N
N
Then
2
2
I( f )=< f > ± < < f > - < f > > .
N -1