Simulation

Monte Carlo

Stochastic simulation

• In stochastic simulation/sampling we observe

values f(X) where X is from distribution of

interest

– Observed simulation results are samples of random

variates

– Goal of a simulation experiment is to get knowledge

about the distribution of the random variable (mean,

variance, higher moments(?))

– ”Right” value is deterministic but it can not be

determined explicitly.

Stochastic simulation

• Two basic scenarios for drawing X:s

– Distribution of X can be constructed so that

independent samples can be generated (plain Monte

Carlo)

• Stochastic numerical integration

– We can only sample a dependent sequence (Markov

Chain) of X:s that has right limit distribution (MCMC,

Markov Chain Monte Carlo)

• Simulation of systems with internal dynamics

Buffons needle

• Classical example of simulation

experiment where ”exact” result is known. • Count of Buffon presented a method to define the

value for p in 1733.

• Throw a needle of length l on a plane that has

parallel lines with distance d.

• Count how often the needle crosses a line.

• Compute experimental probability for hits P=

#hits/#trials

Buffons needle – Needle hits a line if

• The distance from

the center of the

needle to closest line

is less than l sin a,

where a is the angle

between the needle

and the line

• Angle ~ Unif(0, p/2)

• Center ~ Unif

(0,d/2)

d

l a

Buffons needle – Probability of hit can

be computed using the

volume of the area

bounded by the

sinusoidal curve.

• p= 2l/(pd)

• Hence estimate is p =

2l/(pd) for experimental

value for p.

d/2

l/2

p/2

Buffons needle

– Result of a single throw is random • So is the average of N throws.

• What do we know after N throws?

– Can we define the distribution of the average P after N throws.

• Or at least expectation and variance

• P is an average of N independent random variables

• Single attempts obey binomial distribution with mean p (=2l/(pd))

• E(P)=p.

Buffons needle

– Variance of the result in one throw is p(1-p)

(result is a Bin(p) variable)

• Variance of the average of N independent trials is

p(1-p)/N

• I.e. Var(P) = p(1-p)/N

– So now we have observation of a random

variable with known variance.

– We can estimate the relationship between the

sample average and the expectation.

Confidence interval

– Assume we know a sample average of a

random variable

– Where is the true expectation p with, say 99%

probability.

• Define sc. confidence interval for which P( P-d < p<

P+d) >0.99.

• Can be defined if distribution of P is known.

• P is average of N independent Bin variables. For

large N, P is approximately normally distributed.

• d is of form c(p)N^(-1/2).

Monte Carlo -integration

– Buffon’s needle is a sampled variable

• expectation has a formula as definite integral.

– The approach can be used generally to

approximate integrals.

– Integrate f on [a,b] given that 0<f<c

• If x is Unif(a,b) and y is Unif(0,c), determine

experimentally the probability p for y< f(x).

• The sought integral is p(b-a)c.

Monte Carlo -integration

– More experiments lead to more accurate

estimate for p.

– Confidence interval (error) is proportional to

N^(-1/2).

• Not efficient for one dimensional integrals

• Length of confidence interval and asymptotic

behavior depend only on p, not the dimension of

the integral.

• Efficient way to get rough estimates for high

dimensional integrals.

Monte Carlo

– Previous Monte Carlo does not apply directly to all cases

• Unbounded interval or function

– Possible to give up ”y” variable • Compute only E(f(x))

• Cheaper but error analysis is more demanding

– Flat upper bound c can be replaced • Find a pdf g such that f(x)< cg(x)

• Draw x:s from distribution g

• Aim for success rate p ~ 1

Monte Carlo applications

– Typical Monte Carlo case is (very) high dimensional integral arising from modelling ray propagation in material.

– Each collision is modelled with multidimensional integral

• probabilities for absorption, scattering as functions of incident angle, energy, particle shapes, surface properties, adsorption in free path, etc

– For single ray the complexity grows only linearily with number of collisions.

M C Example

• Consider scattering of laser beam from a

material layer

• MSc thesis of Jukka Räbinä 2005

• Goal is to simulate different statistics of

the scattered image using Monte-Carlo

Experimental set up

Scattering

• Simulate propagation

of ray in cloud of

particles

• Basically ray tracing

• Positions and

scattering directions

of particles are

random

Goal of simulation

• Compute the

intensity, center of

mass etc of the

scatter image

captured by camera

• I.e. an integral of a

function involving the

intensity distribution.

Simulation experiment

• Send parallel rays

with normally

distributed intensity

• Collect the (few) rays

scattered to the

camera

Simulation results

• Three different

implementations of M-C

• Each with 100M

simulated rays

• Differences in execution

times and confidence

intervals

• Differences can be

explained after learning

about variance

reduction methods

MCMC

• In many cases it is not possible to sample

efficiently independent samples from equilibrium

distribution

– Like from equilibrium state of discrete event system

– Sampling from Markov chain having right equilibrium

distribution

• X_n depends on X_(n-1) only, there exists an equilibrium

distribution

• Sampling from any MC with right equilibrium distr is enough

(allows simplifying system dynamics sometimes)

MCMC

• Main challenges:

– Finding a chain with right equilibrium

• Correct updating steps

– Ensuring that equilibrium is attained

• Long enough simulation runs

– Coping with non-independence/

autocorrelation of the sampled values

• Doing ones math