Simulation Monte Carlo Monte Carlo simulation Outcome of a single - - PowerPoint PPT Presentation

Simulation

Monte Carlo

Monte Carlo simulation

• Outcome of a single stochastic simulation

run is always random

– A single instance of a random variable – Goal of a simulation experiment is to get knowledge about the distribution of the random variable (mean, variance) – ”Right” value is deterministic but it can not be determined explicitly.

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 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 was sampling a 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
• f ray in cloud of

particles

• Basically ray tracing
• Positions and

scattering directions

• f 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