Monte` Carlo Methods 1 MONTE` CARLO METHODS INTEGRATION and SAMPLING TECHNIQUES

Monte` Carlo Methods 1

MONTE` CARLO METHODSMONTE` CARLO METHODS

INTEGRATION and SAMPLING INTEGRATION and SAMPLING TECHNIQUESTECHNIQUES


THE BOOK by THE BOOK by THE MANTHE MAN


PROBLEM STATEMENTPROBLEM STATEMENT

• System of equations and System of equations and inequalities defines a region in m-inequalities defines a region in m-spacespace

• Determine the volume of the Determine the volume of the regionregion


HISTORYHISTORY• 1919thth C. simple integral like E[X] using straight- C. simple integral like E[X] using straight-

forward samplingforward sampling• System of PDE solved using sample paths of System of PDE solved using sample paths of

Markov ChainsMarkov Chains– Rayleigh 1899Rayleigh 1899– Markov 1931Markov 1931

• Particles through a medium solved using Particles through a medium solved using Poisson Process and Random WalkPoisson Process and Random Walk– Manhattan ProjectManhattan Project

• Combinatorics in the ’80’s in RTP, NCCombinatorics in the ’80’s in RTP, NC


GROOMINGGROOMING

• R = volumetric regionR = volumetric region• R confined to [0,1]R confined to [0,1]mm

(R) = volume(R) = volume• Generalized area-under-the-curve Generalized area-under-the-curve

problemproblem

1

0

1

0

1

02121 ...),...,,(...)( mm dxdxdxxxxfR


ALGORITHMALGORITHM

• for i=1 to nfor i=1 to n– generate x in [0,1]generate x in [0,1]mm

– is x in R?is x in R?•S=S+1S=S+1

• endend(R)=S/n(R)=S/n


MESHMESH

• Generate x’s as a mesh of evenly Generate x’s as a mesh of evenly spaced pointsspaced points

• Each point is 1/k from its nearest Each point is 1/k from its nearest neighborneighbor

• n=kn=kmm

• Many varieties of this method, Many varieties of this method, generally called Multi-Gridgenerally called Multi-Grid


ERROR CONTROLERROR CONTROL

• Define a(R) = the surface Define a(R) = the surface area of Rarea of R

• a(R)/k = volume of a swath a(R)/k = volume of a swath around the surface 1/k thickaround the surface 1/k thick

• a(R)/k=a(R)/(na(R)/k=a(R)/(n1/m1/m) bounds ) bounds errorerror


...more ERROR CONTROL...more ERROR CONTROL


...more ERROR ...more ERROR • If we require error less than If we require error less than ......• the required sample n grows like xthe required sample n grows like xmm

nRanRa

m

m

)(

)(/1


PROBABLY NOT THAT BADPROBABLY NOT THAT BAD

• Reaction: the boundary of R isn’t Reaction: the boundary of R isn’t usually so-alignedusually so-aligned

• Probability statement on the Probability statement on the functions?functions?– this math exists but is only marginally this math exists but is only marginally

helpful with applied problemshelpful with applied problems


ALTERNATIVEALTERNATIVE• Monte` Carlo Method Monte` Carlo Method • for i = 1 to nfor i = 1 to n

– sample x from Uniform[0,1]sample x from Uniform[0,1]mm

– is x in R?is x in R?•S = S + 1S = S + 1

• end end hat = S/nhat = S/n


STATISTICAL TREATMENTSTATISTICAL TREATMENT• S is now a RANDOM VARIABLES is now a RANDOM VARIABLE• P[x in R] =P[x in R] =

– (volume of R)/(volume of unit hyper-cube)(volume of R)/(volume of unit hyper-cube)• S is a sum of Bernoulli TrialsS is a sum of Bernoulli Trials• S is Binomial(n, S is Binomial(n, ))• E[S] = E[S] = nn• VAR[S] = nVAR[S] = n (1- (1-))


ESTIMATORESTIMATOR

n

nSVARnSVAR

nSEnSE

)1(/][]/[

/][]/[2


CHEBYCHEV’S INEQUALITYCHEBYCHEV’S INEQUALITY

• Bounds Tails Bounds Tails of of DistributionsDistributions

• Z~F, E[Z]=0, Z~F, E[Z]=0, VAR[Z]= VAR[Z]= 22, , > 0> 0

2

2

2

2

2

1

ZP

ZP

ZP


• To get an error (statistical) To get an error (statistical) bounded by bounded by ......

2

2

)1(

/)1(

n

nnSP


SIMPLER BOUNDSSIMPLER BOUNDS (1-(1-) is bounded by ¼) is bounded by ¼• n = 1/(4n = 1/(422))• Does not depend on m!Does not depend on m!


SPREADSHEETSPREADSHEET• Find the volume of a sphere Find the volume of a sphere

centered at (0.5, 0.5, 0.5) with centered at (0.5, 0.5, 0.5) with radius 0.5 in [0,1]radius 0.5 in [0,1]33

• Chebyshev bounds look very loose Chebyshev bounds look very loose compared with VAR(compared with VAR(hat)hat)

• Use Use hat for hat for in the sample size in the sample size formulaformula

• Slow convergenceSlow convergence


STRATIFIED SAMPLINGSTRATIFIED SAMPLING

• Best of Mesh and Sampling Best of Mesh and Sampling MethodsMethods

• Very General application of Very General application of Variance ReductionVariance Reduction– survey samplingsurvey sampling– experimental designexperimental design– optimization via simulationoptimization via simulation


PARAMETERS AND DEFINITIONSPARAMETERS AND DEFINITIONS

• n = total number of sample pointsn = total number of sample points• Sample region [0,1]Sample region [0,1]mm is divided into r is divided into r

subregions Asubregions A11, A, A22, ..., A, ..., Arr

• ppii = P[x in A = P[x in Aii]]• k(x) = k(x) =

– 1 if x in R1 if x in R– 0 otherwise0 otherwise– so E[k(x)] = so E[k(x)] =


DENSITY OF SAMPLES xDENSITY OF SAMPLES x

• f(x) is the m-dim density function of f(x) is the m-dim density function of xx– for generalityfor generality– so we keep track of expectationsso we keep track of expectations– in our current scheme, f(x) = 1in our current scheme, f(x) = 1


LAMBDA AYELAMBDA AYE

]|)([

)()(

i

A ii

AxxkE

dxpxfxk

i


STRATIFICATIONSTRATIFICATION

• old method: generate x’s across old method: generate x’s across the whole regionthe whole region

• new method: generate the new method: generate the EXPECTED number of samples in EXPECTED number of samples in each subregioneach subregion

r

iii p

dxxfxkxkEm

1

]1,0[

)()()]([


• let Xlet Xjj be the jth sample in the old be the jth sample in the old methodmethod

n

Xkn

jj

1

)(̂

capitols indicate random samples!


VARIANCE OF THE ESTIMATORVARIANCE OF THE ESTIMATOR

dxXfXkn

dxXfXknVAR

n

Xk

jj

j

n

jj

n

jj

m

m

)(})({/1

)(})({)/1()ˆ(

)(ˆ

2

]1,0[

2

1 ]1,0[

2

1


STRATIFICATION STRATIFICATION

• Generate nGenerate n11, n, n22, ..., n, ..., nrr samples from samples from AA11, A, A22, ..., A, ..., Arr

– on purposeon purpose• nnii = np = npii

• nnii sum to n sum to n• XXi,ji,j is jth sample from A is jth sample from Aii


ii is a conditional expectation is a conditional expectation

21,

2

,

])([

)]([

iii

iji

XkE

XkE


i

r

i

n

jjiiSTRAT nXkp

i

/)(1 1

,


r

i Ajijiiji

r

i Aji

i

jiijii

i

i

r

i

n

jji

i

iSTRAT

i

i

i

XdXfXkn

XdpXf

Xknpnpp

XkVARnpVAR

1,,

2,

1,

,2,22

2

1 1,2

2

)()()(1

)()(

)(

))(()(


r

iii

r

i Ajijiiji

r

i AjijiijiSTRAT

pnVAR

XdXfXkn

XdXfXkn

VAR

i

i

1

2

1,,

2,

1,,

2,

)()/1()ˆ(

)()()(1

)()()(1)(


HOW THAT LAST BIT WORKEDHOW THAT LAST BIT WORKED

22,

2,

2,

2,

][])([

][))()((2])([

)]()([

iji

iijiji

iji

XkE

XkEXkE

XkE


...AND SO......AND SO...• Stratification reduces the variance Stratification reduces the variance

of the estimatorof the estimator• A random quantity (the samples A random quantity (the samples

pulled from Apulled from Aii) is replaced by its ) is replaced by its expectationexpectation

• This only works because of all of This only works because of all of the SUMMATION and no other the SUMMATION and no other complicated functionscomplicated functions


FOR THE SPHERE PROBLEMFOR THE SPHERE PROBLEM• 500 samples500 samples

– Divide evenly in 64 cubesDivide evenly in 64 cubes• 4 X 4 X 44 X 4 X 4• 7 or 8 samples in each cube7 or 8 samples in each cube

– 64 separate 64 separate ’s’s– Add togetherAdd together

• How did we know to start with 500?How did we know to start with 500?


Discussion of applications...Discussion of applications...

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Monte` Carlo Methods 1 MONTE` CARLO METHODS INTEGRATION and SAMPLING TECHNIQUES