MONTE` CARLO METHODS
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Transcript of MONTE` CARLO METHODS
Monte` Carlo Methods 1
MONTE` CARLO METHODSMONTE` CARLO METHODS
INTEGRATION and SAMPLING INTEGRATION and SAMPLING TECHNIQUESTECHNIQUES
Monte` Carlo Methods 2
THE BOOK by THE BOOK by THE MANTHE MAN
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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
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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
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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
0
2121 ...),...,,(...)( mm dxdxdxxxxfR
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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
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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
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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
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...more ERROR CONTROL...more ERROR CONTROL
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...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
nRa
n
Ra
m
m
)(
)(/1
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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
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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
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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-))
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ESTIMATORESTIMATOR
n
nSVARnSVAR
nSEnSE
)1(
/][]/[
/][]/[2
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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
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• To get an error (statistical) To get an error (statistical) bounded by bounded by ......
2
2
)1(
/)1(
n
n
n
SP
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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!
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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
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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
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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)] =
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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
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LAMBDA AYELAMBDA AYE
]|)([
)()(
i
A ii
AxxkE
dxp
xfxk
i
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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[
)()()]([
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• 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!
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VARIANCE OF THE ESTIMATORVARIANCE OF THE ESTIMATOR
dxXfXkn
dxXfXknVAR
n
Xk
jj
j
n
jj
n
jj
m
m
)(})({/1
)(})({)/1()ˆ(
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2
1
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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
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ii is a conditional expectation is a conditional expectation
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XkE
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,
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HOW THAT LAST BIT WORKEDHOW THAT LAST BIT WORKED
22,
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iji
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XkE
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...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
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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?
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Discussion of applications...Discussion of applications...