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MONTE` CARLO METHODS

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

  2. THE BOOK by THE MAN Monte` Carlo Methods

  3. PROBLEM STATEMENT • System of equations and inequalities defines a region in m-space • Determine the volume of the region Monte` Carlo Methods

  4. HISTORY • 19th C. simple integral like E[X] using straight-forward sampling • System of PDE solved using sample paths of Markov Chains • Rayleigh 1899 • Markov 1931 • Particles through a medium solved using Poisson Process and Random Walk • Manhattan Project • Combinatorics in the ’80’s in RTP, NC Monte` Carlo Methods

  5. GROOMING • R = volumetric region • R confined to [0,1]m • l(R) = volume • Generalized area-under-the-curve problem Monte` Carlo Methods

  6. ALGORITHM • for i=1 to n • generate x in [0,1]m • is x in R? • S=S+1 • end • l(R)=S/n Monte` Carlo Methods

  7. MESH • Generate x’s as a mesh of evenly spaced points • Each point is 1/k from its nearest neighbor • n=km • Many varieties of this method, generally called Multi-Grid Monte` Carlo Methods

  8. ERROR CONTROL • Define a(R) = the surface area of R • a(R)/k = volume of a swath around the surface 1/k thick • a(R)/k=a(R)/(n1/m) bounds error Monte` Carlo Methods

  9. ...more ERROR CONTROL Monte` Carlo Methods

  10. ...more ERROR • If we require error less than d... • the required sample n grows like xm Monte` Carlo Methods

  11. PROBABLY NOT THAT BAD • Reaction: the boundary of R isn’t usually so-aligned • Probability statement on the functions? • this math exists but is only marginally helpful with applied problems Monte` Carlo Methods

  12. ALTERNATIVE • Monte` Carlo Method • for i = 1 to n • sample x from Uniform[0,1]m • is x in R? • S = S + 1 • end • lhat = S/n Monte` Carlo Methods

  13. STATISTICAL TREATMENT • S is now a RANDOM VARIABLE • P[x in R] = l • (volume of R)/(volume of unit hyper-cube) • S is a sum of Bernoulli Trials • S is Binomial(n, l) • E[S] = ln • VAR[S] = n l (1- l) Monte` Carlo Methods

  14. ESTIMATOR Monte` Carlo Methods

  15. CHEBYCHEV’S INEQUALITY • Bounds Tails of Distributions • Z~F, E[Z]=0, VAR[Z]= s2, b > 0 Monte` Carlo Methods

  16. To get an error (statistical) bounded by d... Monte` Carlo Methods

  17. SIMPLER BOUNDS • l (1- l) is bounded by ¼ • n = 1/(4e2d) • Does not depend on m! Monte` Carlo Methods

  18. SPREADSHEET • Find the volume of a sphere centered at (0.5, 0.5, 0.5) with radius 0.5 in [0,1]3 • Chebyshev bounds look very loose compared with VAR(lhat) • Use lhat for l in the sample size formula • Slow convergence Monte` Carlo Methods

  19. STRATIFIED SAMPLING • Best of Mesh and Sampling Methods • Very General application of Variance Reduction • survey sampling • experimental design • optimization via simulation Monte` Carlo Methods

  20. PARAMETERS AND DEFINITIONS • n = total number of sample points • Sample region [0,1]m is divided into r subregions A1, A2, ..., Ar • pi = P[x in Ai] • k(x) = • 1 if x in R • 0 otherwise • so E[k(x)] = l Monte` Carlo Methods

  21. DENSITY OF SAMPLES x • f(x) is the m-dim density function of x • for generality • so we keep track of expectations • in our current scheme, f(x) = 1 Monte` Carlo Methods

  22. LAMBDA AYE Monte` Carlo Methods

  23. STRATIFICATION • old method: generate x’s across the whole region • new method: generate the EXPECTED number of samples in each subregion Monte` Carlo Methods

  24. let Xj be the jth sample in the old method capitols indicate random samples! Monte` Carlo Methods

  25. VARIANCE OF THE ESTIMATOR Monte` Carlo Methods

  26. STRATIFICATION • Generate n1, n2, ..., nr samples from A1, A2, ..., Ar • on purpose • ni = npi • ni sum to n • Xi,j is jth sample from Ai Monte` Carlo Methods

  27. li is a conditional expectation Monte` Carlo Methods

  28. Monte` Carlo Methods

  29. Monte` Carlo Methods

  30. Monte` Carlo Methods

  31. HOW THAT LAST BIT WORKED Monte` Carlo Methods

  32. ...AND SO... • Stratification reduces the variance of the estimator • A random quantity (the samples pulled from Ai) is replaced by its expectation • This only works because of all of the SUMMATION and no other complicated functions Monte` Carlo Methods

  33. FOR THE SPHERE PROBLEM • 500 samples • Divide evenly in 64 cubes • 4 X 4 X 4 • 7 or 8 samples in each cube • 64 separate l’s • Add together • How did we know to start with 500? Monte` Carlo Methods

  34. Discussion of applications... Monte` Carlo Methods