1 / 14

Analysis of Monte Carlo Integration Fall 2013

Analysis of Monte Carlo Integration Fall 2013. By Yaohang Li, Ph.D. Review. Last Class Numerical Integration Monte Carlo Integration Crude Monte Carlo Hit-or-Miss Monte Carlo Comparison General Principle of Monte Carlo This Class Analysis of Monte Carlo Integration

bbertram
Download Presentation

Analysis of Monte Carlo Integration Fall 2013

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Analysis of Monte Carlo IntegrationFall 2013 By Yaohang Li, Ph.D.

  2. Review • Last Class • Numerical Integration • Monte Carlo Integration • Crude Monte Carlo • Hit-or-Miss Monte Carlo • Comparison • General Principle of Monte Carlo • This Class • Analysis of Monte Carlo Integration • Variance Reduction Methods • Assignment #2 • Next Class • Random Numbers

  3. Curse of Dimensionality • Curse of Dimensionality • If one needs n points to achieve certain accuracy for an 1-D integral, to achieve the same accuracy one needs ns points for s-dimensional integral • Errors in high-dimensional functions • Rectangle Rule • Trapezoidal Rule • Simpson’s Rule • Convergence Rate • O(n-/s)

  4. High-Dimensional Monte Carlo Integration • Consider the following integral • Definition of expectation of a function on random variable  that is uniformly distributed • Standard error • Independent of Dimension • /n1/2

  5. Convergence Rate • Monte Carlo Integration • Convergence Rate • O(cN-1/2) • Analysis • Very slow convergence • To get 1 digit of accuracy usually requires 100 times more computations

  6. Confidence Interval • Estimator for the standard error • Confidence Interval • 66% • 95% • 99%

  7. Variance Reduction Methods • Variance Reduction Techniques • Employs an alternative estimator • Unbiased • More deterministic • Yields a smaller variance • Methods • Stratified Sampling • Importance Sampling • Control Variates • Antithetic Variates

  8. Stratified Sampling • Idea • Break the range of integration into several pieces • Apply crude Monte Carlo sampling to each piece separately • Analysis of Stratified Sampling • Estimator • Variance • Conclusion • If the stratification is well carried out, the variance of stratified sampling will be smaller than crude Monte Carlo

  9. Importance Sampling • Idea • Concentrate the distribution of the sample points in the parts of the interval that are of most importance instead of spreading them out evenly • Importance Sampling • where g and G satisfying • G(x) is a distribution function

  10. Importance Sampling • Variance • How to select a good sampling function? • How about g=cf? • g must be simple enough for us to know its integral theoretically.

  11. Control Variates • Control Variates • (x) is the control variate with known integral • Estimator • t-t’+’ is the unbiased estimator • ’ is the first integral • Variance • var(t-t’+’)=var(t)+var(t’)-2cov(t,t’) • if 2cov(t,t’)<var(t’), then the variance is smaller than crude Monte Carlo • t and t’ should have strong positive correlation

  12. Antithetic Variates • Main idea • Select a second estimate that has a strong negative correlation with the original estimator • t’’ has the same expectation of t • Estimator • [t+t’’]/2 is an unbiased estimator of  • var([t+t’’]/2)=var(t)/4+var(t’’)/4+cov(t,t’’)/2 • Commonly used antithetic variate • (t+t’’)/2=f()/2+ f(1-)/2 • If f is a monotone function, f() and f(1-) are negatively correlated

  13. Summary • Analysis of Monte Carlo Integration • Curse of Dimensionality • Error Analysis of Monte Carlo Integration • Variance Reduction Methods • Stratified Sampling • Importance Sampling • Control Variates • Antithetic Variates

  14. What I want you to do? • Review Slides • Work on your Assignment 1 if you haven’t finished • Work on your Assignment 2

More Related