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Variance Reduction Fall 2012

Variance Reduction Fall 2012. 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 Variance Reduction Methods Assignment #2 Next Class Random Numbers.

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Variance Reduction Fall 2012

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  1. Variance ReductionFall 2012 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 • Variance Reduction Methods • Assignment #2 • Next Class • Random Numbers

  3. 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

  4. 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

  5. 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

  6. 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.

  7. 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

  8. 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

  9. 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

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

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