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Sampling Techniques MS&E348 Winter 2011/2012 Professor Gerd Infanger

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Sampling Techniques

MS&E348 Winter 2011/2012

Professor Gerd Infanger

Page 2: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Multidimensional integration

Winter 2011/2012 MS&E348/Infanger 2

Page 3: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

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Page 4: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Winter 2011/2012 MS&E348/Infanger 4

Monte Carlo methods

• Monte Carlo Methods are promising for multidimensional integration • For example, determine the average height of a board:

Page 5: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Crude Monte-Carlo

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Page 6: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

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Page 7: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

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Motivational example: importance sampling

F1 F2

F = F1 + F2

z1, assume z2 = 0

Page 8: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Winter 2011/2012 MS&E348/Infanger 8

Winter 2011/2012 MS&E348/Infanger 9

Variance reduction techniques

• Stratified sampling • Importance sampling • Control variates • Antithetic variates

Importance sampling

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Winter 2011/2012 MS&E348/Infanger 11

Page 12: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

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Page 13: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

The best importance distribution

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Page 14: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Additive and multiplicative approximation functions

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Page 15: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Multiplicative marginal cost model

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Page 16: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Variance Estimation

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Page 17: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Additive marginal cost model

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Page 18: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Sample size and variance

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Page 19: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Control variates

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Page 20: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Variance estimation

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Page 21: Sampling Techniques · Winter 2011/2012 MS&E348/Infanger 9 . Variance reduction techniques • Stratified sampling • Importance sampling • Control variates • Antithetic variates

Approximation functions

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