Control variates
The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.^{[1]}^{[2]}^{[3]}
Underlying principleEdit
Let the unknown parameter of interest be , and assume we have a statistic such that the expected value of m is μ: , i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic such that is a known value. Then
is also an unbiased estimator for for any choice of the coefficient . The variance of the resulting estimator is
It can be shown that choosing the optimal coefficient
minimizes the variance of , and that with this choice,
where
is the correlation coefficient of and . The greater the value of , the greater the variance reduction achieved.
In the case that , , and/or are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.
When the expectation of the control variable, , is not known analytically, it is still possible to increase the precision in estimating (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating is significantly cheaper than computing ; 2) the magnitude of the correlation coefficient is close to unity. ^{[3]}
ExampleEdit
We would like to estimate
using Monte Carlo integration. This integral is the expected value of , where
and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as . Then the estimate is given by
Now we introduce as a control variate with a known expected value and combine the two into a new estimate
Using realizations and an estimated optimal coefficient we obtain the following results
Estimate | Variance | |
Classical estimate | 0.69475 | 0.01947 |
Control variates | 0.69295 | 0.00060 |
The variance was significantly reduced after using the control variates technique. (The exact result is .)
See alsoEdit
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NotesEdit
- ^ Lemieux, C. (2017). "Control Variates". Wiley StatsRef: Statistics Reference Online: –. doi:10.1002/9781118445112.stat07947.
- ^ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0-387-00451-3 (p. 185)
- ^ ^{a} ^{b} Botev, Z.; Ridder, A. (2017). "Variance Reduction". Wiley StatsRef: Statistics Reference Online: –. doi:10.1002/9781118445112.stat07975.
ReferencesEdit
- Ross, Sheldon M. (2002) Simulation 3rd edition ISBN 978-0-12-598053-1
- Averill M. Law & W. David Kelton (2000), Simulation Modeling and Analysis, 3rd edition. ISBN 0-07-116537-1
- S. P. Meyn (2007) Control Techniques for Complex Networks, Cambridge University Press. ISBN 978-0-521-88441-9. Downloadable draft (Section 11.4: Control variates and shadow functions)