- Control variate
In
Monte Carlo methods , one or more control variates may be employed to achievevariance reduction by exploiting thecorrelation between statistics.Example
Let the parameter of interest be mu, and assume we have a statistic m such that mathbb{E}left [m ight] =mu. If we are able to find another statistic t such that mathbb{E}left [t ight] = au and ho_{mt}= extrm{corr}left [m,t ight] are known values, then
:m^{star}=m-cleft(t- au ight)
is also unbiased for mu for any choice of the constant c. It can be shown that choosing
:c=frac{sigma_m}{sigma_t} ho_{mt}
minimizes the variance of m^{star}, and that with this choice,
:extrm{var}left [m^{star} ight] =left(1- ho_{mt}^2 ight) extrm{var}left [m ight] ;
hence, the term
variance reduction . The greater the value of vert ho_{tm}vert, the greater the variance reduction achieved.In the case that sigma_m, sigma_t, and/or ho_{mt} 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.References
* Averill M. Law & W. David Kelton, "Simulation Modeling and Analysis", 3rd edition, 2000, ISBN 0-07-116537-1
* S. P. Meyn. "Control Techniques for Complex Networks", Cambridge University Press, 2007. ISBN-13: 9780521884419. Online: http://decision.csl.uiuc.edu/~meyn/pages/CTCN/CTCN.html
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