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| 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.<ref>Glasserman, P. (2004). ''Monte Carlo Methods in Financial Engineering''. New York: Springer. ISBN 0-387-00451-3 (p. 185)</ref>
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| ==Underlying principle==
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| Let the unknown [[Parameter#Statistics_and_econometrics|parameter]] of interest be <math>\mu</math>, and assume we have a [[statistic]] <math>m</math> such that the [[expected value]] of ''m'' is μ: <math>\mathbb{E}\left[m\right]=\mu</math>, i.e. ''m'' is an [[bias of an estimator|unbiased estimator]] for μ. Suppose we calculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> is a known value. Then
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| :<math>m^\star = m + c\left(t-\tau\right) \, </math>
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| is also an unbiased estimator for <math>\mu</math> for any choice of the coefficient <math>c</math>.
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| The [[variance]] of the resulting estimator <math>m^{\star}</math> is
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| :<math>\textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{Cov}\left(m,t\right);</math> | |
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| It can be shown that choosing the optimal coefficient
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| :<math>c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)}; </math>
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| minimizes the variance of <math>m^{\star}</math>, and that with this choice,
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| :<math>\begin{align}
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| \textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[\textrm{Cov}\left(m,t\right)\right]^2}{\textrm{Var}\left(t\right)} \\
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| & = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right);
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| \end{align} </math>
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| where
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|
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| :<math>\rho_{m,t}=\textrm{Corr}\left(m,t\right); \, </math>
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| is the [[Pearson product-moment correlation coefficient|correlation coefficient]] of ''m'' and ''t''. The greater the value of <math>\vert\rho_{m,t}\vert</math>, the greater the [[variance reduction]] achieved.
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| In the case that <math>\textrm{Cov}\left(m,t\right)</math>, <math>\textrm{Var}\left(t\right)</math>, and/or <math>\rho_{m,t}\;</math> 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'''.
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| ==Example==
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| We would like to estimate
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| :<math>I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x</math>
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| using [[Monte Carlo integration]]. This integral is the expected value of <math>f(U)</math>, where
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| :<math>f(x) = \frac{1}{1+x}</math>
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| and ''U'' follows a [[uniform distribution (continuous)|uniform distribution]] [0, 1].
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| Using a sample of size ''n'' denote the points in the sample as <math>u_1, \cdots, u_n</math>. Then the estimate is given by
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| :<math>I \approx \frac{1}{n} \sum_i f(u_i); </math> | |
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| Now we introduce <math>g(x) = 1+x</math> as a control variate with a known expected value <math>\mathbb{E}\left[g\left(U\right)\right]=\int_0^1 (1+x) \, \mathrm{d}x=\frac{3}{2} </math> and combine the two into a new estimate
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| :<math>I \approx \frac{1}{n} \sum_i f(u_i)+c\left(\frac{1}{n}\sum_i g(u_i) -3/2\right). </math>
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| Using <math>n=1500</math> realizations and an estimated optimal coefficient <math> c^\star \approx 0.4773 </math> we obtain the following results
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| {| cellspacing="1" border="1"
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| | align="right" | '''Estimate'''
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| | align="right" | '''Variance'''
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| |-
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| | ''Classical estimate''
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| | align="right" | 0.69475
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| | align="right" | 0.01947
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| |-
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| | ''Control variates ''
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| | align="right" | 0.69295
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| | align="right" | 0.00060
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| |}
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| The variance was significantly reduced after using the control variates technique. (The exact result is <math>I=\ln 2 \approx 0.69314718</math>.)
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| ==See also==
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| :* [[Antithetic variates]]
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| :* [[Importance sampling]] | |
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| {{refimprove|date=August 2011}}
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| ==Notes==
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| <references/>
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| ==References==
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| * Ross, Sheldon M. (2002) ''Simulation'' 3rd edition ISBN 978-0-12-598053-1
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| * Averill M. Law & W. David Kelton (2000), ''Simulation Modeling and Analysis'', 3rd edition. ISBN 0-07-116537-1
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| * S. P. Meyn (2007) ''Control Techniques for Complex Networks'', Cambridge University Press. ISBN 978-0-521-88441-9. [https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html Downloadable draft] (Section 11.4: Control variates and shadow functions)
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| [[Category:Monte Carlo methods]]
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| [[Category:Randomness]]
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| [[Category:Computational statistics]]
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