# Control variates

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{{Short description|Technique for increasing the precision of estimates in Monte Carlo experiments}}
The '''control variates''' method is a [variance reduction](/source/variance_reduction) technique used in [Monte Carlo methods](/source/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 name="lemieux17">{{cite book|last1= Lemieux |first1=C. |title=Wiley StatsRef: Statistics Reference Online |chapter=Control Variates |date=2017|pages=1–8|doi= 10.1002/9781118445112.stat07947 |isbn=9781118445112 }}</ref>
<ref>Glasserman, P. (2004). ''Monte Carlo Methods in Financial Engineering''. New York: Springer. {{ISBN|0-387-00451-3}} (p. 185)</ref><ref name="varred17">{{cite book|last1=Botev|first1=Z.|last2=Ridder|first2=A. |title=Wiley StatsRef: Statistics Reference Online |chapter=Variance Reduction |date=2017|pages=1–6|doi=10.1002/9781118445112.stat07975|isbn=9781118445112 |hdl=1959.4/unsworks_50616|hdl-access=free}}</ref>

==Underlying principle==
Let the unknown [parameter](/source/Parameter) of interest be <math>\mu</math>, and assume we have a [statistic](/source/statistic) <math>m</math> such that the [expected value](/source/expected_value) of ''m'' is &mu;: <math>\mathbb{E}\left[m\right]=\mu</math>, i.e. ''m'' is an [unbiased estimator](/source/bias_of_an_estimator) for &mu;. Suppose we calculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> is a known value. Then

:<math>m^\star = m + c\left(t-\tau\right) \, </math>

is also an unbiased estimator for <math>\mu</math> for any choice of the coefficient <math>c</math>. 
The [variance](/source/variance) of the resulting estimator <math>m^{\star}</math> is

:<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>

By differentiating the above expression with respect to <math>c</math>, it can be shown that choosing the optimal coefficient

:<math>c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)} </math>

minimizes the variance of <math>m^{\star}</math>. (Note that this coefficient is the same as the coefficient obtained from a [linear regression](/source/linear_regression).) With this choice,

:<math>\begin{align}
\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)} \\
& = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right)
\end{align} </math>

where
        
:<math>\rho_{m,t}=\textrm{Corr}\left(m,t\right) \, </math>

is the [correlation coefficient](/source/Pearson_product-moment_correlation_coefficient) of <math>m</math> and <math>t</math>. The greater the value of <math>\vert\rho_{m,t}\vert</math>, the greater the [variance reduction](/source/variance_reduction) achieved.

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](/source/least_squares) system; therefore this technique is also known as '''regression sampling'''.

When the expectation of the control variable, <math>\mathbb{E}\left[t\right]=\tau</math>, is not known analytically, it is still possible to increase the precision in estimating <math>\mu</math> (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating <math>t</math> is significantly cheaper than computing <math>m</math>; 2) the magnitude of the correlation coefficient <math>|\rho_{m,t}| </math> is close to unity. <ref name="varred17"/>

==Example==

We would like to estimate
:<math>I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x</math>
using [Monte Carlo integration](/source/Monte_Carlo_integration). This integral is the expected value of  <math>f(U)</math>,  where
:<math>f(U) = \frac{1}{1+U}</math>
and ''U'' follows a [uniform distribution](/source/uniform_distribution_(continuous))&nbsp;[0,&nbsp;1].
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
:<math>I \approx \frac{1}{n} \sum_i f(u_i). </math>

Now we introduce <math>g(U) = 1+U</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=\tfrac{3}{2} </math> and combine the two into a new estimate
:<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>

Using <math>n=1500</math> realizations and an estimated optimal coefficient <math> c^\star \approx 0.4773 </math> we obtain the following results

{| class="wikitable"
|
| align="right" | '''Estimate'''
| align="right" | '''Variance'''
|-
| ''Classical estimate''
| align="right" | 0.69475
| align="right" | 0.01947
|-
| ''Control variates ''
| align="right" | 0.69295
| align="right" | 0.00060
|}

The variance was significantly reduced after using the control variates technique. (The exact result is   <math>I=\ln 2 \approx 0.69314718</math>.)

==See also==
* [Antithetic variates](/source/Antithetic_variates)
* [Importance sampling](/source/Importance_sampling)

{{refimprove|date=August 2011}}

==Notes==
<references/>

==References==
* 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}}.  [https://web.archive.org/web/20100619011046/https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html  Downloadable draft] (Section 11.4: Control variates and shadow functions)

Category:Monte Carlo methods
Category:Statistical randomness
Category:Computational statistics
Category:Variance reduction

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