VARIANCE REDUCTION
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VARIANCE REDUCTION VARIANCE REDUCTION
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VARIANCE REDUCTION. CALCULATIONS ON VARIANCES: SOME BASICS. Let X and Y be random variables. COV=0 if X and Y are independent. COMMON RANDOM NUMBERS. Built for distinguishing among two systems d i = y i – x i Variance reduced by COV(X, Y) Streaming induces MORE Covariance. STREAMING. - PowerPoint PPT Presentation
Transcript of VARIANCE REDUCTION
VARIANCE VARIANCE REDUCTIONREDUCTION
CALCULATIONS ON CALCULATIONS ON VARIANCES: SOME BASICSVARIANCES: SOME BASICSLet X and Y be random variablesLet X and Y be random variables
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YXCOVYVARXVARYXVAR
XVARccXVAR
YEXEXYEYXCOV
YXCOVYVARXVARYXVAR
XEXEXVAR
COV=0 if X and Y are independent.
COMMON RANDOM NUMBERSCOMMON RANDOM NUMBERS
Built for distinguishing among two systemsBuilt for distinguishing among two systems
ddii = y = yii – x – xii
Variance reduced by COV(X, Y)Variance reduced by COV(X, Y)
Streaming induces MORE CovarianceStreaming induces MORE Covariance
STREAMINGSTREAMING
Segregate the random number generation Segregate the random number generation task into streams connected to task into streams connected to phenomenaphenomena
seed1 seed2
Inter-arrivaltimes
Servicetimes
Zi=aZi-1 mod m
1. Change features of the service.2. Use exact same arrival stream forcomparing each service setting.
ANTITHETIC VARIATESANTITHETIC VARIATES
Use Uniforms U1, U2, ... to generate a sampleUse Uniforms U1, U2, ... to generate a sample
Use Uniforms 1-U1, 1-U2, ... to generate a Use Uniforms 1-U1, 1-U2, ... to generate a second samplesecond sample
Combine the samplesCombine the samples
Extreme values get canceled outExtreme values get canceled out
Depends on... Depends on... effective streamingeffective streaming straightforward straightforward FF-1-1(U) method of variate generation(U) method of variate generation
spreadsheet...spreadsheet...
CONTROL VARIATESCONTROL VARIATES
X is your output variableX is your output variable You seek the Expected Value of XYou seek the Expected Value of X
Y is a random variableY is a random variable Y is one of the variables that we are generatingY is one of the variables that we are generating We know the Expected Value of YWe know the Expected Value of Y
ExampleExample X is the total waiting time of a customerX is the total waiting time of a customer Y is the inter-arrival time before he entered serviceY is the inter-arrival time before he entered service
...more CONTROL VARIATES...more CONTROL VARIATES
Xc is a random variable with less Variance and Xc is a random variable with less Variance and the same Expected Valuethe same Expected Value
pick pick to minimize VAR(Xc) to minimize VAR(Xc)
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OPTIMAL CONTROLOPTIMAL CONTROL
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YVAR
YXCOV
YXCOVYVARXcVAR
YXCOVYVARXVARXcVAR
YEYXXc
IMPORTANT IMPORTANT CALCULATIONSCALCULATIONS
Fusing many results in statisticsFusing many results in statistics
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XYXVAR
YVARXVAR
YXCOVXVAR
YVAR
YXCOVXVAR
YXCOVYVAR
YXCOVYVAR
YVAR
YXCOVXVARXcVAR
ALSO KNOWN AS...ALSO KNOWN AS...
We are regressing X vs. YWe are regressing X vs. Y
* is the parameter that a regression * is the parameter that a regression package would calculatepackage would calculate
= SQRT[COV(X,Y)= SQRT[COV(X,Y)22/VAR(X)VAR(Y)] /VAR(X)VAR(Y)] is the correlation coefficient of X and Yis the correlation coefficient of X and Y
=1 or -1 implies =1 or -1 implies Y completely explains X and Y completely explains X and VAR(Xc)=0VAR(Xc)=0
