Variance reduction. Michel Bierlaire. Transport and Mobility Laboratory. Variance reduction p. 1/18

Transcription

1 Variance reduction p. 1/18 Variance reduction Michel Bierlaire Transport and Mobility Laboratory

2 Variance reduction p. 2/18 Example Use simulation to compute I = 1 0 e x dx We know the solution: e 1 = Simulation: consider draws two by two Let r 1,...,r R be independent draws from U(0,1). Let s 1,...,s R be independent draws from U(0,1). I 1 R R i=1 e r i +e s i 2 Use R = (that is, a total of draws) Mean over R draws from (e r i +e s i )/2: 1.720,variance: 0.123

3 Variance reduction p. 3/18 Example Now, use half the number of draws Idea: if X U(0,1), then (1 X) U(0,1) Let r 1,...,r R be independent draws from U(0,1). I 1 R R i=1 e r i +e 1 r i 2 Use R = Mean over R draws of (e r i +e 1 r i )/2: ,variance: Compared to: mean of (e r i +e s i )/2: 1.720,variance: 0.123

4 Variance reduction p. 4/18 Example Independent Antithetic

5 Variance reduction p. 5/18 Antithetic draws Let X 1 and X 2 i.i.d r.v. with mean θ Then ( ) X1 +X 2 Var = (Var(X 1)+Var(X 2 )+2Cov(X 1,X 2 )) If X 1 and X 2 are independent, then Cov(X 1,X 2 ) = 0. If X 1 and X 2 are negatively correlated, then Cov(X 1,X 2 ) < 0, and the variance is reduced.

6 Variance reduction p. 6/18 Back to the example Independent draws X 1 = e U, X 2 = e U Var(X 1 ) = Var(X 2 ) = E[e 2U ] E[e U ] 2 = 1 0 e 2x dx (e 1) 2 = e (e 1) 2 = Cov(X 1,X 2 ) = 0 ( ) X1 +X 2 Var 2 = 1 4 ( )) =

7 Variance reduction p. 7/18 Back to the example Antithetic draws X 1 = e U, X 2 = e 1 U Var(X 1 ) = Var(X 2 ) = Var Cov(X 1,X 2 ) = E[e U e 1 U ] E[e U ]E[e 1 U ] = e (e 1)(e 1) = ( ) X1 +X 2 = 1 ( )) =

8 Variance reduction p. 8/18 Antithetic draws: generalization Suppose that X 1 = h(u 1,...,U m ) where U 1,...U m are i.i.d. U(0,1). Define X 2 = h(1 U 1,...,1 U m ) X 2 has the same distribution as X 1 If h is monotonic in each of its coordinates, then X 1 and X 2 are negatively correlated. If h is not monotonic, there is no guarantee that the variance will be reduced.

9 Variance reduction p. 9/18 Another example Antithetic draws: X 1 = I = ( 1 0 ( x 1 2 dx 2) ( U 2) 1 2, X 2 = (1 U) 1 2 ) 2 The covariance is positive: Cov(X 1,X 2 ) = > 0. The variance will therefore be (slightly) increased!

10 Variance reduction p. 10/18 Another example Independent Antithetic

11 Variance reduction p. 11/18 Control variates We use simulation to estimate θ = E[X], where X is an output of the simulation Let Y be another output of the simulation, such that we know E[Y] = µ We consider the quantity: By construction, E[Z] = E[X] Its variance is Z = X +c(y µ). Var(Z) = Var(X +cy) = Var(X)+c 2 Var(Y)+2cCov(X,Y) Find c such that Var(Z) is minimum

12 Variance reduction p. 12/18 Control variates First derivative: Zero if Second derivative: We use Its variance 2cVar(Y)+2Cov(X,Y) c = Cov(X,Y) Var(Y) 2Var(Y) > 0 Z = X Cov(X,Y) (Y µ). Var(Y) Var(Z ) = Var(X) Cov(X,Y)2 Var(Y) Var(X)

13 Variance reduction p. 13/18 Control variates In practice... Cov(X, Y) and Var(Y) are usually not known. We can use their sample estimates: Ĉov(X,Y) = 1 n 1 R (X r X)(Y r Ȳ) r=1 and Var(Y) = 1 n 1 R (Y r Ȳ)2. r=1

14 Variance reduction p. 14/18 Control variates In practice... Alternatively, use linear regression X = ay +b+ε where ε N(0,σ 2 ). The least square estimators of a and b are â = R r=1 (X r X)(Y r Ȳ) R r=1 (Y r Ȳ)2 ˆb = X â Ȳ. Therefore c = â

15 Variance reduction p. 15/18 Control variates Moreover, ˆb+âµ = X â Ȳ +âµ = X â(ȳ µ) = X +c (Ȳ µ) = θ Therefore, the control variate estimate θ of θ is obtained by the estimated linear model, evaluated at µ.

16 Variance reduction p. 16/18 Back to the example Use simulation to compute I = X = e U 1 0 Y = U, E[Y] = 1/2, Var(Y) = 1/12 Cov(X,Y) = (3 e)/ Therefore, the best c is e x dx c = Cov(X,Y) Var(Y) = 6(3 e) 1.69 Test with R = Result of the regression: â = , ˆb = Estimate: ˆb+â/2 = , Variance: (compared to 0.24)

17 Variance reduction p. 17/18 Back to the example No control Control

18 Variance reduction p. 18/18 Variance reductions techniques Conditioning Stratified sampling Importance sampling Draw recycling In general: correlation helps!