Variance reduction techniques

Transcription

1 Variance reduction techniques Lecturer: Dmitri A. Moltchanov moltchan/modsim/

2 OUTLINE: Simulation with a given confidence; Variance reduction techniques; Antithetic variates technique; Control variates technique; Method of conditioning; Validation of simulations. Lecture: Variance reduction techniques 2

3 1. Simulation with a given confidence Two inverse tasks: we considered: what are the confidence intervals given N observation: is it OK to say that mean is 50 ± 30? what we are asked: provide a kind of assurance: we perhaps would like to say 50 ± 3... question: how many experiments are needed to achieve that? General notes: recall, width of confidence intervals is proportional to 1/ N; N: number of iid observations; the larger N the smaller is the interval. to halve the confidence interval: increase N four times: ( ) ŝ Ê[X] z α/2, Ê[X]+z ŝ α/2. (1) N N Lecture: Variance reduction techniques 3

4 Planning prior to simulations: we do not know how many observations are needed: what accuracy may N experiments give? e.g. how many observations are needed to get 50 ± 3? solution 1: carry out pilot experiment: aim 1: to get overall idea how things go; aim 2: obtain rough estimate of N providing required accuracy. solution 2: sequential in-simulation checking: test is carried out periodically to check whether required accuracy is achieved. We consider these methods for: batch mean method; method of replications. Lecture: Variance reduction techniques 4

5 1.1. Method of replications Use of pilot experiments: we want to estimate statistics a with the intervals ±0.1â: pilot experiments is carried out to collect N 1 replications; let â 1 be a point estimator of a; let 1 be the width of confidence intervals; check the following: if 1 0.1â 1 we stop; if 1 > 0.1â 1 we carry out main simulation with N 2 =( 1 /0.1â 1 ) 2 N 1 replications; we obtain new â 2 and 2 : they may not be exactly what we wanted (0.1â 2 ) due to randomness. if 1 > 0.1â 1 carry out new attempt. Lecture: Variance reduction techniques 5

6 Sequential in-simulation checking: we want to estimate statistics a with the intervals ±0.1â: we carry out N replications; we calculate â 1 and 1 out of these replications; check the following: if 1 0.1â 1 we stop; if 1 > 0.1â 1 we carry out N additional replications; we calculate new â 2 and 2 based on total 2N replications: if 2 0.1â 2 we stop; if 2 > 0.1â 2 we carry out N additional replications; repeat... Note: N may be set to 10. Lecture: Variance reduction techniques 6

7 1.2. Method of batch means Use of pilot experiments: we want to estimate statistics a with the intervals ±0.1â: gather k 1 batches; estimate â 1 and 1 ; check the following: if 1 0.1â 1 we stop; if 1 > 0.1â 1 we carry out k 2 k 1 additional batches have to be simulated; simply re-run for k 2 =( 1 /0.1â 1 ) 2 k 1 batches. calculate new â 2 and 2 based on total k 1 and k 2 k 1 batches; they may not be exactly what we wanted (0.1â 2 ) due to randomness. we may either: simulate k 2 k 1 batches (preferred); simulate k 2 + k 1 batches. Lecture: Variance reduction techniques 7

8 Use of pilot experiments: we want to estimate statistics a with the intervals ±0.1â: gather k batches; estimate â 1 and 1 ; check the following: if 1 0.1â 1 we stop; if 1 > 0.1â 1 gather k additional batches. calculate new â 2 and 2 based on total 2k batches; check the following: if 2 0.1â 2 we stop; if 2 > 0.1â 2 gather k additional batches. repeat... Note: N may be set to Lecture: Variance reduction techniques 8

9 2. Variance reduction techniques Suppose we have to estimate mean given N iid observations: point estimate of the mean is: Ê[X] = 1 N N X i, (2) i=1 confidence intervals for the mean are given by: ( ) ŝ Ê[X] z α/2, Ê[X]+z ŝ α/2. (3) N N where ŝ 2 is the estimate of the variance. How to shorten the confidence intervals: accuracy of an estimate can be increased by increasing the number of observations; shortcoming: may require very long simulations. accuracy can also be achieved by reducing variance. Lecture: Variance reduction techniques 9

10 General notes: techniques that tries to reduce variance: variance reduction techniques; requires additional computational complexity; it is not known in advance (prior to simulation) whether they actually reduce variance; To decide whether a technique helps to reduce variance: pilot experiments; in-simulation check. We consider variance reduction techniques: antithetic variates techniques; control variates technique; method of conditioning Lecture: Variance reduction techniques 10

11 3. Antithetic variates technique General notes: very simple technique; requires only few additions to the program; no guarantees of effectiveness Major shortcomings: no guarantees of effectiveness; no information in advance how much the variance will reduce. Take the following assumptions: iid observations (x (1) 1,x (1) 2,...,x (1) n ) are obtained in the first simulation; iid observations (x (2) 1,x (2) 2,...,x (2) n ) are obtained in the second simulation. Lecture: Variance reduction techniques 11

12 The idea of the method: define a new random variable Z =(X (1) + X (2) ): z i = x(1) i + x (2) i, 2 i =1, 2,...,n. (4) for mean of Z we have: ( ) X (1) + X (2) E[Z] =E = 1 ( E[X (1) + E[X (2) ] ) = E[X]. 2 2 (5) for variance of Z we have: ( ) X (1) + X (2) Var[Z] =Var 2 = 1 4 ( Var[X (1) ]+Var[X (2) ]+2Cov(X (1),X (2) ) ). (6) recall that Var[X (1) ]=Var[X (2) ]=Var[X], we have: Var[Z] = 1 2 (Var[X]+Cov(X(1),X (2) ). (7) Lecture: Variance reduction techniques 12

13 since Cov(X, Y )=ρ Var[X]Var[Y ], we have: where ρ is the correlation coefficient. Var[Z] = 1 Var[X](1 + ρ). (8) 2 next, use Z (instead of X) to determine confidence intervals as usual. How the reduction is achieved: observe Var[Z] and see that: also note the special case: Var[Z] Var[X] when ρ 1, Var[Z] 0 when ρ 1. (9) Var[Z] = 1 Var[X] when ρ =0. (10) 2 The idea: construct (x (1) 1,x (1) 2,...,x (1) n )and(x (2) 1,x (2) 2,...,x (2) n )suchthatρ<0. Lecture: Variance reduction techniques 13

14 Negative correlation: queuing system: X waiting times, Y interarrival times: Y is small and W is large and vice versa. How to create negative correlation in this example: let F (t) andg(s) be the cdf of interarrival and service time, respectively; let r i and v i be pseudo random numbers with U(0, 1); let t i = F 1 (r i )ands i = G 1 (v i ) be interarrival and service times associated with ith arrival; to determine whether queue increases or decreases consider d i = t i s i : negative: busy period, positive: empty period. consider the second run and associate numbers r i and s i with ith arrival so that: d i = t i s i has an opposite sign compared to d i ; this can be achieved using r i =1 r i and v i =1 v i. we have negative correlation in two runs! Lecture: Variance reduction techniques 14

15 How to implement: make the first run and get (x (1) 1,x (1) 2,...,x (1) n ); make the second run using r i =1 r i and v i =1 v i to get (x (2) 1,x (2) 2,...,x (2) n ); construct point and interval estimates using Z =(X (1) + X (2) )/2. Straightforward simulation: sampling every 10th customer. Method of antithetic variates: ± 1.76 for n = 300 (in overall ) Lecture: Variance reduction techniques 15

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17 Important note: this technique does not always provide better results; example: M/M/2 queue: results are only slightly better. Lecture: Variance reduction techniques 17

18 4. Control variates technique Also known as the method of control variable. Assume the following: X: endogenously created variable whose mean we have to estimate; Y : endogenously created variable whose mean is known in advance; Y is strongly correlated with X. When Y and X are negatively correlated: define Z = X + Y E[Y ], we have: E[Z] =E[X + Y E[Y ]] = E[X], Var[Z] =Var[X]+Var[Y ]+2Cov(X, Y ). (11) since Y and X are negatively correlated we have that Cov(X, Y ) < 0; if Var[Y ]+2Cov(X, Y ) < 0 we reduce the variance. Lecture: Variance reduction techniques 18

19 When Y and X are positively correlated: define Z = X Y + E[Y ], we have: E[Z] =E[X Y + E[Y ]] = E[X], Var[Z] =Var[X]+Var[Y ] 2 Cov(X, Y ). (12) since Y and X are positively correlated we have that Cov(X, Y ) > 0; if Var[Y ] 2Cov(X, Y ) < 0 we reduce the variance. Example: queuing system: X waiting times, Y interarrival times: Y is small and X is large and vice versa: negative correlation: get observations (x 1,x 2,...,x n )and(y 1,y 2,...,y n ) and let: z i = x i + y i E[Y ], i =1, 2,...,n. (13) construct confidence intervals for E[X] usingê[z] ± 1.96s/ n, where Ê[Z] = 1 n z i, ŝ 2 [Z] = 1 n (z i n n 1 Ê[Z])2. (14) i=1 i=1 Lecture: Variance reduction techniques 19

20 General view of the approach: RV Z can be obtained as follows: a is some constant to be estimated; X and Y are positively or negatively correlated RVs. we have that E[Z] =E[X] and for Var[Z]: Z has a smaller variance than X if: we have to select a to minimize RHS, we have: Z = X a(y E[Y ]), (15) Var[Z] =Var[X]+a 2 Var[Y ] 2aCov(X, Y ). (16) a 2 Var[Y ] 2aCov(X, Y ) < 0. (17) 2aV ar[y ] 2Cov(X, Y )=0, a = Cov(X, Y ). (18) Var[Y ] Lecture: Variance reduction techniques 20

21 substituting into expression for Var[Z] we have: Var[Z] =Var[X] [Cov(X, Y )]2 Var[Y ] =(1 ρ 2 XY )Var[X]. (19) note: if X and Y are correlated, we always get reduction of variance if a is optimal: detection of a requires knowledge of Var[Y ]andcov(x, Y ); pilot experiments can be used to get estimates of Var[Y ]andcov(x, Y ). we can also further generalize to get: m Z = X a i (Y i E[Y i ]), (20) i=1 where a i, i =1, 2,...,m are any real numbers. Important notes: this method may give assurance after pilot experiment; complexity if a little bit higher than that of the antithetic variates. Lecture: Variance reduction techniques 21

22 5. Method of conditioning The basis of the method: X and Y are arbitrary random numbers; for X and Y we have: E[X] =E[E[X Y ]], Var[X] =E[Var[X Y ]] + Var[E[X Y ]] Var[E[X Y ]]. (21) note that RV E[X Y ] has the same mean and smaller variance than RV X. The idea of the method: let E[X] be the performance measure we have to estimate; if RV Y is such that Y = E[X Y = y] is known: we simulate Y ; we use E[X Y ] as an estimate. Lecture: Variance reduction techniques 22

23 Example: M/M/1/N queuing system: performance measure E[X] the mean number of lost customers at time t; direct simulation: K runs until time t; of X i is the number of lost customers in run i, we estimate mean as: Ê[X] = 1 K K X i. (22) i=1 we reduce the variance of the point estimator as follows: let Y i be the total amount of time in (0,t] that there are N customers in ith run; since we have Poisson arrival process we have: E[X Y i ]=λy i. (23) estimator with reduced variance is then: Ê[Y ]= 1 K K Y i. (24) i=1 Lecture: Variance reduction techniques 23

24 Example: numerical results: λ =0.5, µ =1,N =3,t = 1000; we reduce standard deviation by 1.3. Figure 1: Estimation of E[X] without conditioning. Figure 2: Estimation of E[X] using conditioning. Lecture: Variance reduction techniques 24

25 6. Validation of the simulation model General notes: how close we are to the reality? how confident we are that the simulation results are accurate? vary important issue and very often neglected. Example: develop a certain switch: we have to know its performance in advance; the only way: model the system either analytically of vie simulation studies: almost no way to make validation. Example: extend a switch: we have to know its performance in advance; we can use the following approach: simulate the real system and compare results; extend the simulation model and simulate. Lecture: Variance reduction techniques 25

26 The following checks can be used: check the pseudo-random numbers generators; independence, uniformity, etc. check the generator of arbitrary random numbers; independence or dependence tests, fitting criteria (e.g. χ 2 ), etc. check the logic of the simulation program; print out small samples of variables, event list, data structures for further hand checking. check the relationship validity; print out all variables for small parts of run and check whether they are correctly updated. check the output validity; possible only if empirical data are available. Some important notes: these check must be performed if you get what was not expected; but: do not try to get what you expect!!! Lecture: Variance reduction techniques 26