Variance reduction techniques

Transcription

1 Variance reduction techniques Lecturer: Dmitri A. Moltchanov

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

3 1. Simulation with a given accuracy Two inverse tasks: we considered: what are the confidence intervals given N observation: is it ok if we got mean estimate, say, 50 ± 30 with confidence 95%? what we are always asked: provide a kind of assurance/guarantees: say what would be the value within intervals 50 ± 3 with confidence 95%. question: how to get this? 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 How did we decide how many replications are needed? N 2 = ( 1 0.1â 1 ) 2 N 1. (2) recall the interval estimate of the mean ( Ê[X] z α/2 ) ŝ, Ê[X] + z ŝ α/2. (3) N N assuming ŝ, z α/2 are constant (they are not!) for different N 1, N N1 0.1a 1 1 N2. (4) solving for N 2 1 N2 = 0.1a 1 N1 1 N 1 1 = 0.1a 1 N2 N 2 = ( 1 0.1â 1 ) 2 N 1. (5) Lecture: Variance reduction techniques 6

7 Sequential in-simulation checking: we want to estimate statistics a with the intervals ±0.1â: we get 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 can be set to 10. Lecture: Variance reduction techniques 7

8 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. in the re-run we may either: simulate k 2 k 1 batches (preferred); simulate k 2 + k 1 batches. Lecture: Variance reduction techniques 8

9 Sequential in-simulations checking: 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 can be set to Lecture: Variance reduction techniques 9

10 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, (6) i=1 confidence intervals for the mean are given by: ( ) ŝ Ê[X] z α/2, Ê[X] + z ŝ α/2. (7) 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 10

11 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 checking. We consider variance reduction techniques: antithetic variates technique; control variates technique. Lecture: Variance reduction techniques 11

12 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 12

13 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. (8) for mean of Z we have: ( ) X (1) + X (2) E[Z] = E = 1 ( E[X (1) + E[X (2) ] ) = E[X]. 2 2 (9) for variance of Z we have: ( ) X (1) + X (2) V ar[z] = V ar 2 = 1 4 ( V ar[x (1) ] + V ar[x (2) ] + 2Cov(X (1), X (2) ) ). (10) recalling that V ar[x (1) ] = V ar[x (2) ] = V ar[x], we have: V ar[z] = 1 2 (V ar[x] + Cov(X(1), X (2) ). (11) Lecture: Variance reduction techniques 13

14 since Cov(X, Y ) = Cov(X,Y ) V ar[x]v ar[y ] = ρ V ar[x]v ar[y ], we have: V ar[x]v ar[y ] where ρ is the correlation coefficient. V ar[z] = 1 V ar[x](1 + ρ). (12) 2 next, use Z (instead of X) to determine confidence intervals as usual. How the reduction is achieved: observe V ar[z] and see that: also note the special case: V ar[z] V ar[x] when ρ 1, V ar[z] 0 when ρ 1. (13) V ar[z] = 1 V ar[x] when ρ = 0. (14) 2 How to benefit: construct (x (1) 1, x (1) 2,..., x (1) n ) and (x (2) 1, x (2) 2,..., x (2) n ) such that ρ < 0. Lecture: Variance reduction techniques 14

15 Let be interested in X waiting times: we can control Y and Z interarrival and service times; (Y small, Z large) W large, (Y large, Z small) W small. How to create negative correlation in this example: let F (t) and G(s) be CDFs 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 ) and s 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 15

16 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 16

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18 Important note: this technique does not always provide better results; example: M/M/2 queue: results are only slightly better. Lecture: Variance reduction techniques 18

19 4. Control variates technique Also known as the method of control variable. Assume the following: X: variable whose mean we have to estimate; Y : 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], V ar[z] = V ar[x] + V ar[y ] + 2Cov(X, Y ). (15) since Y and X are negatively correlated we have that Cov(X, Y ) < 0; if V ar[y ] + 2Cov(X, Y ) < 0 we reduce the variance. Lecture: Variance reduction techniques 19

20 When Y and X are positively correlated: define Z = X Y + E[Y ], we have: E[Z] = E[X Y + E[Y ]] = E[X], V ar[z] = V ar[x] + V ar[y ] 2Cov(X, Y ). (16) since Y and X are positively correlated we have that Cov(X, Y ) > 0; if V ar[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. (17) construct confidence intervals for E[X] using Ê[Z] ± 1.96s/ n, where Ê[Z] = 1 n z i, ŝ 2 [Z] = 1 n (z i n n 1 Ê[Z])2. (18) i=1 i=1 Lecture: Variance reduction techniques 20

21 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 V ar[z]: note that Z has a smaller variance than X if: Z = X a(y E[Y ]), (19) V ar[z] = V ar[x] + a 2 V ar[y ] 2aCov(X, Y ). (20) a 2 V ar[y ] 2aCov(X, Y ) < 0. (21) we have to select a to minimize RHS of (20), we have to find minimum: 2aV ar[y ] 2Cov(X, Y ) = 0, a = Cov(X, Y ). (22) V ar[y ] Lecture: Variance reduction techniques 21

22 substituting into expression for V ar[z] we have: V ar[z] = V ar[x] [Cov(X, Y )]2 V ar[y ] = (1 ρ 2 XY )V ar[x]. (23) note: if X and Y are correlated, we always get reduction of variance if a is optimal: detection of a requires knowledge of V ar[y ] and Cov(X, Y ); pilot experiments can be used to get estimates of V ar[y ] and Cov(X, Y ). we can also further generalize to get: m Z = X a i (Y i E[Y i ]), (24) i=1 where a i, i = 1, 2,..., m are 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 22

23 5. 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 step and very often just skipped. Example: develop a certain switch: we have to know its performance in advance; the only way: model the system either analytically of using 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 23

24 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 24