MONTE CARLO VARIANCE REDUCTION METHODS

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1 MONTE CARLO VARIANCE REDUCTION METHODS M. Ragheb /4/3. INTRODUCTION The questio arises of whether oe ca reduce the variace associated with the samplig of a radom variable? Ideed we ca, but we eed to be somehow sophisticated i samplig the radom variable. There is a wealth of methods ad techiques that allow us to reduce the variace: these are called variace reductio methods. We shall cosider some of them here. These methods are sometimes problem-depedet, but their usage i some cases would make possible the solutio of problems that would be otherwise itractable, because of the tremedous amout of computer time that would be eeded i a direct simulatio. This is the cause of the miscoceptio of the Mote Carlo method beig expesive. Ideed, if variace reductio methods are ot used i some cases, the umerical experimet may become more costly tha buildig a real physical experimet. The kowledge of variace reductio methods is thus a requiremet for the clever ad ecoomical use of the Mote Carlo method.. VARIANCE REDUCTION BY MODIFICATION OF THE SAMPLING SCHEME Every radom variable has a uderlyig variace associated with its probability desity fuctio. For istace, the radom variable: : Poits obtaied i throwig a sigle die : () The mea value ad the variace of this radom variable are: ( ) ( ) (3.5)

3 3. ANALOG, HIT OR MISS, OR POOR MAN'S SAMPLING We cosider variace reductio for the estimatio of simple itegrals sice the basic priciples ivolved geeralize quite easily to more complex problems, such as particle trasport ad fluid flow. Suppose we eed to estimate the itegral: with: f( x), whe x. Let us draw the curve: f ( x) dx (3) y f ( x), withi the uit square, as show i Fig.. We may write: where: gx ( ) =, if f ( x) y =, if f ( x) y. g( x, y) dy f ( x), (4) Simply stated, we accept ay sampled poits that lie below the curve for the mea estimatio process ad reject those that do ot. We the ca estimate θ as a double itegral: usig the estimator: g( x, y) dxdy, g i * g i, i (5) where: * is the umber of occasios o which f, i i i is a uiformly sampled y-coordiate poit over the iterval [,], is a uiformly sampled x-coordiate poit over the uit iterval, i i, i thus determies a poit uiformly distributed over the uit square.

4 I fact, we sample poits at radom i the uit square, ad cout the proportio of them which lie below the curve y f ( x). This is i retrospect the rejectio samplig method. Figure. Hit-or-Miss, Aalog, or Poor Ma's Mote Carlo estimatio for the estimatio of a itegral. As a umerical example, let us estimate the itegral: I x e dx (6) which has a exact aalytical value: I e x To have f ( x) whe x, we perform a simple scalig:

5 I e x e dx e e g( x) dx e ' (7) ad we the estimate the value of θ' from which we ca evaluate θ. A procedure usig the Aalog Mote Carlo Itegratio method is show i Fig..! aalog.f9! Aalog, Hit or Miss, or Poor Ma's Mote Carlo! Estimatio of the value of itegral usig Aalog Mote Carlo! I=Itegral e**x over the iterval [,] =.78888! M. Ragheb program aalog real mea real :: e= real :: trials=! Defie sampled fuctio; fuctio is scaled to lie withi the! uit square g(x) = exp(x)/e! Iitialize score score =.! Sample poits o uit square do i=,trials call radom(rr) x=rr f=g(x) call radom(rr) y=rr if(y.le.f)the score =score+. ed if ed do! Estimate mea value mea=score/trials! rescale the mea value mea = mea * e! Write results write(*,*) mea, trials ed Figure. Procedure for itegratio usig Aalog, Hit-or-miss, or Poor Ma s Mote Carlo. I the case of the hit ad miss Mote Carlo, the variace is that of the biomial distributio give by: p( p), p( p I our case, p is the umber of hits to the umber of trials ratio: * p Some computatioal results are show i Table. As a example to estimate the variace ad the stadard deviatio of the biomial distributio:

6 * score.7633 p trials ( p) p( p) p( p Table. Mea values ad Fractioal Stadard Deviatio usig Aalog, Hit-or-miss, or Poor Ma s Mote Carlo. fsd / , , It ca be oticed that a excessive umber of trials is required to get a low fractioal stadard deviatio. The stadard deviatio is that of the biomial distributio. 4. CRUDE MONTE CARLO SAMPLING If ξ, ξ, ξ3,, ξ are idepedet radom umbers uiformly distributed over the uit iterval [.], the the quatities: f i f( ) (8) i are idepedet radom variates with expectatio θ. Therefore, accordig to the Cetral Limit Theorem: f i f i (9) Table. Mea Values ad Stadard Deviatios usig Crude Mote Carlo. fsd / , ,

7 is a ubiased estimator of θ, ad its variace is: The stadard error of f is: f x f dx (). f From Table, it appears that Crude Mote Carlo does ot improve i term of the achieved error estimate over Hit or Miss Mote Carlo. I Fig. 3 we show how simple the procedure for crude Mote Carlo for the estimatio of the itegral is. I Fig. 4 the procedure is modified for the estimatio of the variace usig the covetioal estimatio formula. Figure 5 shows a alterative way of estimatig the variace usig the earlier derived recursive variace estimatio formula.! crude f9! Crude Mote Carlo! I=Itegral[e**x,,]=.78888! M. Ragheb program crude real mea real :: trials=! Defie sampled fuctio g(x) = exp(x)! Iitialize score score =.! Sample fuctio do i=,trials call radom(rr) score =score+g(rr) ed do! Estimate mea value mea=score/trials! Write results write(*,*) mea, trials ed Figure 3. Procedure for itegratio usig Crude Mote Carlo.! crude.f9! Crude Mote Carlo! Variace estimatio usig covetioal formula! I=Itegral[e**x,,]=.78888! M. Ragheb program crude real mea real :: trials=! Defie sampled fuctio g(x) = exp(x)! Iitialize scores score =. sum_squares =.! Sample fuctio do i=,trials

8 call radom(rr) s = g(rr) score =score + s sum_squares = sum_squares + s *s ed do! Estimate mea value, variace, stadard deviatio,! ad fractioal stadard deviatio mea=score/trials variace = sum_squares/trials - mea*mea variace = (trials/(trials-.)) * variace std_dev = sqrt (variace) std_error = std_dev/sqrt(trials) frac_sta_dev = std_error/mea! Write results write(*,*) mea, std_error, frac_sta_dev, trials ed Figure 4. Procedure for itegratio usig Crude Mote Carlo, with covetioal formula for the estimatio of the variace.! crude.f9! Crude Mote Carlo! Variace estimatio usig recursive formula! I=Itegral[e**x,,]=.78888! M. Ragheb program crude real mea_value, t, ta real :: trials=! Defie sampled fuctio g(x) = exp(x)! Iitialize variables t=. s_square_old =.! Sample fuctio do k=,trials xk=k call radom(rr) s = g(rr) if (k.ge.) the ta = t/(xk-.) & ed if t=t+s s_square_ew = (xk-.)/(xk-.)*s_square_old & + (ta-s)*(ta-s)/xk s_square_old = s_square_ew ed do! Estimate mea value, variace, stadard deviatio,! ad fractioal stadard deviatio mea_value = t/trials std_dev = sqrt (s_square_ew) std_error = std_dev/sqrt(trials) frac_sta_dev = std_error/mea_value! Write results write(*,*) mea_value, std_error, frac_sta_dev, trials ed Figure 5. Procedure for itegratio usig Crude Mote Carlo, with recursive formula for the estimatio of the variace. 5. CORRELATED SAMPLING OR CONTROL VARIATES The equatio:

9 f ( x) dx is broke ito two parts by addig the subtractio a easy fuctio ( x) that approximates the fuctio f(x): ( x) dx f ( x) ( x) dx which are itegrated separately, the first part by mathematical theory ad the secod part by crude Mote Carlo. This amouts to estimatig a correctio [f(x)-(x)] to the fuctio ( x), which approximates f(x). Deotig: ( x) dx, the used estimator would be: f ( x) ( x) ux ( ) () px ( ) with p(x) as a uiformly distributed probability desity fuctio. The followig choices for ( x) ca be cosidered: x ( x) x ( ) x x x 3 x x 3 x x x x x 4 x x 6 4 which are i fact successive approximatios to the itegrad e x. Samplig from these s ca be carried out by use of the rejectio method. The correspodig s are:

10 A procedure for the applicatio of the correlated sample tevhique is show i Fig. 6.! correlated f9! Correlated Samplig or Cotrol Variates Mote Carlo! Variace estimatio usig recursive formula! I=Itegral[e**x,,]=.78888! M. Ragheb program correlated real mea_value, t, ta real :: trials=5! Defie sampled fuctio g(x) = exp(x)! Defie cotrol variate fuctio f(x) =. + x! Aalytically itegrated value of f(x) phi =.5! Ope output file ope(44, file = 'radom_out')! Iitialize variables t=. s_square_old =.! Sample fuctio do k=,trials xk=k call radom(rr)! Evaluate cotrol variate s = (g(rr) - f(rr)) + phi if (k.ge.) the ta = t/(xk-.) & ed if t=t+s s_square_ew = (xk-.)/(xk-.)*s_square_old & + (ta-s)*(ta-s)/xk s_square_old = s_square_ew ed do! Estimate mea value, variace, stadard deviatio,! ad fractioal stadard deviatio mea_value = t/trials std_dev = sqrt (s_square_ew) std_error = std_dev/sqrt(trials) frac_sta_dev = std_error/mea_value! Write results write(*,) mea_value, std_error, frac_sta_dev, trials write(44,) mea_value, std_error, frac_sta_dev, trials format(x,'mea value =',e4.8,/,x,'stadard error =',e4.8,/, & & x,'fractioal stadard deviatio =',e4.8,/, & & ed x,'umber of trials =',e4.8) Figure 6. Procedure for the applicatio of the correlated samplig techique. A drastic decrease i variace compared with aalog ad crude samplig was detected by goig from a good to a better approximatio for f(x), some of the results beig show i the followig Table 3. Table 3. Mea Values ad Stadard Deviatios usig Correlated Samplig. fsd /

11 ( x) x ( ) x.738. x x x x 3 x x ,, x x x 4 x x IMPORTANCE SAMPLING We have: f ( x) dx f( x) gx ( ) g( x) dx f( x) dg ( x ) gx ( ) (3) for ay fuctios g ad G where: dg( x) g( x) dx ; ad if g is positive valued ad ormalized such that: G() g( x) dx. If η is a radom umber sampled from the distributio G, the: f ( ) E g ( ) (4) Thus, a equivalet estimator would be: f( x) vx ( ). ( x) (5) with the samplig carried out from the probability desity fuctio: ( x) px ( ).

12 ! importace f9! Importace Samplig Mote Carlo! Variace estimatio usig expoetial easy fuctio ad recursive formula! I=Itegral[e**x,,]=.78888! M. Ragheb program importace real mea_value, t, ta real :: alpha=.9 real :: trials=5! Defie sampled fuctio f(x) = exp(x)! Defie importace fuctio(ormalized) g(x) =(alpha/(exp(alpha)-.))*exp(alpha*x)! Ope output file ope(44, file = 'radom_out')! Iitialize variables t=. s_square_old =.! Sample fuctio do k=,trials xk=k! Sample Importace Fuctio g(x) call radom(rr) x=log(.+((exp(alpha)-.)*rr))/alpha! Evaluate importace estimator s = f(x)/g(x) if (k.ge.) the ta = t/(xk-.) s_square_ew = (xk-.)/(xk-.)*s_square_old & & + (ta-s)*(ta-s)/xk s_square_old = s_square_ew ed if t=t+s ed do! Estimate mea value, variace, stadard deviatio,! ad fractioal stadard deviatio mea_value = t/trials std_dev = sqrt (s_square_ew) std_error = std_dev/sqrt(trials) frac_sta_dev = std_error/mea_value! Write results write(*,) mea_value, std_error, frac_sta_dev, trials write(44,) mea_value, std_error, frac_sta_dev, trials format(x,'mea value =',e4.8,/,x,'stadard error =',e4.8,/, & & x,'fractioal stadard deviatio =',e4.8,/, & & x,'umber of trials =',e4.8) ed Figure 7. Procedure for the applicatio of the Importace Samplig techique usig the expoetial fuctio. Results of computatios are show i the followig Table 4 for the cases: ( x) x ( ) x x x 3 x x 3 x x 6

13 3 4 x x x 4 x x 6 4 as successive approximatios of the itegral. The rejectio method was used to sample these fuctios. Better results tha for correlated samplig were obtaied. Table 4. Mea Values ad Stadard Deviatios for Importace Samplig. ( x) x ( ) x x x x x 3 x x 6 / fsd , x x x 4 x x WEIGHTED UNIFORM SAMPLING We use the estimator: w,,..., 3. i i g i i N D (6) where the ξi s are idepedet radom umbers with idetical probability desity fuctio p(s): ad: f ( s) g( s) where ps ( ) # p( s) where p ( s) () s () s where ps ( ) # ps () where p ( s) The followig approximatig fuctios were used: ( x) x

14 ( ) x x x 3 x x 3 x x x x x 4 x x 6 4 Results of computatios are show i Table 5 below. Table 5. Mea Values ad Stadard Deviatios for Weighted Uiform Samplig. ( x) x ( ) x x x x x x x / fsd , x x x 4 x x ANTITHETIC VARIATES I the Atithetic Variates method, variace reductio is achieved by symmetrizatio of the itegral through a group trasformatio. A estimator that ca be used is: t f ( ) f ( ) (7) which amouts to a average of the fuctio f() ad its mirror image f(-). As show i Fig., less variatio from the mea ca be expected from the estimator t tha by usig f() or f(-) aloe.

15 Figure 8. Variace reductio through the use of the Atithetic Variates estimator: t f f 9. STRATIFICATION Aother estimator based o the Atithetic Variates priciple uses stratificatio of the itegratio iterval such as: t f f f f 4 (8)! atithetic f9! Atithetic Variate Mote Carlo! Variace estimatio usig recursive formula! I=Itegral[e**x,,]=.78888! M. Ragheb program importace real mea_value, t, ta

16 real :: trials=5! Defie sampled fuctio f(x) = exp(x)! Ope output file ope(44, file = 'radom_out')! Iitialize variables t=. s_square_old =.! Sample fuctio do k=,trials xk=k! Sample Importace Fuctio g(x) usig rejectio call radom(rr)! Evaluate atithetic variates estimator! Atithetic variate! s = (f(rr)+f(.-rr))/.! Stratificatio s=(f(rr/.)+f(.5-rr/.)+f(.5+rr/.)+f(.-rr/.))/4. if (k.ge.) the ta = t/(xk-.) s_square_ew = (xk-.)/(xk-.)*s_square_old & & + (ta-s)*(ta-s)/xk s_square_old = s_square_ew ed if t=t+s ed do! Estimate mea value, variace, stadard deviatio,! ad fractioal stadard deviatio mea_value = t/trials std_dev = sqrt (s_square_ew) std_error = std_dev/sqrt(trials) frac_sta_dev = std_error/mea_value! Write results write(*,) mea_value, std_error, frac_sta_dev, trials write(44,) mea_value, std_error, frac_sta_dev, trials format(x,'mea value =',e4.8,/,x,'stadard error =',e4.8,/, & & x,'fractioal stadard deviatio =',e4.8,/, & & x,'umber of trials =',e4.8) ed Figure 9. Procedure for the applicatio of the Atithetic Variates ad Stratificatio Samplig techique with the t ad t estimators. Results are show i Tables 6 ad 7, ad demostrate the cosiderable advatage obtaied over other variace reductio methods. Table 6. Mea Values ad Stadard Deviatios for Atithetic Variates. Estimator t t f f / fsd , , Table 7. Mea Values ad Stadard Deviatios for Stratificatio.

18 For aother samplig scheme, the efficiecy is: (3) T These efficiecies are cosidered to be iversely proportioal to the product of the sample variace ad the amout of labor expeded i the computatio expressed i terms of the computatioal time T. This extra labor is ormally caused by the use of a more sophisticated samplig scheme. To compare two samplig methods ad, oe ca defie a figure of merit F or efficiecy ratio i terms of the ratio of their respective efficiecies as: T (4) VL T F It ca be oticed that the efficiecy ratio is the product of the variace ad labor ratios. The compariso i terms of the efficiecy ratio F ca be exteded to the compariso of the Mote Carlo method to aother umerical method if its error estimate ca be obtaied. A alterate defiitio of the efficiecy ratio ca be writte i terms of the stadard deviatios rather tha the variaces as: F ' T S L (5) T with a stadard deviatio ratio defied as: S (6). OVER-BIASING AND UNDER-BIASING: CHARACTERISTIC MONTE CARLO PITFALLS I modifyig the samplig for variace reductio, i.e. improved efficiecy or makig the best use of computig time to stimulate evets which are most sigificat to the fial aswer, it is possible to overshoot the mark ad produce a samplig scheme that is so strogly biased as to be less efficiet tha crude samplig. This is over-biasig or over-samplig. The opposite situatio is also ecoutered as uder-samplig or uder-biasig. This occurs i crude or a slightly modified samplig scheme whe the result depeds heavily o ifrequet evets ad ot eough observatios occurred for good statistics. It is a geeral characteristic of both cases that most of the time the geerated aswers are too small. This produces a apparetly cosistet bias i the result. Variace estimates are also geerally small so that the cofidece itervals calculated i the simulatio will ted to idicate that the results are much more accurate tha they

19 really are. This geerates a false feelig of security ad faith i the obtaied results, which are actually cosistetly bad. McGrath ad Irvig demostrate these ideas by a extremely simplified example: Cosider a simulatio with two types of evets. Oe type (X) occurs frequetly (f(x) =.999) but cotributes oly a small amout (g(x) =.) to the fial result, while the other type (X) is rare (f(x) =.) but makes a large cotributio (g(x) = ) whe it occurs. The quatity I beig estimated has the correct value: I =f(x )g(x ) + f(x )g(x ) =.9999x. +.x =. (7) Usig Crude Mote Carlo with a moderate umber of histories, evet X would rarely occur ad a uderbiased aswer would be recorded as: Iu=I g(x)=. (8) If importace samplig is used so that the X evets occurred frequetly; sice they cotribute so much to the aswer, (to a excess), say ew probabilities of f*(x) =.9999 ad f*(x) =., were used, the the X evets will ever occur i a ru of moderate size ad the overbiased result will be: I f( X). g X.. f ( X ).9999 (9) * The proper modificatio for that case is to let X ad X happe with equal probabilities, f * (X) = f * (X) =.5 (accordig to their cotributio to the fial aswer), the the cotributio from each history is: f( X ).9999 g X f X ( ).. * ( ).5 f( X ). g X ( ). * f ( X).5 (3) ad the fial estimate from a small sample would be: I =. The uderlyig distributio is highly skewed with the large majority of cases makig little or o cotributios to the fial aswer, while a small umber of cases ca make large cotributios. I both cases the fial estimates were smaller tha the correct aswer, which is a geeral characteristic of such cases. If a set of histories, say, was simulated by crude Mote Carlo, the most likely there would be o X evets observed ad the (icorrect) estimate will be.. Oce every sets of histories, a sigle evet X will happe ad the estimate will be:

20 ' 99. I u., (3) a umber very much larger tha the correct value. Eve though the estimatio procedure is right ad averages come out correctly i the log ru, most users would thik this was the result of some iput mistake or computatio error, ad throw out the ru. Therefore cautio is recommeded i simulatios where most histories cotribute a small bit to the aswer but a few histories cotribute a large value. Complete faith should ot be placed i estimates of variace especially whe the results are smaller tha expected or if the possibility of overbiasig is suspected. EXERCISES. Usig the Crude Mote Carlo method, write a procedure to estimate the value of the itegral: ( x) dx. Use a equatio for the estimatio of the variace of your choice. Plot the mea value ad the variace as a fuctio of the umber of experimets N.. Usig the correlated samplig variace reductio method, write a procedure to estimate the value of the itegral ( x) dx usig x e dx as a easy fuctio. Plot the mea value ad the variace as a fuctio of the umber of experimets N. Compare the result to that obtaied from the crude samplig Mote Carlo method. 3. Compare the results for the Correlated Samplig method usig the differet approximatio fuctios discussed. 4. Derive a higher level Atithetic Variates fuctio t3, ad compare its results to those obtaied usig the fuctios t ad t. 5. Write a procedure usig weighted Uiform Samplig for estimatig a itegral. Discuss ay produced bias i the result. 6. Compare the mea values, stadard errors, fractioal stadard deviatios, variace ratios, ad efficiecy ratios for the Crude Mote Carlo Samplig method for the estimatio of a itegral of your choice for differet umbers of trials. For istace you could use the itegral of x e dx over the iterval [,]. Now use the Atithetic Variates estimators t ad t, ad compare their results to each other ad to those obtaied usig the Crude Mote Carlo estimator. Choose the variace estimatio formula of your choice. Use the differece betwee two calls to the computer clock at the begiig ad ed of a calculatio to estimate the labor time.

Chapter 2 The Monte Carlo Method

Chapter 2 The Monte Carlo Method Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful