1 Introduction to reducing variance in Monte Carlo simulations

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Basil Higgins
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1 Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by collectig iid samples from the distributio, X 1,, X ad usig the sample mea (1) X() = 1 X j This is justified by the strog law of large umbers (SLLN), which asserts that this estimate coverges with probability oe (wp1) to the desired µ = E(X), as But the SLLN does ot tell us how good the approximatio is; we cosider this ext Lettig σ = V ar(x) deote the variace of the distributio, we coclude that () V ar(x()) = σ The cetral limit theorem asserts that as, the distributio of def Z = σ (X() µ) teds to N(0, 1), the uit ormal distributio Lettig Z deote a N(0, 1) rv, we coclude that for sufficietly large, Z Z i distributio From here we obtai for ay z 0, P ( X() µ > z σ ) P ( Z > z) = P (Z > z) (We ca obtai ay value of P (Z > z) by referrig to tables, etc) For ay α > 0 o matter how small (such as α = 005), lettig z α/ be such that P (Z > z α/ ) = α/, we thus have P ( X() µ > z α/ σ ) α, which implies that the ukow mea µ lies withi the iterval X() ± z α/ σ with (approximately) probability 1 α This allows us to costruct cofidece itervals for our estimate: we say that the iterval X() ± z α/ σ is a 100(1 α)% cofidece iterval for the mea µ Typically, we would use (say) α = 005 i which case z α/ = z 005 = 196, ad we thus obtai a 95% cofidece iterval X() ± (196) σ The legth of the cofidece iterval is (196) σ which of course teds to 0 as the sample size gets larger I practice we would ot actually kow the value of σ ; it would be ukow (just as µ is) But this is ot really a problem: we istead use a estimate for it, the sample variace s () defied by s () = 1 (X j X ) 1 1

2 It ca be show that s () σ, with probability 1, as ad that E(s ()) = σ, So, i practice we would use s() is place of σ whe costructig our cofidece itervals For example, a 95% cofidece iterval is give by X() ± (196) s() The followig recursios ca be derived; they are useful whe implemetig a simulatio requirig a cofidece iterval: X( + 1) = X() + X +1 X(), + 1 ( S( + 1) = 1 1 ) S() + ( + 1)(X( + 1) X()) 1 Applicatio to Mote Carlo simulatio I Mote Carlo simulatio, istead of collectig the iid data X 1,, X, we simulate it Moreover, we ca choose as large as we wat; = 10, 000 for example, so the cetral limit theorem justificatio for costructig cofidece itervals ca safely be used (eg, i statistics out i the field applicatios, oe might oly have = 5 or = 50 samples!) Thus we ca immediately obtai cofidece itervals for Mote Carlo estimates But simulatio also allows us to be clever: We ca purposely try to iduce egative correlatio amog the variables X 1,, X, or geerate copies that while havig the same mea, have a smaller variace, so that the variace of the estimator i (1) becomes smaller tha σ resultig i a smaller cofidece iterval The idea is to try to get eve better estimates by reducig the ucertaity i our estimate I the ext sectios, we explore ways of doig this 13 Atithetic variates method Let X i deote our copies of X Let = m, for some m 1, that is, is eve Note that (3) X() = 1 m where m X j = 1 m m Y j = Y (m), Y 1 = X 1 + X Y = X 3 + X 4 Y m = X 1 + X, ad E(Y i ) = E(X) = µ This meas that for purpose of estimatig µ = E(X), we ca view each Y i as the ed copy that we wish to simulate from (istead of the X i ) We let Y = X 1+X deote a geeric Y i The problem of estimatio ca be re-cast as we are tryig to estimate µ = E(Y )

3 Computig variaces, (4) (5) (6) (7) (8) V ar(y ) = (1/4)(σ + σ + Cov(X 1, X )) = (1/)(σ + Cov(X 1, X )), hece V ar(y (m)) = 1 (σ + Cov(X 1, X )) = σ / + Cov(X 1, X ))/ = V ar(x()) + Cov(X 1, X ))/ I the case whe the X i are iid, Cov(X 1, X ) = 0 ad thus V ar(y (m)) = σ = V ar(x()), ad we get back to where we started i () But if Cov(X 1, X ) < 0, the V ar(y (m)) < σ ; variace is reduced So it is i our iterest to somehow create some egative correlatio withi each pair (X 1, X ), (X 3, X 4 ),, but keep the pairs iid so that the Y i are iid (ad thus the CLT still applies); for the V ar(y (m)) will be lowered from what it would be if we simply used iid copies of the X i To motivate how we might create the desired egative correlatio, recall that we ca geerate a expoetially distributed rv X 1 = (1/λ) l (U) with U uiformly distributed o (0, 1) Now istead of usig a ew idepedet uiform to geerate a secod such copy, use 1 U which we well kow is also uiformly distributed o (0, 1); that is, defie X = (1/λ) l (1 U) Clearly X 1 ad X are egatively correlated sice if U icreases, the 1 U decreases ad the fuctio l(y) is a icreasig fuctio of y: X 1 icreases iff U icreases iff 1 U decreases iff X decreases More geerally, for ay distributio F (x) = P (X x) with iverse F 1 (y) we could geerate a egatively correlated pair via X 1 = F 1 (U), X = F 1 (1 U) sice F 1 (y) is a mootoe icreasig fuctio of y The radom variables U ad 1 U have a correlatio coefficiet ρ = 1, they are egatively correlated (to the largest extet), thus the mootoicity preserves the property of egative correlatio; ρ X1,X < 0 (ot ecessarily 1 though) I a geeral Mote Carlo simulatio our X is of the form X = h(u 1,, U k ), for some (perhaps very complicated) fuctio h, ad some k (perhaps large), that is, we eed k iid U i to geerate each copy of X For example, if we are cosiderig X = D for the FIFO GI/GI/1 queue, by usig the iverse trasform method for the service (G 1 ) ad iterarrival times (A 1 ), the we eed k = 4 because we eed to geerate service times ad iterarrival times; assumig D 0 = 0 we ca write this as D 1 = (S 0 T 0 ) + = (G 1 (U 1 ) A 1 (U )) + ad the D = (D 1 + S 1 T 1 ) + or D = Thus D = h(u 1, U, U 3, U 4 ) where h(y 1, y, y 3, y 4 ) = [ (G 1 (U 1 ) A 1 (U )) + + G 1 (U 3 ) A 1 (U 4 )] + [ (G 1 (y 1 ) A 1 (y )) + + G 1 (y 3 ) A 1 (y 4 )] + As log as the fuctio h is mootoe (either icreasig or decreasig) i each variable, the it ca be show that X 1 = h(u 1,, U k ) ad X 1 = h(1 U 1,, 1 U k ) are ideed egatively correlated, ad are referred to as atithetic variates I geeral, as log as the fuctio h is mootoe (either icreasig or decreasig) i each variable, the it ca be show that X 1 = h(u 1,, U k ) ad X = h(1 U 1,, 1 U k ) are ideed egatively correlated, ad are referred to as atithetic variates Agai, because the vectors (U 1, U, U k ) ad (1 U 1, 1 U, 1 U k ) have the same distributio, so do X 1 ad X ; i particular they have the same mea E(X) But because of the iduced 3

4 egative correlatio (whe h is mootoe) the two are themselves egatively correlated copies I the above example for D, this is easily established sice each iverse fuctio is mootoe; h(y 1, y, y 3, y 4 ) icreases i y 1 ad y 3 ad decreases i y ad y 4 As aother example, if we are cosiderig X = C = ( 1 S i K) +, i=1 the payoff at time T = of a Asia call optio uder the biomial lattice model, S = S 0 Y 1 Y, the re-writig 1 S i = (1/)S 0 Y 1 [1 + Y ], i=1 where the Y i are the iid up-dow rvs, we have h(u 1, U ) = ( (1/)S 0 (ui{u 1 p} + di{u 1 > p})[1 + (ui{u p} + di{u > p})] K) + This fuctio is mootoe decreasig i U 1 ad U : as either variable icreases, they will exceed the value p ad hece the idicators will ted towards the lower value d as opposed to the higher value u > d Because the vectors (U 1, U ) ad (1 U 1, 1 U ) are idetically distributed, so are the rvs X 1 = h(u 1, U ) ad X = h(1 U 1, 1 U ); i particular they have the same mea E(X) But the mootoicity of h results i egative correlatio betwee them, Cov(X 1, X ) < 0 We summarize (without proof): Propositio 11 If the fuctio h for geeratig X = h(u 1,, U k ) is mootoe i each variable, the X 1 = h(u 1,, U k ) ad X = h(1 U 1,, 1 U k ) with the U i iid uiform o (0, 1) are i fact egatively correlated; Cov(X 1, X ) < 0 (Equivaletly E(X 1 X ) < E(X 1 )E(X ) = E (X)) Algorithm for usig atithetic variates to estimate µ = E(X), whe X = h(u 1,, U k ) is mootoe i the U i : The method of simulatig our pairs is straightforward: 1 Geerate U 1, U k Costruct a first pair: Set X 1 = h(u 1,, U k ) ad X = h(1 U 1,, 1 U k ) Defie Y 1 = [X 1 + X ]/ ad ote that E(Y 1 ) = E(X) = µ Now idepedetly geerate k ew iid uiforms to costruct aother pair X 3, X 4 ad so o pair by pair util reachig m (large) desired pairs, ad m iid radom variables Y j = [X j 1 + X j ]/, 1 j m These Y j have the same mea E(X) = µ, but have a smaller variace because of (8) 3 Use the estimate m Y (m) = Y j, 4

5 where Y 1 = X 1 + X Y = X 3 + X 4 Y m = X m 1 + X m To costruct our (ew ad better) cofidece iterval: The sample variace for these Y j is give by s (m) = 1 m 1 m (Y j Y m ) The the iterval Y (m) ± z α/ s(m) is a 100(1 α)% cofidece iterval for the mea µ Examples 1 Estimatig π: As a very simple example, ote that we ca estimate π by observig that π = the area of a disk of radius 1 ({(x, y) : x + y 1}); π/4 = x dx = E( 1 U ) So Mote Carlo ca be used to estimate π by geeratig copies of X = 1 U ad averagig Sice h(x) = 1 x is mootoe decreasig i x, we ca use atithetic variates Thus we would use X 1 = 1 U1, X = 1 (1 U 1 ) for our first pair, X 3 = 1 U, X 4 = 1 (1 U ) ad so o Customer delay i a FIFO sigle-server queue: As aother example, cosider the delay recursio for a FIFO GI/GI/1 queue: D +1 = (D + S T ) +, 0, where {T : 0} are iid customer iterarrival times distributed as A(x) = P (T x), x 0 ad idepedetly {S : 0} are iid customer service times distributed as G(x) = P (S x), x 0 We assume here that both A 1 (y) ad G 1 (y) are explicitly kow so that the iverse trasform method ca be applied The D = h(u 1,, U ) ca be writte as a mootoe i each variable fuctio For example, D 1 = h(u 1, U ) = (D 0 +G 1 (U 1 ) A 1 (U )) +, which is mootoe icreasig i U 1 ad mootoe decreasig i U We ca the write D = (D 1 + G 1 (U 3 ) A 1 (U 4 )) +, which is thus mootoe icreasig i both U 1 ad U 3, ad mootoe decreasig i U ad U 4 This same idea exteds to D for ay Thus if we wated to estimate (say) E(D 10 ), the expected delay of the 10 th customer (whe (say) D 0 = 0), we could do so as follows: Geerate (U 1, U 10 ) ad (V 1, V 10 ) as iid uiforms over (0, 1) Costruct a copy X 1 = D 10 via usig the followig recursio for 0 9: D +1 = (D + G 1 (U +1 ) A 1 (V +1 )) + 5

6 Now repeat the costructio usig (1 U 1, 1 U 10 ) ad (1 V 1, 1 V 10 ) to get the secod (atithetic) copy X = D 10 : D +1 = (D + G 1 (1 U +1 ) A 1 (1 V +1 )) + X 1 ad X are thus egatively correlated copies of D 10 as desired Remark 11 I a real simulatio applicatio, computig exactly Cov(X 1, X ) whe X 1 ad X are atithetic is ever possible i geeral; after all, we do ot eve kow (i geeral) either E(X) or V ar(x) But this is ot importat sice our objective was oly to reduce the variace, ad we accomplished that 14 Atithetic ormal rvs I may fiace applicatios, the fudametal rvs eeded to costruct a desired output copy X are uit ormals, Z 1, Z, (As opposed to uiforms) For example, whe usig geometric Browia motio for asset pricig, our payoffs typically ca be writte i the form X = h(z 1,, Z k ) Notig that Z is also a uit ormal if Z is, ad that the correlatio coefficiet betwee them is ρ = 1, the followig is the Gaussia aalogue to Propositio 11 Propositio 1 If the fuctio h for geeratig X = h(z 1,, Z k ) is mootoe i each variable, the X 1 = h(z 1,, Z k ) ad X = h( Z 1,, Z k ) with the Z i iid N(0, 1) are i fact egatively correlated; Cov(X 1, X ) < 0 6

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%