Simulation and Random Number Generation

Transcription

1 Smulaton and Random Number Generaton Summary Dscrete Tme vs Dscrete Event Smulaton Random number generaton Generatng a random sequence Generatng random varates from a Unform dstrbuton Testng the qualty of the random number generator Some probablty results Evaluatng ntegrals usng Monte-Carlo smulaton Generatng random numbers from varous dstrbutons Generatng dscrete random varates from a gven pmf Generatng contnuous random varates from a gven dstrbuton

2 Dscrete Tme vs. Dscrete Event Smulaton Solve the fnte dfference equaton x = mn, x +.x + log x + u { } k+ k k k k Gven the ntal condtons x, x - and the nput functon u k, k=,, we can smulate the output! Ths s a tme drven smulaton. For the systems we are nterested n, ths method has two man problems. x(k) k Issues wth Tme-Drven Smulaton System Input System Dynamcs f parcel arrves at t u () t = otherwse f truck arrves at t u2 () t = otherwse u (t) xt () + f u() t =, u2() t =, + xt ( ) = xt () f u() t =, u2() t = xt () otherwse t u 2 (t) x(t) Ineffcent: At most steps nothng happens Accuracy: Event order n each nterval t t t

3 Tme Drven vs. Event Drven Smulaton Models Tme Drven Dynamcs Event Drven Dynamcs xt () + f u() t =, u2() t =, + xt ( ) = xt () f u() t =, u2() t = xt ( ) otherwse f ( x, e ) x+ f e = a = x f e = d In ths case, tme s dvded n ntervals of length t, and the state s updated at every step State s updated only at the occurrence of a dscrete event Pseudo-Random Number Generaton It s dffcult to construct a truly random number generator. Random number generators consst of a sequence of determnstcally generated numbers that look random. A determnstc sequence can never be truly random. Ths property, though seemngly problematc, s actually desrable because t s possble to reproduce the results of an experment! Multplcatve Congruental Method x k+ = a x k modulo m The constants a and m should follow the followng crtera For any ntal seed x, the resultant sequence should look lke random For any ntal seed, the perod (number of varables that can be generated before repetton) should be large. The value can be computed effcently on a dgtal computer.

4 Pseudo-Random Number Generaton To ft the above requrements, m should be chosen to be a large prme number. Good choces for parameters are m= and a=7 5 for 32-bt word machnes m= and a=5 5 for 36-bt word machnes Mxed Congruental Method x k+ =(a x k + c) modulo m A more general form: n xk = ax j k j mod m j= where n s a postve constant x k /m maps the random varates n the nterval [,) Qualty of the Random Number Generator (RNG) The random varate u k =x k /m should be unformly dstrbuted n the nterval [,). Dvde the nterval [,) nto k subntervals and count the number of varates that fall wthn each nterval. Run the ch-square test. The sequence of random varates should be uncorrelated Defne the auto-correlaton coeffcent wth lag k > C k and verfy that E[C k ] approached for all k=,2, n k ( )( ) Ck = u u 2 + k 2 n k = where u s the random varate n the nterval [,).

5 Some Probablty Revew/Results Probablty mass functons Cumulatve mass functons Pr{=}=p F ( x) = Pr{ x} Pr{=2}=p x 2 Pr{=3}=p = Pr 3 { = } = 2 3 p + p 2 + p 3 = { } Pr = = p + p 2 p 2 3 Some Probablty Results/Revew Probablty densty functons Cumulatve dstrbuton f (x) ( x) = Pr{ } F x Pr{x=x }=! x x x = f ( y) dy Pr { x ε x ε} + = x + ε = x ε f ( x) dx F (x) x

6 Independence of Random Varables Jont cumulatve probablty dstrbuton. { } F ( x, y) = Pr x, Y y Y Independent Random Varables. { C Y D} = { C} { Y D} Pr, Pr Pr For dscrete varables { = x Y = y} = { = x} { Y = y} Pr, Pr Pr For contnuous varables f ( x, y) = f ( x) f ( y) F ( x, y) = F ( x) F ( y) Y Y Y Y Condtonal Probablty Condtonal probablty for two events. Bayes Rule. { A B} Pr { } { A} Total Probablty = Pr Pr { AB} { B} { = x} = { = x Y = y } { Y = y } Pr Pr Pr k k { } { } Pr{ B} Pr AB Pr B A Pr A Pr { B A} = Pr { A B} = Pr k

7 Expectaton and Varance Expected value of a random varable E [ ] = x Pr{ = x} = E [ ] = xf ( xdx ) Expected value of a functon of a random varable E[ g ( )] = g( x )Pr{ = x} = Varance of a random varable var[ ] = E E[ ] E [ g ( )] = g( x) f ( xdx ) ( ) 2 Covarance of two random varables The covarance between two random varables s defned as cov [ Y, ] = E ( E[ ] )( Y E[ Y] ) [ ] = [ [ ] [ ] + [ ] [ ]] cov Y, E Y E Y YE E E Y = E[ Y ] E[ ] E[ Y ] The covarance between two ndependent random varables s (why?) [ Y ] = j Pr { =, = j} E x y x Y y j { } Pr{ } [ ] [ Y ] = x Pr = x y Y = y = E E j j j

8 Markov Inequalty If the random varable takes only non-negatve values, then for any value a >. Pr { a} E [ ] a Note that for any value <a < E[], the above nequalty says nothng! Chebyshev s Inequalty If s a random varable havng mean µ and varance σ 2, then for any value k >. Pr { µ kσ} 2 σ k f (x) kσ Ths may be a very conservatve bound! Note that for ~N(,σ 2 ), for k=2, from Chebyshev s nequalty Pr{.}<.25, when n fact the actual value s Pr{.}<.5 µ kσ x

9 The Law of Large Numbers Weak: Let, 2, be a sequence of ndependent and dentcally dstrbuted random varables havng mean µ. Then, for any ε > Pr n µ > ε, as n n Strong: wth probablty, lm n n n = µ The Central Lmt Theorem Let, 2, n be a sequence of ndependent random varables wth mean µ and varance σ ι2. Then, defne the random varable, Let σ = n = E[ ] = E[ n] = µ µ 2 n 2 ( µ ) 2 = E = σι Then, the dstrbuton of approaches the normal dstrbuton as n ncreases, and f are contnuous, then f ( x) e 2π = ( x µ ) 2 /2σ n =

10 Ch-Square Test Let k be the number of subntervals, thus p =/k, =,,k. Let N be the number of samples n each subnterval. Note that E[N ]=Np where N s the total number of samples Null Hypothess H : The probablty that the observed random varates are ndeed unformly dstrbuted n (,). Let T be k k 2 k ( ) = = N T = N Np = N Np N k Defne p-value= P H {T>t} ndcate the probablty of observng a value t assumng H s correct. For large N, T s approxmated by a ch-square dstrbuton wth k- degrees of freedom, thus we can use ths approxmaton to evaluate the p-value The H s accepted f p-value s greater than.5 or. 2 Monte-Carlo Approach to Evaluatng Integrals Suppose that you want to estmate θ, however t s rather dffcult to analytcally evaluate the ntegral. θ = g( x) dx Suppose also that you don t want to use numercal ntegraton. Let U be a unformly dstrbuted random varable n the nterval [,) and u are random varates of the same dstrbuton. Consder the followng estmator. n ˆ θ = g u n = ( ) Also note that: E[ g ( U )] as n [ ] θ = g( u) du = E g ( U ) Strong Law of Large Numbers

11 Monte-Carlo Approach to Evaluatng Integrals Use Monte-Carlo smulaton to estmate the followng b ntegral. θ = g( x) dx a Let y=(x-a)/(b-a), then the ntegral becomes θ = ( b a) g ( b a) y+ a dy = h y dy What f θ = g( x) dx Use the substtuton y= /(+x), ( ) ( ) What f ( ) θ =... g x,..., x dx... dx n n Example: Estmate the value of π. (-,) Area of Crcle π = Area of square 4 (,) (-,-) (,-) SOLUTION Let, Y, be ndependent random varables unformly dstrbuted n the nterval [-,] The probablty that a pont (,Y) falls n the crcle s gven by 2 2 π Pr{ + Y } = 4 Generate N pars of unformly dstrbuted random varates (u,u 2 ) n the nterval [,). Transform them to become unform over the nterval [-,), usng (2u -,2u 2 -). Form the rato of the number of ponts that fall n the crcle over N

12 Dscrete Random Varates Suppose we would lke to generate a sequence of dscrete random varates accordng to a probablty mass functon N Pr = x = p, j =,,..., N, p = u { } Inverse Transform Method j j j j= x x f u p x f p u p + p M M = j j xj f p u p = = M M Dscrete Random Varate Algorthm (D-RNG-) Algorthm D-RNG- Generate u=u(,) If u<p, set =x, return; If u<p +p, set =x, return; Set =x n, return; Recall the requrement for effcent mplementaton, thus the above search algorthm should be made as effcent as possble! Example: Suppose that {,,,n} and p = p = = p n = /(n+), then = ( n + ) u

13 Dscrete Random Varate Algorthm Assume p =., p =.2, p 2 =.4, p 3 =.3. What s an effcent RNG? D-RNG-: Verson Generate u=u(,) If u<., set =x, return; If u<.3, set =x, return; If u<.7, set =x2, return; Set =x3, return; D-RNG-: Verson 2 Generate u=u(,) If u<.4, set =x2, return; If u<.7, set =x3, return; If u<.9, set =x, return; Set =x, return; More Effcent Geometrc Random Varables Let p the probablty of success and q=-p the probablty of falure, then s the tme of the frst success wth pmf Pr{ = } = pq Usng the prevous dscrete random varate algorthm, =j f j = j Pr { = } U < Pr{ = } = = j { } { } Pr = = Pr > j = q { j } = mn j: q < U { ( ) ( )} = mn j: jlog q < log U log ( U ) = mn j: j > log ( q) j As a result: j < q U q j j q < U q log ( U ) = + log ( q) j

14 Posson and Bnomal Dstrbutons Posson Dstrbuton wth rate λ. { } λ e λ Pr = =, =,,...! λ Note: p = p + + The bnomal dstrbuton (n,p) gves the number of successes n n trals gven that the probablty of success s p. n! n Pr { = } = p ( p), =,,..., n! ( n )! Note: p+ = n p p + p Accept/Reject Method (D-RNG-AR) Suppose that we would lke to generate random varates from a pmf {p j, j } and we have an effcent way to generate varates from a pmf {q j, j }. Let a constant c such that p j c for all j such that pj q > j In ths case, use the followng algorthm D-RNG-AR:. Generate a random varate Y from pmf {q j, j }. 2. Generate u=u(,) 3. If u< py/(cqy), set =Y and return; 4. Else repeat from Step.

15 Accept/Reject Method (D-RNG-AR) Show that the D-RNG-AR algorthm generates a random varate wth the requred pmf {p j, j }. p = Pr{ = } = Pr{ not stop for,..., k } Pr { = and stop at teraton k} k = { = k} = p p { = Y = } { Y = } { Y = } = = Pr and stop after Pr and s accepted Pr Y accepted Pr p /cq q Pr{ Not stop up to k } = Pr Y not accepted Y = j Pr Y = j j k k p j = q j = j cq j c Therefore k p p = = p c c k = { } { } q cq k c D-RNG-AR Example Determne an algorthm for generatng random varates for a random varable that take values,2,.., wth probabltes.,.2,.9,.8,.2,.,.9,.9,.,. respectvely. p c = max =.2 q D-RNG-: Generate u=u(,) k=; whle(u >cdf(k)) k=k+; x()=k; D-RNG-AR: u=u(,), u2=u(,) Y=floor(*u + ); whle(u2 > p(y)/c) u= U(,); u2=u(,); Y=floor(*rand + ); y()=y;

16 The Composton Approach Suppose that we have an effcent way to generate varates from two pmfs {q j, j } and {r j, j } Suppose that we would lke to generate random varates for a random varable havng pmf, a (,). { } Pr = j = p = aq + ( a) r, j j j j Let have pmf {q j, j } and 2 have pmf {r j, j } and defne wth probablty a = 2 wth probablty - a Algorthm D-RNG-C: Generate u=u(,) If u <= a generate Else generate 2 Contnuous Random Varates Inverse Transform Method Suppose we would lke to generate a sequence of contnuous random varates havng densty functon F (x) Algorthm C-RNG-: Let U be a random varable unformly dstrbuted n the nterval (,). For any contnuous dstrbuton functon, the random varate s gven by = F ( U ) F (x) u x

17 Example: Exponentally Dstrbuted Random Varable Suppose we would lke to generate a sequence of random varates havng densty functon Soluton f x ( x) = λe λ Fnd the cumulatve dstrbuton x λ y F ( x) = λe dy = e Let a unformly dstrbuted random varable u λx u = F ( x) = e ln ( u) = λx x = ln ( u) λ Equvalently, snce -u s also unformly dstrbuted n (,) x = ln ( u ) λ λx Convoluton Technques and the Erlang Dstrbuton Suppose the random varable s the sum of a number of ndependent dentcally dstrbuted random varables Algorthm C-RNG-Cv: n = Y = Generate Y,,Yn from the gven dstrbuton =Y+Y2+ +Yn. An example of such random varable s the Erlang wth order n whch s the sum of n d exponental random varables wth rate λ. n λx ( λx) e f ( x) = n!

18 Accept/Reject Method (C-RNG-AR) Suppose that we would lke to generate random varates from a pdf f (x) and we have an effcent way to generate varates from a pdf g (x). Let a constant c such that f ( x) for all g ( x) c x In ths case, use the followng algorthm C-RNG-AR:. Generate a random varate Y from densty g (x). 2. Generate u=u(,) 3. If u< f (Y)/(cg (Y)), set =Y and return; 4. Else repeat from Step. Accept/Reject Method (C-RNG-AR) The C-RNG-AR s smlar to the D-RNG-AR algorthm except the comparson step where rather than comparng the two probabltes we compare the values of the densty functons. Theorem The random varates generated by the Accept/Reject method have densty f (x). The number of teratons of the algorthm that are needed s a geometrc random varable wth mean c Note: The constant c s mportant snce s mples the number of teratons needed before a number s accepted, therefore t s requred that t s selected so that t has ts mnmum value.

19 C-RNG-AR Example Use the C-RNG-AR method to generate random varates that are normally dstrbuted wth mean and varance, N(,). Frst consder the pdf of the absolute value of. 2 2 x 2 f ( x) = e 2π We know how to generate exponentally dstrbuted random varates Y wth rate λ=. x g ( x) = e, x Y Determne c such that t s equal to the maxmum of the rato f ( ) 2 x 2 x x 2 = e 2e c = g ( x) 2π π Y C-RNG-AR Example C-RNG-AR for N(,): u=u(,), u2=u(,); Y= -log(u); whle(u2 > exp(-.5(y-)*(y-))) u= U(,); u2=u(,); Y= -log(u); u3= U(,); If u3 <.5 =Y; Else = -Y; Suppose we would lke Z~N(µ, σ 2 ), then Z : = σ + µ

20 Generatng a Homogeneous Posson Processes A homogenous Posson process s a sequence of ponts (events) where the nter-even tmes are exponentally dstrbuted wth rate λ (The Posson process wll be studed n detal durng later classes) Let t denote the th pont of a Posson process, then the algorthm for generatng the frst N ponts of the sequence {t, =,2,,N} s gven by Algorthm Posson-λ: k=, t(k)=; Whle k<n k= k+; Generate u=u(,) t(k)= t(k-) log(u)/lambda; Return t. Generatng a Non-Homogeneous Posson Processes Suppose that the process s non-homogeneous.e., the rate vares wth tme,.e., λ(t) λ, for all t<t. Let t denote the th pont of a Posson process, and τ the actual tme, then the algorthm for generatng the frst N ponts of the sequence {t, =,2,,N} s gven by Algorthm Thnnng Posson-λ: k=, t(k)=, tau= ; Whle k<n Generate u=u(,); tau= tau log(u)/lambda; Generate u2= U(,); If(lambda(tau)\lambda < u2) k= k+, t(k)= tau; Return t.

21 Generatng a Non-Homogeneous Posson Processes Agan, suppose that the process s non-homogeneous.e., the rate vares wth tme,.e., λ(t) λ, for all t<t but now we would lke to generate all ponts t drectly, wthout thnnng. Assumng that we are at pont t, then the queston that we need to answer s what s the cdf of S where S s the tme untl the next event s = { < = } = exp λ ( t + ) F () s Pr S s t t S { } y dy Thus, to smulate ths process, we start from t and generate S from F S to go to t =t +S. Then, from t, we generate S 2 from F S2 to go to t 2 =t +S 2 and so on. Example of Non-Homogeneous Posson Processes Suppose that λ(t)= /(t+α), t, for some postve constant a. Generate varates from ths non-homogeneous Posson process. Frst, let us determne the rate of the cdf s s s + t + a λ ( t ) + y dy = dy = log ( a+ t + y) t a + s+ t + a t + a s F = exp log S = t + a = s + t + a s+ t + a Invertng ths yelds ( t + a) u F = S u

22 Example of Non-Homogeneous Posson Processes Inverse Transform F = ( t ) + S a u u Thus we start from t = ( t + a) u au t = t + = u u t 2 t ( ) u 2 t + a t + au = t + = u u 2 ( ) 2 2 t + a u t + au = t + = u u The Composton Approach Suppose that we have an effcent way to generate varates from cdfs G (x),, G n (x). Suppose that we would lke to generate random varates for a random varable havng cdf n F ( x) = rg ( x), r =, r >, =,..., n = = n Algorthm C-RNG-C: Generate u=u(,) If u<p, get from G(x), return; If u<p +p 2, get from G2(x), return;

23 Polar Method for Generatng Normal Random Varates Let and Y be ndependent normally dstrbuted random varables wth zero mean and varance. Then the jont densty functon s gven by 2 2 x y ( x + y ) fy ( x, y) = e e = e 2π 2π 2π Y R θ Then make a varable change r x y θ = 2 2 = + arctan y x The new jont densty functon s Unform n the nterval [,2π] frθ (, r θ ) = e 2π 2 2 r Exponental wth rate /2 Polar Method for Generatng Normal Random Varates Algorthm C-RNG-N: Generate u=u(,), u2=u(,); R= -2*log(u); W= 2*p*u2; = sqrt(r) cos(w); Y= sqrt(r) sn(w); But, sne and cosne evaluatons are neffcent! Algorthm C-RNG-N2:. Generate u=u(,), u2=u(,); 2. Set V= 2*u-, V2= 2*u2-; (-,) 3. S=V*V+V2*V2; 4. If S >, Go to 5. R= sqrt(-2*log(s)/s); 6. = R*V; 7. Y= R*V2; (-,-) Generates 2 ndependent RVs (V,V 2 ) (,) (,-)

24 Smulaton of Dscrete Event Systems INITIALIZE STATE Update State x =f(x,e ) EVENT CALENDAR e t e 2 t 2 TIME Update Tme t =t CLOCK STRUCTURE RNG Delete Infeasble Events Add New Feasble Events Verfcaton of a Smulaton Program Standard debuggng technques Debug modules or subroutnes Create smple specal cases, where you know what to expect as an output from each of the modules Often choosng carefully the system parameters, the smulaton model can be evaluated analytcally. Create a trace whch keeps track of the state varables, the event lst and other varables.