Simulation and Monte Carlo integration


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1 Chapter 3 Simulatio ad Mote Carlo itegratio I this chapter we itroduce the cocept of geeratig observatios from a specified distributio or sample, which is ofte called Mote Carlo geeratio. The ame of Mote Carlo was applied to a class of mathematical methods first by scietists workig o the developmet of uclear weapos i Los Alamos i the 1940s. For history of Mote Carlo see Kalos ad Whitelock (1986), Mote Carlo Methods, Vol. I: Basics, Wiley. 3.1 The law of large umbers There are two types of laws of large umbers, a strog oe ad a weak oe. They may be thought of as the theoretical backgroud for Mote Carlo methods. Below is a strog law of large umbers. Theorem 1. If X 1, X 2,..., X,... are idepedet ad idetically distributed radom variables with EX k µ, k 1, 2,..., the ( k1 P lim X ) k µ 1. Proof. See page 85 of Billigsley, P. (1995), Probability ad Measure, 3rd editio, Wiley. The ext oe is a weak law of large umbers. Theorem 2. If X 1, X 2,..., X,... are idepedet ad idetically distributed radom variables with EX k µ ad E(X k µ) 2 σ 2, k 1, 2,..., the, for ay positive costat ε, ( lim P k1 X ) k µ ε 0. Proof. It is ot so hard, ad is left as a homework. 41
2 42 CHAPTER 3. SIMULATION AND MONTE CARLO INTEGRATION 3.2 The hitormiss method Cosider a evet A whose probability is p. Defie a sequece of idepedet ad idetically distributed Beroulli radom variables X 1, X 2,..., X by 1, if evet A occurs, X k k 1, 2,... 0, otherwise, You may imagie {X 1 as hit ad {X 0 as miss. The momet estimator ad maximum likelihood estimator of p based o X 1, X 2,..., X are the same, with k1 ˆp X k. This estimator is ubiased, that is, E ˆp p. More importatly, it follows from the strog law of large umbers that ( k1 P lim X ) k p 1. The earliest documeted use of radom samplig to fid the solutio to a itegral seems to be that of Comte de Buffo, which is ofte called Buffo s eedle problem. Here we estimate π by the hitor miss method. Example 1 (Estimatio of π). Suppose that a disk with radius 1 meter is put iside a square whose legth is 2 meters. We toss a little eedle to hit the square. The questio is: what is the probability that the eedle lies withi the disk? The aswer is π 4. To perform the Mote Carlo method, let U 1 ad U 2 be idepedet radom variables uiformly distributed o the iterval [ 1, 1], ad they may be treated as two sides of a square. Defie a radom variable that represets the evet of hittig the disk, { 1, if U 2 X 1 + U2 2 1, 0, otherwise. The EX P (U U 2 2 1) π 4. For a sequece of such i.i.d. variables, X 1,..., X, by the strog law of large umbers we obtai with probability 1. lim k1 X k π 4,
3 3.3. MONTE CARLO INTEGRATION 43 pimcfuctio(){ # is the umber of simulatios u1ruif(, 1, 1) # geerate a uiform variable U_1 u2ruif(, 1, 1) # U_2 xrep(0, ) # format of X x[u1^2 + u2^2 <1]1 # X variable pimc4*mea(x) # pi estimator pimc > pimc(10) [1] 4 > pimc(100) [1] 3.28 > pimc(1000) [1] 3.06 > pimc(10000) [1] > pimc(100000) [1] > pimc( ) [1] Mote Carlo itegratio Suppose that g(x), x [0, 1], is a real ad cotiuous fuctio. The questio is: how to estimate the itegral g(x)dx? There may be may approximatios for this itegral. Here we use the Mote Carlo method. 1 0 Example 2 (Estimatio of π). Cosider the fuctio It is easy to show that 1 0 g(u)du π. g(u) 4 1 u 2, u [0, 1]. To use the Mote Carlo method, suppose that U 1, U 2,..., U are idepedet radom variables uiformly distributed o the iterval [0, 1]. The X 1 g(u 1 ), X 2 g(u 2 ),..., X g(u ) are i.i.d radom variables, with mea EX k Eg(U k ) 1 0 g(u)du π. pimcfuctio(){ # is the umber of simulatios uruif() # geerate a sequece of uiform U radom umbers g4*sqrt(1u^2) # g(u) pimcmea(g) # pi estimator pimc
4 44 CHAPTER 3. SIMULATION AND MONTE CARLO INTEGRATION 3.4 Geeratig radom umbers This sectio describes some approaches to geerate radom variables or vectors with specified distributios. Whe talkig about geerate a radom object, we mea a algorithm whose output is a object of the desired type. To geerate a radom variable (or vector) havig a target distributio F (x) o R d, we typically start from a sequece of i.i.d. uiform radom umbers. Thus, our task is: give a i.i.d. sequece U 1, U 2,... that follow a uiform distributio o the iterval [0, 1], fid m d ad a determiistic fuctio g : [0, 1] m R d such that the distributio fuctio of g(u 1,..., U m ) is F (x) The iverse method Theorem. Let X be a radom variable with distributio fuctio F (x). (1) For every t [0, 1], P (F (X) t) t. I particular, if F (x) is cotiuous o the real lie, the U F (X) is uiformly distributed o [0, 1]. (2) Defie the iverse of F as F 1 (y) if{x, F (x) y, 0 < y < 1. If U is a uiform radom variable o [0, 1], the F 1 (U) has distributio fuctio F (x). Example 1. A expoetial desity fuctio is of the form { λ exp( λx), x > 0, f(x) 0, x 0, where λ is a positive costat. The cumulative distributio fuctio { 1 exp( λx), x > 0, F (x) 0, x 0, is strictly icreasig ad cotiuous o [0, ), with the iverse F 1 (x) 1 l(1 x), 0 < x < 1. λ Thus, we could use the followig procedure to geerate expoetial radom umbers. rexp fuctio(, lambda){ u ruif() # geerate uiform radom umbers g 1/lambda*log(1u) # expoetial radom umbers g
5 3.4. GENERATING RANDOM NUMBERS 45 Oe way to see whether a data set comes from a particular distributio is the qq plot, which is based o sample order statistics (or sample quatiles) ad theoretical quatiles. For a set of observatios x 1, x 2,..., x, we put them i order, x 1: x 2: x :. ( ) k If the target distributio fuctio is F (x), the the quatile for +1 is F 1 k +1, k 1,...,. A qq plot is a scatterplot of poits ( ( )) k x k:, F 1, k 1,...,. + 1 These poits would be close to a straight lie if the data set actually comes from F ( ). Below is a example for a expoetial case. x rexp(50, 1) # geerate 50 expoetial radom umbers sq sort(x) # sample quatiles tq fuctio(u){log(1u) # expoetial quatile fuctio with lambda1 plot(sq, tq(1:50/51), xlab"sample quatiles", ylab"theoretical quatiles") The ext example is for geeratig ormal observatios, but ot usig the iverse method. Example 2 (Geeratig ormal observatios). To simulate ormal radom variables, Box ad Muller (1958) suggested the followig procedure. Let U 1 ad U 2 be idepedet radom variables uiformly distributed o the iterval (0, 1). Defie X 1 ad X 2 by X 1 2 l U 1 cos(2πu 2 ), X 2 2 l U 1 si(2πu 2 ). It ca be show that X 1 ad X 2 are idepedet stadard ormal radom variables. rormalfuctio(){ u1ruif() u2ruif() x (2*log(u1))^(1/2)*cos(2*pi*u2) x qqorm(rormal(100)) qqlie(rormal(100)) Rejectio method We wat to geerate radom umbers from a target distributio with the desity f(x).
6 46 CHAPTER 3. SIMULATION AND MONTE CARLO INTEGRATION Suppose that f(x) ca be rewritte as f(x) h(x)g(x), x R, h(u)h(u)du where h(x) is a oegative fuctio, ad g(x) is aother desity fuctio that has a simple form. The rejectio procedure is: Step 1. Geerate Y g(x). Step 2. Geerate U uiform(0, 1). Step 3. Accept Y is U h(y ) 1, otherwise go back to Step 1. There are two discrete evets ivolved i the above procedure: a success (Y is accepted), ad o success (Y is rejected). To cofirm the procedure makes sese, we eed evaluate the coditioal probability P (Y x success). Note that P (Y xad success) P (Y x, U h(y ) 1) (joit desity of Y ad U) dydu y x y x x y x,u h(y) 1 y x,u h(y) 1 ( du u h(y) 1 h(y)g(y)dy g(y) 1dydu ) g(y)dy h(y)g(y)dy, x R, from which we obtai P (success) lim P (Y x ad success) h(y)g(y)dy. x Thus, P (Y x success) P (Y x ad success) P (success) x h(y)g(y)dy, x R. h(y)g(y)dy This meas the above procedure works theoretically.
7 3.4. GENERATING RANDOM NUMBERS 47 Example 1. Let f(x) 4 π 1 1+x 2, 0 < x < 1, 0, otherwise. We will use the rejectio method to geerate radom umbers from this distributio. Choose g(x) to be a uiform desity, 1, 0 < x < 1, g(x) 0, otherwise, ad h(x) f(x). The, the algorithm is re fuctio(){ hfuctio(x){4/pi*(1+x^2)^(1) y ruif() u ruif() re y[u < h(y) & h(y) <1] #if(u>h(y)) repeat re > re(100) [1] [8] [15] [22] [29] [36] [43] Example 2. Cosider the followig algorithm that was proposed by Marsaglia ad Bray 1 : (1) Geerate U ad V idepedet uiform ( 1, 1) radom variables. (2) Set W U 2 + V 2. (3) If W > 1, go to Step (1). (4) Set Z ( 2 l W )/W, ad let X 1 UZ ad X 2 V Z. Show that the radom variables X 1 ad X 2 are idepedet ad ormally distributed with mea zero ad variace 1. Use this algorithm to geerate 100 ormal radom umbers, ad obtai a ormal QQ plot. 1 Marsaglia ad Bray (1964), A coveiet method for geeratig ormal variables, SIAM Review, vol. 6,
8 48 CHAPTER 3. SIMULATION AND MONTE CARLO INTEGRATION 3.5 Importace samplig Suppose that we have a dimesioal itegral I g(x)f(x)dx R that eeds to evaluate, where f(x) is a desity fuctio. The Mote Carlo procedure is to take a radom sample X 1,..., X from this distributio, ad the to form the mea g 1 g(x k ). Whe R g 2 (x)f(x)dx <, the strog large umber law asserts that with probability 1. Sice g estimates I, we ca write 1 k1 g(x k ) I, k1 g I + error. This error is a radom variable, with mea 0 ad variace var(g ) 1 { g 2 (x)f(x)dx I 2. R What we wat is to reduce the error (variace). But, f(x) is ot ecessarily the best desity fuctio to use i the Mote Carlo simulatio eve though it appears i the itegrad. A differet desity fuctio, f(x), ca be itroduced ito the itegral as follows: g(x)f(x) I f(x)dx, R f(x) where f(x) 0, f(x)dx R 1, ad g(x)f(x) < except for perhaps a (coutable) set of poits. How to f(x) choose a appropriate f is a trick i importace samplig.
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