General Principles in Random Variates Generation

Transcription

1 General Principles in Random Variates Generation E. Moulines and G. Fort Telecom ParisTech June 2015 Bibliography : Luc Devroye, Non-Uniform Random Variate Generator, Springer-Verlag (1986) available on the webpage of the author http ://luc.devroye.org/books-luc.html

2 Outline Uniform Random number generators Pesudo-Random sequences Inversion Methods for distributions on R Cumulative distribution function Algorithm Sampling real valued random variables When F 1 (u) is not explicit Simulation of Gaussian variables Box-Müller Marsaglia-Bray Inversion method The rejection method (on R d ) Sampling under a curve Conditional distribution Algorithm Design parameters Example 1 : a normal generator based on the rejection algorithm Example 2 : Gaussian from Cauchy

3 Uniform Random number generators Uniform Random number generators Pesudo-Random sequences Inversion Methods for distributions on R Cumulative distribution function Algorithm Sampling real valued random variables When F 1 (u) is not explicit Simulation of Gaussian variables Box-Müller Marsaglia-Bray Inversion method The rejection method (on R d ) Sampling under a curve Conditional distribution Algorithm Design parameters Example 1 : a normal generator based on the rejection algorithm Example 2 : Gaussian from Cauchy

4 Uniform Random number generators Pesudo-Random sequences Pseudo-random sequences (1/3) The key ingredient for Monte Carlo methods is a generator of random number. We will see in this lesson that the generator of uniform random variable on [0,1] is a central tool for sampling more general distributions. Therefore, how to sample from a uniform distribution on [0, 1]? i.e. how to obtain a sequence of numbers u 1,, u n, that can be considered as a path of the random sequence (U 1,, U n, ) where the r.v. (U n) n are independent and with the same distribution U([0, 1]). Well, we are only able to produce a sequence (u 1,, u n, ) from a machine which is a pseudo random sequence.

5 Uniform Random number generators Pesudo-Random sequences Pseudo-random sequences (2/3) Let us start with the question of sampling a sequence of {0, 1}-valued r.v. with a (fair) coin. Examples (L = 12) All of these sequences of length L occur with probability 1/2 L. But are they random?

6 Uniform Random number generators Pesudo-Random sequences Pseudo-random sequences (2/3) Let us start with the question of sampling a sequence of {0, 1}-valued r.v. with a (fair) coin. Examples (L = 12) All of these sequences of length L occur with probability 1/2 L. But are they random? Champernowne sequence : }{{ 0 } }{{} 3 1 } {{ 0 0 } 4 1 } {{ 0 1 } 5

7 Uniform Random number generators Pesudo-Random sequences Pseudo-random sequences (3/3) For any length L and any binary motif (a 1,, a L), the Law of Large Numbers for i.i.d. r.v. implies 1 n n k=1 1 (a1,,a L )(U (k 1)L+1,, U kl ) n 1 2 L w.p.1 A sequence (u n) n is said -uniform if for any L 1 and any (a 1,, a L) {0, 1} L, 1 n n k=1 1 (a1,,a L )(u (k 1)L+1,, u kl ) 1 2 L It can be proved that the Champernowne sequence is -uniform...

8 Uniform Random number generators Pesudo-Random sequences Pseudo-random generator The definition of a generator consists in a finite sequence of states (x 1,, x M ), a mapping T : x k+1 = T (x k ) a mapping S : u k = S(x k ) an output sequence (u n) n. The output sequence is called the random sequence produced by this generator. Example : The best known and still most widely used generators are the simple linear congruential generators. x 0 (seed) x k = (ax k 1 + b) modm u k = x k M The properties (e.g. cycles) of this generator depends on a, b, M (M = 2 32 )

9 Uniform Random number generators Pesudo-Random sequences Bibliography More complex generators exist, with improved properties. See e.g. Cours d E. Moulines, sur le site pédagogique du cours MDI 345, rubriques : Introduction aux méthodes de Monte Carlo et à la Simulation Méthodes de simulation de v.a. uniformes Cours de B. Ycart méthodes de Monte Carlo, http ://ljk.imag.fr/membres/bernard.ycart/polys/polys.html Livre de S. Asmussen and P.W. Glynn Stochastic Simulation, Springer. Chapter II section 2.1. Livre de P. Glasserman Monte Carlo methods in Financial Engineering, Springer. Chapter II section 2.1.

10 Inversion Methods for distributions on R Uniform Random number generators Pesudo-Random sequences Inversion Methods for distributions on R Cumulative distribution function Algorithm Sampling real valued random variables When F 1 (u) is not explicit Simulation of Gaussian variables Box-Müller Marsaglia-Bray Inversion method The rejection method (on R d ) Sampling under a curve Conditional distribution Algorithm Design parameters Example 1 : a normal generator based on the rejection algorithm Example 2 : Gaussian from Cauchy

11 Inversion Methods for distributions on R Cumulative distribution function Cumulative distribution function Let F be a cumulative distribution function on R, F (x) = P(X x). Properties of F : The function x F (x) is non-decreasing. At every point, F has left limits lim F (y) y x exists F is continuous to the right For any x R, F (x) = lim y x + F (y) F (x + ) F (x ) = P(X = x).

12 General Principles in Random Variates Generation Inversion Methods for distributions on R Cumulative distribution function The quantile function Here are examples of cumulative distribution functions F : F is not necessarily invertible. The generalized inverse of the function F, denoted F 1, is defined on (0, 1) by F 1 (p) = inf{y R, F (y) p}, 0 < p < 1. This function is called the quantile function. On the figures above : [left] F 1 : (0, 1) R ; [center] F 1 : (0, 1) (0, ) ; [right] F 1 : (0, 1) (2, 8)

13 Inversion Methods for distributions on R Cumulative distribution function Properties of the quantile function F 1 (p) = inf{y R, F (y) p}, 0 < p < 1. F 1 is a proper inverse if and only if F is continuous and strictly increasing. Properties : F 1 is nondecreasing. F F 1 (p) p ; equality can fail only if F is discontinuous at F 1 (p). F 1 F (x) x ; equality fails iff x is in the interior or at the right end of a flat of F. F 1 (p) x if and only if p F (x).

14 Inversion Methods for distributions on R Cumulative distribution function 1 p 0.8 p=f(x) F(F 1 (p)) p x F 1 (p) F 1 (p) =F 1 (F(x)) F 1 (p)

15 Inversion Methods for distributions on R Algorithm Simulation using the quantile transformation Theorem (Inverse CDF method) Let F be a cumulative distribution function and U U([0, 1]). Then the cumulative distribution function of the r.v. F 1 (U) is F. This is called the inverse transform sampling method (also known as the inverse probability integral transform). The interest for this method stems from the fact that many programming languages have the ability to generate pseudorandom numbers which are (almost) distributed according to i.i.d. standard uniform r.v. see the first section of these slides, anf refs therein Theorem If the cumulative distribution function F of the r.v. X is continuous, then F (X) U([0, 1]). This is called the probability integral transformation

16 Inversion Methods for distributions on R Sampling real valued random variables Sampling discrete random variables Case 1 : X takes values in {a 1,, a n} with probabilities p 1,, p n 0 if x < a 1 F (x) = p p j 1 if a j 1 x < a j 1 if x a n and for any p (0, 1), { F 1 a1 if 0 < p p 1 (p) = a j ifp p j 1 < p p p j

17 Inversion Methods for distributions on R Sampling real valued random variables Sampling discrete random variables Case 1 : X takes values in {a 1,, a n} with probabilities p 1,, p n 0 if x < a 1 F (x) = p p j 1 if a j 1 x < a j 1 if x a n and for any p (0, 1), { F 1 a1 if 0 < p p 1 (p) = a j ifp p j 1 < p p p j Hence, the algorithm is Draw U U([0, 1]) Find J {1,, n} such that p p J 1 < U p p J. Return a J.

18 Inversion Methods for distributions on R Sampling real valued random variables Case 2 : X takes values in the countable set {a 1,, a j, } with probabilities p 1,, p j,. The algorithm is Draw U U([0, 1]) Find J 1 such that p p J 1 < U p p J. Return a J. Remark : strategies have ben developed to solve efficiently find the index J such that p p J 1 < U p p J (see TD and the book Non-uniform Random Variate Generation by L. Devroye)

19 Inversion Methods for distributions on R Sampling real valued random variables Case 3 : Sampling random variable with continuous (and explicit) cdf support f F (x) X = F 1 (U) Exponential x > 0 λe λx 1 e λx λ 1 log(u) σ 1 Cauchy R + 1 atan ( ) x σtan(πu) π(x 2 +σ ) 2 π σ x Rayleigh R σ e x2 /2σ 2 1 e x2 /2σ 2 σ 2 log(u) Pareto x b > 0 1 ( b x ab a x a+1 ) a b U 1/a The cdf of a Gaussian random variable does not have an explicit expression. We will see later how to sample such a distribution from a generator of uniform random variables.

20 Inversion Methods for distributions on R When F 1 (u) is not explicit When F 1 is not explicit (1/2) The inversion method is exact when an explicit form of F 1 is known. In other cases, we must solve the equation F (x) = p and this requires an infinite amount of time if F is continuous. Any stopping rule that we use with the numerical method leads necessarily to an inexact algorithm...

21 Inversion Methods for distributions on R When F 1 (u) is not explicit When F 1 is not explicit (2/2) : The Bissection Method When F 1 (u) is not explicit, the solution x such that F (x) = u can be numerically approximated by the Bissection method Algorithm : Find an initial interval [a, b] to which the solution belongs repeat X (a + b)/2 if F (X) u then a X else b X end if until b a 2δ Return X Rmk : This algorithm may never work if, for fixed u, there is an interval of solution to F (x) = u. Nevertheless, it can be proved that the set is of null Lebesgue-measure. {u (0, 1) : x < y s.t.f (x) = F (y) = u}

22 Simulation of Gaussian variables Uniform Random number generators Pesudo-Random sequences Inversion Methods for distributions on R Cumulative distribution function Algorithm Sampling real valued random variables When F 1 (u) is not explicit Simulation of Gaussian variables Box-Müller Marsaglia-Bray Inversion method The rejection method (on R d ) Sampling under a curve Conditional distribution Algorithm Design parameters Example 1 : a normal generator based on the rejection algorithm Example 2 : Gaussian from Cauchy

23 Simulation of Gaussian variables Box-Müller Box-Müller algorithm (1/2) Set Z 1, Z 2, R, Θ such that Z 1 = R cos(θ) Z 2 = R sin(θ) The algorithm is based on the following property : (Z 1, Z 2) N (0, I 2) iff (R, Θ) are independent random variables, R has the distribution of the square root of a chi-square with two degrees of freedom or, equivalently, R is a Rayleigh distribution with parameter σ = 1 Θ has the uniform distribution [0, 2π].

24 Simulation of Gaussian variables Box-Müller Box-Müller algorithm (1/2) Set Z 1, Z 2, R, Θ such that Z 1 = R cos(θ) Z 2 = R sin(θ) The algorithm is based on the following property : (Z 1, Z 2) N (0, I 2) iff (R, Θ) are independent random variables, R has the distribution of the square root of a chi-square with two degrees of freedom or, equivalently, R is a Rayleigh distribution with parameter σ = 1 Θ has the uniform distribution [0, 2π]. Algorithm Generate a pair (U 1, U 2) of independent variables uniform on [0, 1]. R 2 log(u 1) V 2πU 2 Z 1 R cos(v ), Z 2 R sin(v ) Return Z 1, Z 2.

25 Simulation of Gaussian variables Box-Müller Box-Müller algorithm (2/2) Show that 1 R is a Rayleigh distribution. 2 V is a uniform distribution on [0, 2π]. 3 R and V are independent. and prove that (Z 1, Z 2) are independent r.v. with distribution N (0, 1). The main drawback of this method is that it requires the use of the sinus and cosinus functions, which can be expensive to evaluate. The following method is a way to overcome this problem.

26 Simulation of Gaussian variables Marsaglia-Bray Marsaglia-Bray algorithm (1/2) repeat Sample two independent uniform [0, 1] random variables U 1 and U 2 U 1 2U 1 1, U 2 2U 2 1 until U1 2 + U2 2 1 Y 2 log(u1 2 + U 2 2) U Z 1 Y 1 U, Z U 2 1 +U2 2 2 Y 2. U 2 1 +U2 2 Return Z 1, Z 2.

27 Simulation of Gaussian variables Marsaglia-Bray Marsaglia-Bray algorithm (2/2) Set T = U 1 U U 2 2 W = U U What is the distribution of (U 1, U 2) at the end of the repeat/until loop? 2 Show that T and W are independent, W is a uniform distribution on [0, 1] and T has the same distribution as cos(θ) when Θ U([0, 1]). 3 By using the result of the Box-Müller algorithm, show that Z 1, Z 2 are independent r.v. with normal distribution N (0, 1). 4 What is the acceptance probability of the repeat/until test? rmk : π/ This method substitutes the call to the sinus, cosinus functions for the following algorithm : instead of computing (cos(θ), sin(θ)) from Θ, a point on the unit ball is drawn, then projected on the unit sphere and its cartesian coordinates are then considered.

28 Simulation of Gaussian variables Inversion method Inversion Method The cumulative distribution function Φ of a standard Gaussian random variable satisfies Φ 1 (1 u) = Φ 1 (u), 0 < u < 1, and it suffices to approximate Φ 1 on [1/2, 1]. We may approximate Φ 1 using a rational function 3 Φ 1 n=0 an(u 1/2)2n+1 (u) 3 n=0 bn(u, 1/2)2n where 0.5 u 0.92 with a precision better than 10e 5 using the value computed in Springer Beasley (1977) For 0.92 u 1, Moro suggest to approximate Φ 1 using Φ 1 (u) 8 c n [log (log(1 u))] n, 0.92 u 1, n=0

29 The rejection method (on R d ) Uniform Random number generators Pesudo-Random sequences Inversion Methods for distributions on R Cumulative distribution function Algorithm Sampling real valued random variables When F 1 (u) is not explicit Simulation of Gaussian variables Box-Müller Marsaglia-Bray Inversion method The rejection method (on R d ) Sampling under a curve Conditional distribution Algorithm Design parameters Example 1 : a normal generator based on the rejection algorithm Example 2 : Gaussian from Cauchy

30 The rejection method (on R d ) Sampling under a curve Tools 1 and 2 : Sampling under the curve Theorem (i) Let X be a random vector with density g on R d, and let U be a uniform r.v. on [0, 1], independent of X. Let c > 0 be an arbitrary constant. Then, (X, cug(x)) is uniformly distributed on the set A = {(x, v) : x R d, 0 v cg(x)}. (ii) Conversely, if (X, V ) is a random vector in R d+1 uniformly distributed on this set A, then X has density g on R d Function x > cg (x) 0.8 Function x > g(x) Point with coordinates: (X, c g(x)) X

31 The rejection method (on R d ) Conditional distribution Tool 3 : First draw hitting a subset (1/2) Theorem Let X 1, X 2,... be a sequence of i.i.d. random vectors taking values in A R d, and let B A be a Borel set such that P(X 1 B) = p > 0. Let Y be the first r.v. X i taking values in A. Then (i) The distribution of Y is given by P(Y A) = P(X 1 A X 1 B) (ii) Let I = inf{i 1, X i B}. Then I is a geometric r.v. with parameter p i.e. P(I = k) = (1 p) k 1 p k 1.

32 The rejection method (on R d ) Conditional distribution Tool 3 : First draw hitting a subset (2/2) repeat Draw i.i.d. samples X k with distribution π until X k B Return X k. 1 If π is the uniform distribution on A, then the algorithm returns a sample with a uniform distribution on B. 2 The number of loops to return a value is a Geometric random variable with parameter p (and expected value 1/p).

33 The rejection method (on R d ) Algorithm Rejection Method for sampling under the distribution f on X R d below : either f, g are densities w.r.t. Lebesgue on R d ; or f, g are probability mass functions on an (at most) countable set X of R d The basic version of the rejection algorithm assumes the existence of a distribution g on R d and the knowledge of a constant c 1 such that f(x) cg(x) for all x X Algorithm : repeat Generate independently X with distribution g and U U([0, 1]) until U f(x) cg(x) Return X. Key result : The distribution of the r.v. X is f.

34 The rejection method (on R d ) Design parameters On the design parameters The three things we need before running the rejection algorithm are 1 a dominating distribution g 2 a simple method for generating random variates with distribution g 3 knowledge of c. Basically g must have heavier tails and sharper peaks than f. The dominating measure should be chosen with care! The number of iterations in order to return a value X is a Geometric r.v. with parameter 1/c : we should keep c as small as possible!

35 The rejection method (on R d ) Design parameters Development of good rejection algorithms Generally speaking, g is chosen from a class of classical densities. This class includes the uniform density, triangular densities and most densities that can be generated quickly by the inversion method. One generally starts with a family of dominating densities g (say a parametric family {g θ, θ Θ}) and chooses the density within this class for which c is the smallest. This approach sometimes leads to some difficult optimization problem.

36 The rejection method (on R d ) Example 1 : a normal generator based on the rejection algorithm Example 1 : A normal generator by rejection from the Laplace density The Laplace density is given by g(x) exp( x ) x R and the Normal density is given by f(x) = 1 2π exp( 0.5 x 2 ) x R Show that the following algorithm is a normal generator by rejection from the Laplace density repeat Generate independently an exponential random variate X (with parameter λ = 1) two r.v. U and V with distribution U([0, 1]). If U < 1/2 set X X. until V e 1/2 X e X2 /2 return X

37 The rejection method (on R d ) Example 2 : Gaussian from Cauchy Example 2 : Gaussian from Cauchy (1/2) We want to sample a standard Gaussian distribution on R. The family of dominating densities is the Cauchy family with scale parameter θ g θ (x) = θ 1 π θ 2 + x. 2 There is no need to consider a translation parameter as well because both f and g θ are unimodal with peak at 0. The optimal rejection constant is defined by It is given by c θ = c θ = sup x { 2π eθ eθ2 /2 f(x) g θ (x). θ < 2 θ π/2 θ 2 The function c θ has only one minimum at θ = 1, and the minimal value is c 1 = 2π/e.

38 The rejection method (on R d ) Example 2 : Gaussian from Cauchy An example : Gaussian from Cauchy (2/2) Show that the following algorithm is a normal generator by rejection from the Cauchy distribution. set α e/2 repeat Sample independently the r.v. U and V with uniform distribution on [0, 1]. Set X tan(πv ) until U α(1 + X 2 )e X2 /2 Return X The rejection constant is near 1.52 ; it is no match for most normal generators developed further on.