Random numbers and Monte Carlo

Inmetro, 8/7/2003

João R. T. de Mello NetoIF - UFRJ

References

• The Nature of Mathematical Modeling, N. Gershenfelder, Cambridge, 1999;

• Numerical Recipes in C, Second Edition, W.H Press et al., Cambridge, 1992;

• Statistical Data Analysis, G. Cowan, Oxford, 1998• Computational Physics, Dean Karlen (online), 1998

Random

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Random numbers

Important task: generate random variables from known probability distributions.

random numbers: produced by the computer in a strictly deterministic way – pseudorandom (→ Monte Carlo Method)

random number generators: ),...,,( 111 xxxfx jjj

linear congruential generators: mcaxx jj mod)( 1

ex: c=3, a=5, m=16 00 x 3163051 mod)(x2163351 mod)(x

0,2,3,13,4,7,6,1,8,11,10,5,12,15,14,9,0,2….

0110 xxxx m

Generators

Arguments from number theory: good values for a, c and m.

Good generators: • longest possible period• individual elements within a period should follow each other “randomly”

Ex. 1 RANDU

a = 65539, m=231, c=0

311 265539 modnn xx

3112 296 mod)( ttt xxx

Generators

Generators

Generators

Generators

“Random numbers fall mainly in the planes” Marsaglia, Proc. Acad. Sci. 61, 25, 1968

What is seen in RANDU is present in any multiplicative congruential generator.

In 32 bit machines: maximum number of hyperplanes in the space of d-dimensions is:

d=3 2953d=4 566d=6 120d=10 41

RANDU has much less than the maximum!

Generators

Ex. 2 Minimal standard generator (Num. Recp. ran0)

a = 75 =16807, m=231-1 RAN1 and RAN2, given in the first edition, are “at best mediocre”

Generators

Ex. 3 RANLUXcernlib (mathlib V115, F.James )A portable high-quality random generator for lattice field theory simulation, M. Lüscher, Comp. Phys. Comm. 79, 100, 1994

period ≥ 10165

Recommendations:1. Do some simple tests.2. Check the results with other generator

Lots of literature about random generators testing. See a set of programs (Die Hard) http://stat.fsu.edu/~geo/diehard.html

Functional definition: an algorithm that generates uniform numbers in acceptable if it is not rejected by a set of tests.

Inverse transform

p(x) uniform probability distribution

otherwise 0

10 xdxdxxp )(

x: uniform deviate Generate y according to g(y).

dxxpdyyg )()( dy

dxxpyg )()( )(yg

dy

dx

y

dyygx0

// )( )(yGx )(xGy 1

0

1

x

y

uniformdeviate in

out

analyticallyor numerically.

G(y)

g(y)

Inverse transform

1

0.5

0 1 2 x

F(x)

3

Ex. 4 discrete case

f(x=0)=0.3f(x=1)=0.2f(x=2)=0.5 u

0.0 ≤ u ≤ 0.3 x=00.3 ≤ u ≤ 0.5 x=10.5 ≤ u ≤ 1.0 x=2

for 2000 deviates:x=0 0.2880X=1 0.2075X=2 0.5045

Inverse transform

Exponential• amount of time until a specific event occurs;• time between independent events (time until a part fails);

0 0, x, xexf );( xexF 1)(

)(xFu

ue x 1

)log( ux 11

ux log1

Acceptance-rejection method

ymax

0

f(x)

xmin xmax

minminmax )( xuxxx

uyy max

if (x,y) under the curve, accept the point, else, discard.

pN

n

A

dxxf

E acc

x

xf

max

min

)(maxminmax )( yxxA

AN

nI acc

N

ApNpI )( 1

p

p

NI

I 11

NI

I 1

Acceptance-rejection method

Importance sampling

g(x): random numbers are easy to generate

g(x)

f(x)

•generate random x according to g(x)• generate u uniformily between 0 and g(x);• if u < f(x), accept x, if not, reject.

Multivariate Normal

Very useful

Starts with a d-dimensional vector of standard normal Random numbers;

Transformed to the desired distribution using:

zRx T

z

dX1 vector of standard normal random numbers

dX1 vector representing the mean

R dXd matrix such that RRT covariancematrix desired

(Cholesky factorization of the cov. matrix)

Multivariate Normal

170

701

.

.

3 2

Random walk

10 simulations1K points each