UNIFORM RANDOM NUMBER GENERATION

The goal is to generate a sequence of Uniformly distributed Independent X1, X2, …. IID Unif[0, 1]

These are the basis of generating all random variates in simulations

UNIFORM [0,1]

0 1

1

f(x)

x

Every real number beteween 0 and 1 is equally likely…

EARLY METHODS USED PHYSICAL PHENOMENON

Spinning disks with digits on them

Dice (generate on 1/6, 2/6, …, 1)

Picking people out of the phone book

Picking the nth (n = 1200, 1201, …) digit in p

Gamma ray photon counting

SEQUENCIAL COMPUTER-BASED METHODS

Antique Mid-Square Method (Von Neumann)

Z0 is a four-digit number

Z1 = middle four digits of Z02

Z2 = middle four digits of Z1

Ui = Zi/10000

SAMPLE MID-SQUARE

i Zi Zi^2 Ui

0 7182 51581124  

1 5811 33767721 0.5811

2 7677 58936329 0.7677

3 9363 87665769 0.9363

4 6657 44315649 0.6657

5 3156 9960336 0.3156

MID-SQUARE PROPERTIES

Once you know a Z, you can accurately predict the entire future sequence (not really random)

If a Z ever repeats, the whole sequence starts over

Only generate 9999 numbers (not dense in [0, 1]

Turns out, we LIKE some of these properties for computer simulations (Repeatability enables debugging)

DESIRABLE PROPERTIES

Appear Unif[0, 1]

Fast algorithm

Reproducible stream Debugging

Contrast in comparison (Variance Reduction)

Lots of numbers before a repeat

PRELIM: MODULUS ARITHMETIC

mod is a mathematical operator producing a result from two inputs (+, x,

^)

mod == “remainder upon division” 10 mod 6 = 4

12 mod 6 = 0

68 mod 14 = 56 mod 14 + 12 mod 14 = 12

LINEAR CONGRUENTIAL GENERATOR (Lehmer, 1954)

Z0 is the SEED

m is the MODULUS

a is the MULTIPLIER

c is the INCREMENT (forget this one)

mcaZZ ii mod)( 1

m

ZU ii

U’s in (0, 1)

PROPERTIES

Can generate at most m-1 samples before repeat Length of non-repeating sequence called the

PERIOD of the generator

If you get m-1, you have a full cycle generator

Divides [0, 1] into m equal slices

unif.xls

observe the basic functions of the seed, multiplier, and modulus

experiment with multipliers for m = 17

m = 16

HISTORY Necessary for full cycle

a and m relatively prime

q (prime) divides m and a-1

m = 2,147,483,648 = 231-1 is everybody’s favorite 32-bit generator (SIMAN, SIMSCRIPT, GPSS/H, Arena)

Fishman, G. S. (1972). An Exhaustive Study of Multipliers for Modulus 231-1, RAND Technical Series. a = 630,360,647

HISTORICAL COMPETITION FOR LINEAR CONGRUENTIAL

GENERATORS

add or multiply some combination of variates from the stream’s recent history

mZaZaZaZ niniii mod)...( 2211

WHAT IS GOOD

Full, Long Cycle

Seemingly Independent We can test this, but our simple tests

stink

c2 Test for U[0, 1]

U1, U2, ...Un a sequence of U[0, 1] samples

Let us divide [0, 1] into k equally-sized intervals

Let oi = observations in [(i-1)/k, i/k]

ei = n/k is the expected number of Ui’s

that fall in [(i-1)/k, i/k] for each i

i

iin e

eo 221

)(

TESTING

cn-12 follows the Chi-Squared distribution

with n-1 degrees of freedom

DUMB TEST any full-cycle generator is exactly AOK

expand to two or more dimensions using n-

tuples (Ui, Ui+1, ..., Ui+n)

maybe a picture would be better?