1 / 16

UNIFORM RANDOM NUMBER GENERATION

UNIFORM RANDOM NUMBER GENERATION. The goal is to generate a sequence of Uniformly distributed Independent X 1 , X 2 , …. IID Unif[0, 1] These are the basis of generating all random variates in simulations. UNIFORM [0,1]. f(x). Every real number beteween 0 and 1 is equally likely…. 1.

ada
Télécharger la présentation

UNIFORM RANDOM NUMBER GENERATION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. UNIFORM [0,1] f(x) Every real number beteween 0 and 1 is equally likely… 1 x 1 0

  3. 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

  4. 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

  5. SAMPLE MID-SQUARE

  6. 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)

  7. DESIRABLE PROPERTIES • Appear Unif[0, 1] • Fast algorithm • Reproducible stream • Debugging • Contrast in comparison (Variance Reduction) • Lots of numbers before a repeat

  8. 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

  9. LINEAR CONGRUENTIAL GENERATOR (Lehmer, 1954) • Z0 is the SEED • m is the MODULUS • a is the MULTIPLIER • c is the INCREMENT (forget this one) U’s in (0, 1)

  10. 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

  11. unif.xls • observe the basic functions of the seed, multiplier, and modulus • experiment with multipliers for • m = 17 • m = 16

  12. 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

  13. HISTORICAL COMPETITION FOR LINEAR CONGRUENTIAL GENERATORS • add or multiply some combination of variates from the stream’s recent history

  14. WHAT IS GOOD • Full, Long Cycle • Seemingly Independent • We can test this, but our simple tests stink

  15. 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

  16. 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?

More Related