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UNIFORM RANDOM NUMBER GENERATION

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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.

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UNIFORM RANDOM NUMBER GENERATION
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  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?