Test for randomness : Chi square test We can say how probable or improbable certain types of events are. The difference between observed Ys and expected nps V = (Y2 – np2) 2 + (Y3 – np3) 2 + … + (Y12 – np12) 2 What is the probability what V is this high using true dice? December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Suppose that every observation can fall into one of k categories. We take n independent observations. Let ps be the probability that each observation falls into category s, and let Ys be the number of observations that actually do fall into category s. Weighted by the prob. of occurrence nps December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Expanding (Ys – nps) 2 = Y2 – 2nps Ys + n2 ps 2 and Y1 + Y2 + … Yk = n p1 + p2 + … pk = 1 v = k –1 the number of degree of freedom is k – 1. December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Chi-square distribution table says “The quantity V will be less than or equal to x with approximate probability p, if n is large enough”. How large should n be? Rule of thumb is nps >= 5 December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Example five Chi-square test on three data of four generators. December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana B : x0 = 0, a = 3141592653, c = 2718281829, m = 2 35 C : x0 = 0, a = 2 7 + 1 , c = 1, m = 2 35 D : x0 = 47194118, a = 23, c = 0, m = 10 8 + 1 F : x0 = 314159265, a = 2 18 + 1, c = 1, m = 2 35 Conclusion, B and D are satisfactory, C is on the borderline, F is unsatisfactory. December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Other tests Kolmogorov-Smirnov test (available in SPSS package) Frequency test, serial test, gap test, Poker test (partition test), Coupon’s collector test, permutation test, run test, maximum-of-t test, collision test, serial correlation test, tests on subsequences Spectral test December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Advise for RNG The seed number X0 may be chosen arbitrarily. The number m should be large. The computation (aX + c) mod m must be done exactly with no roundoff error. If m is the power of 2, pick a so that a mod 8 = 5 to ensure max period and high potency. December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana The muliplier a should preferably be chosen between .01m and .99m and is binary digits should not have a simple, regular pattern. The value of c is immaterial when a is a good multiplier, except that c must be no factor in common with m. The least significant (right-handed) digits of X are not very random. It is generally best to think of X as a random fraction X/m between 0 and 1. December 1999 Prabhas Chongstitvatana

Prabhas Chongstitvatana Some good RNG a, m 137, 256 3141592221, 235 5 15, 2 35 17059465, 2 35 1664525, 2 32 314159269, 2 31 - 1 31167285, 2 48 6364136223846793005, 2 64 December 1999 Prabhas Chongstitvatana