CS 450 – Modeling and Simulation

Dr. X

Topics

• What Does Randomness Mean?• Randomness in games• Generating Random Values• Random events in real life: measuring

randomness

What is Randomness?

• Unpredictability?• Does it characterize a single event or a

sequence of events?• From Wikipedia:

Randomness means lack of pattern or predictability in events. Randomness suggests a non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination.

Randomness in games

Probability 101

If you roll the dice twice, what is the probability of rolling two sixes?

Probability vs odds

• Odds = Favorable Outcomes / Unfavorable Outcomes

• Probability = Favorable Outcomes / All Outcomes

Probability 101

If you toss two coins what is the probability of HH?

Probability 101

• Independent events• Logical relations:– AND– OR

Defining Randomness

• How can a computer generate random numbers?

• Can we measure randomness?

Measuring Randomness

• Runs Test• Frequency test• Autocorrelation test• Chi-square goodness of fit

Statistical Hypothesis test

• H0: Null Hypothesis, ex., a sequence of numbers is random

• Hα: Alternative Hypothesis, ex., a sequence of numbers is NOT random

Real SituationDecision

H0 is not rejected H0 is rejected

H0 is true Valid Type I Error

H0 is not true Type II Error Valid

Runs test

• Sequence of numbers X1, …, XN

• Subtract Xi+1 – Xi

• Collect only the signs of the subtraction results

• Do the signs alternate ~50% of the time?

Frequency Test (Monobit test)

• Create a string Sn = X1 + X2 + … + Xn

• Calculate Sobs = Sn/n• Calculate P-Value:

P-value = erfc(Sobs/2)where erfc: complementary error function

• If P-value > 0.01 accept the null hypothesis else reject

Autocorrelation Test

Intuition: if the sequence of bits in e is random, then it will be different from another bit string obtained by shifting the bits of e by d positions.

Chi-Square Goodness of Fit Test

• Intuition: test checks whether a sequence of pseudo-random numbers in [0,1] are uniformly distributed

• If you have n numbers and split them in k groups, then if in the groups you have n/k numbers the distribution is normal

• Compare all observed values fi inside your kth group with the mean n/k

• If chi-square > theoretical x2 obtained from table with k degrees of freedom, then null hypothesis rejected

Generating Random Numbers

• Congruential method• Tausworthe generators• Lagged Fibonacci generators• The Mercenne Twister

Congruential Method

xi+1 = axi + c (mod m)• a, c, m: non-negative• If m is large, period of number repetition is

large• Conditions for full period:– m and c have no common divisor– a = 1 (mod r) if r is a prime factor of m – a = 1 (mod 4) if m is a multiple of 4

Tausworthe generators

• Additive congruential generators • Modulus m is equal to 2• xi = (a1xi-1 + a2xi-2 + ...+ anxi-n) (mod 2)

where xi: 0 or 1• Mod 2 is equivalent to XOR

Lagged Fibonacci Generators

• Based on Fibonacci series: xn = xn-1 + xn-2

• LFG: xn = xn-j O xn-k (mod m)where O: addition, or subtraction, or multiplication, or XOR

Mersenne Twister

xk+n = xk+m (⊕ xku | xk+1

k )A, k≥0

Where x is the block of numbers of w bits

CS 450 21

References

• The Guide to Computer Simulations and Games, By Katrin Becker and J.R.Parker, Wiley (Chapter 6)

• Computer Simulation Techniques: The Definitive Introduction, by Harry Perros, http://www4.ncsu.edu/~hp/simulation.pdf (Chapter 2)