Download Test of Randomness

Randomness Test
Fall 2013
By Yaohang Li, Ph.D.
Review
• Last Class
– Random Number Generation
– Uniform Distribution
• This Class
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Test of Randomness
Chi Square Test
K-S Test
10 empirical tests
• Next Class
– Nuclear Simulation
Chi-square test
• Introduced by Karl Pearson in 1900
• Test for discrete distributions e.g. binomial and Poisson distributions
• Implementation:
- assume we have k possible categories
- P = sequence size / k
- expected sample size = P * n trials
- suppose category i occurs Yi times
- error = Yi – nPi
- chi-square statistic
- chi-square percentile = proportion of samples from a "true“ distribution
having a chi-square statistic (function of errors) less than the percentile.
Example
• Given two “true” dice for 144 trials we get:
• s = 2
3
4
5
6
7 8
9 10 11 12
• Ps = 1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36
• Ys = 2
4
10 12 22 29 21 15 14 9
6
• nPs = 4
8
12 16 20 24 20 16 12 8
4
• V = (Y2 – nP2)² / nP2 + (Y3 – nP3)² / nP3 +………+ (Y12 – nP12)² / nP12
• V = (2 – 4)² / 4 + (4 – 8)² / 8 +……+ (9 – 8)² / 8 + (6 – 4)² / 4 = 7 7/48
Chi-square Table
Kolmogorov-Smirnov test
• Introduced in 1933
• Test for continuous distributions e.g. normal and
Weibull distributions
• based on ECDF defined as,
K-S Test
K-S Table
Empirical Tests
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Equidistribution Test
Serial Test
Gap Test
Poker Test
Coupon Collector’s Test
Permutation Test
Run Test
Maximum of t test
Collision Test
Serial Correlation Test
Summary
• Chi-Square Test
• KS Test
• Empirical Tests
What I want you to do?
• Review Slides
• Review basic probability/statistics concepts