Statistical Randomness Metric

A Statistical Randomness Metric is a statistical metric of a numeric sequence.

• Example(s):
  • a frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, 2s, 3s, etc.
  • aserial test, did the same thing but for sequences of two digits at a time (00, 01, 02, etc.), comparing their observed frequencies with their hypothetical predictions were they equally distributed.
  • a poker test, tested for certain sequences of five numbers at a time (aaaaa, aaaab, aaabb, etc.) based on hands in the game poker.
  • a gap test, looked at the distances between zeroes (00 would be a distance of 0, 030 would be a distance of 1, 02250 would be a distance of 3, etc.).
  • …
• Counter-Example(s):
  • Algorithmic Randomness.
• See: Randomness, Stochastic Process.

---

---

References

2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Statistical_randomness
  • A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll, or the digits of π exhibit statistical randomness.

    Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability. Pseudorandomness is sufficient for many uses, such as statistics, hence the name statistical randomness.

    Global randomness and local randomness are different. Most philosophical conceptions of randomness are global—because they are based on the idea that "in the long run" a sequence looks truly random, even if certain sub-sequences would not look random. In a "truly" random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but repeating numbers, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths in which random distributions are approximated. Long stretches of the same numbers, even those generated by "truly" random processes, would diminish the "local randomness" of a sample (it might only be locally random for sequences of 10,000 numbers; taking sequences of less than 1,000 might not appear random at all, for example).

    A sequence exhibiting a pattern is not thereby proved not statistically random. According to principles of Ramsey theory, sufficiently large objects must necessarily contain a given substructure ("complete disorder is impossible").

    Legislation concerning gambling imposes certain standards of statistical randomness to slot machines.

• http://en.wikipedia.org/wiki/Statistical_randomness#Tests
  • The first tests for random numbers were published by M.G. Kendall and Bernard Babington Smith in the Journal of the Royal Statistical Society in 1938. They were built on statistical tools such as Pearson's chi-squared test that were developed to distinguish whether experimental phenomena matched their theoretical probabilities. Pearson developed his test originally by showing that a number of dice experiments by W.F.R. Weldon did not display "random" behavior.

    Kendall and Smith's original four tests were hypothesis tests, which took as their null hypothesis the idea that each number in a given random sequence had an equal chance of occurring, and that various other patterns in the data should be also distributed equiprobably.

    • The frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, 2s, 3s, etc.
    • The serial test, did the same thing but for sequences of two digits at a time (00, 01, 02, etc.), comparing their observed frequencies with their hypothetical predictions were they equally distributed.
    • The poker test, tested for certain sequences of five numbers at a time (aaaaa, aaaab, aaabb, etc.) based on hands in the game poker.
    • The gap test, looked at the distances between zeroes (00 would be a distance of 0, 030 would be a distance of 1, 02250 would be a distance of 3, etc.).
  • If a given sequence was able to pass all of these tests within a given degree of significance (generally 5%), then it was judged to be, in their words "locally random". Kendall and Smith differentiated "local randomness" from "true randomness" in that many sequences generated with truly random methods might not display "local randomness" to a given degree — very large sequences might contain many rows of a single digit. This might be "random" on the scale of the entire sequence, but in a smaller block it would not be "random" (it would not pass their tests), and would be useless for a number of statistical applications.

    As random number sets became more and more common, more tests, of increasing sophistication were used. Some modern tests plot random digits as points on a three-dimensional plane, which can then be rotated to look for hidden patterns. In 1995, the statistician George Marsaglia created a set of tests known as the diehard tests, which he distributes with a CD-ROM of 5 billion pseudorandom numbers.

---