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I have a sequence of numbers generated by a programmed computer using pseudorandom functions. Is there a formula which will allow me to see how random this series is?

Here is my idea, are there any more elegant approaches?

  1. State the number of elements in the array (For example, array IsThisRandom(1,2,3) returns 3 for this).
  2. Record the lowest and the highest numbers returned by each (array IsThisRandom returns 1 and 3 for this).
  3. Compare the expected number of outcomes X for element in array Y: for IsThisRandom this returns 1, 1, and 1. This can be generalized by (total number of outcomes recorded in step 1)/something else, not sure what to call it.

EDIT: if i have an array of 1 element, for example 1Array(1) , there is no way to know whether it is actually random- does certainty about randomness increase with number of elements given?

Thomas Shera
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    I searched a lot (minimal correlation of different order, in different basis) and I didn't find any way to extract (by gradient descent) from a signal $x_n$ a perfectly random signal $w_n$ such that $\min_w ||x - w||^2$, Honestly, this is a huge lack in signal processing that there is no way to define what is a "perfect white noise" (of finite length). – reuns Jan 04 '16 at 17:18
  • Why do you try to find the minimum? I have precalculus level math education, I googled "gradient descent" but it didn't seem to explain anything. – Thomas Shera Jan 04 '16 at 17:20
  • so by random you mean every digit is extracted by a discrete uniform distribution on ${0, \dots, 9}$ and the number of digit is extracted by a poisson distribution? Something like that? – Ant Jan 04 '16 at 17:21
  • There are several tests based mostly on statistic analysis, check this article for example https://www.random.org/analysis/ – z100 Jan 04 '16 at 17:23
  • just to say that the problem is really complicated : all these tests are possible to fake with sequences which are the opposite of white noise. – reuns Jan 04 '16 at 17:28
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    The first problem is how define randomness.

    Once done, there are many methods for measuring the randomness. For example Kolmogorov's test.

     – sinbadh Jan 04 '16 at 17:58

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