I have built a piece of hardware that produce random samples from a Gaussian distribution. Now that I have the hardware I want to empirically evaluate how random the samples are.
I am aware that the NIST and Diehard randomness tests exist but as far as I am aware they are for uniformly distributed random numbers and therefore not directly applicable to Gaussian distributed random numbers. Is there a standard empirical test for the randomness of Gaussian distributed random numbers?
Dear James, as first you must distinguish between randomness linked to the lack of any order dependent structure and randomness simply linked to probability distribution, I suggest you to read this paper that is even linked to R software routine to solve your problem:
Article RandseqR: An R Package for Describing Performance on the Ran...
Use a standard uniform random number generator to generate N random numbers, x, between 0 and 1, then use the following inverse cumulative distribution function to map them into a Gaussian distribution:
ICDF(x)=sqrt(2).IERF(2x-1)
Above, IERF stands for inverse error function. Now compare the reduced chi square obtained form this distribution with that obtained from your sample of N points. If they are comparable then you can claim you've got the same level of randomness, otherwise you should search for some hidden spectral issues in your hardware/generator.
Hello Alessandro,
Thanks for the answer. Does the method in the paper you have linked work random samples from any probability distribution (not just uniform)?
Is it superior to the NIST and Diehard tests? If so how?
I understand that I can get a time series of samples that follow a Gaussian distribution without having any randomness in the time series. Just because the samples follow a Gaussian distribution it does not mean that there is no correlation between successive samples.
Hello Mohammad,
As far as I understand it, the method you propose would only check that my random number generator produced the correct probability distribution. It would not check that there was no correlation between successive samples. Is this correct?
Hello James,
It allows you to both check whether your random numbers have a correct probability distribution, and more importantly, map them into a uniform distribution which you can then use to trace any correlation or spectral issues.
Hello Mohammad
Thank for the clarification. How would the correlation or spectral issues manifest themselves in the uniform distribution output and how would I quantitatively measure this?
In the attached paper you will find many measures of internal correlation of the series, note the difference between order dpendent (correlation) and order independent (population statistics) measures.
Since the mapping enforces a one-to-one correspondence between the original set with a gaussian distribution and the new one with a uniform distribution, any correlation in the former must be traceable in the latter as well.
As for how to qualitatively measure it, let's say you generate N numbers with a gaussian distribution, x_G, you then map these numbers into their corresponding set with a uniform probability distribution, x_u, using the cumulative distribution function:
x_u=0.5(1+erf(x_G/sqrt(2))
Mind that here I assume your Gaussian probability distribution has sigma=1 and is clustered around x=x0=0. If this is not the case, you should use the general expression:
x_u=0.5(1+erf( (x_G-x0) / (sigma*sqrt(2)) )
Once you've got the uniformly distributed set x_u, you can then check its randomness using any standard toolkit like NIST test suite.
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