Test of random sampling

[claudio-ICL]

This PR is about the tests of sampling functions from lib/cpp/random.

Three issues are addressed:

• chisquare test and power_divergence test of scipy.stats requires that the sum of expected frequencies and the sum of observed frequencies be the same. Previously, we were using probabilities in place of expected frequencies.
  https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chisquare.html
  https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.power_divergence.html#scipy.stats.power_divergence

• Usage of p-values in Kolmogorov-Smirnov tests and Chi square tests are made more explicit, and we give benchmarks from the performance of these test by using sampling methods from scipy.

• We visualise the performance of the sampling functions from lib/cpp/random in tick.random.tests.qqplots

[PhilipDeegan]

@Mbompr some of the random tests have been failing since updating things for reasons unknown to me, please have a look and tell us what you think

[claudio-ICL]

Hi @Mbompr
Could you please give some clarity about the tests on the random number generators?

My understanding is that we should do the following.
Assume we want to test that samples in the vector x are drawn from the exponential distribution, with unit rate. The null hypothesis would be that the samples in x are indeed drawn from Expon(1). Then, we should use kolmogorov smirnoff test and compare the empirical cdf of x with the theoretical cdf. The test would return a p-value. If indeed x is drawn from Expon(1), then the p-value should indicate that the null hypothesis cannot be rejected: the higher the p-value, the more confident we are in saying that the null hypothesis is true.

As a benchmark, one could do the following:

>>> from scipy import stats
>>> stats.kstest('expon', 'expon')
KstestResult(statistic=0.17823022417117995, pvalue=0.49370343902508884)

Or:

>>> rvs = stats.expon.rvs(size=10000)
>>> stats.kstest(rvs, 'expon')
KstestResult(statistic=0.005253711831793573, pvalue=0.9439901815922778)

Therefore, we should assert that the p-value calculated on our sample x is greater than some threshold t.

[Mbompr]

Sorry that is a bit old and I am not very proud of these tests that simply consist in checking that the C++ random number generator is working as expected... The tiny extra robustness they add to the project does not compensate the extra complexity they bring...
From what I remember from statistical tests, your remarks look correct. But we can as well skip them if that makes things easier.

[claudio-ICL]

Hi @Mbompr
It is good to have those tests. They were however a bit confusing and / or with too stringent requirements.

I would say that we modify the statistical tests based on Kolmogorov- Smirnov and on Chi-square as in this PR. In addition, I wrote the script tick.random.tests.qqplots which helps visualise how good the sampling from uniform distribution, gaussian distribution, exponential distribution, and Poisson distribution are. I think that overall the sampling is good.