Two finite-sampled continuous random variables $X$ and $Y$ are transformed so that they are uniformly distributed, $U$ and $V$. With these as marginals, the empirical copula of $X$ and $Y$, denoted $C(u,v)$, can be estimated. The estimation approach being asked here is by discretization via histogram binning.

Since alot of copula theory sticks to just that, theory (integrals, therefore, non-discrete), the discretization implementation is hardly covered. Could someone provide a source from a copula textbook that explains this formula found in the numpy histogram2d documentation:

probability density function at the bin = bin_count / sample_count / bin_area.

By 'at the bin' does this mean there is a distribution formed for each bin? and why the triple fraction? I'm trying to make sense of it for the following:

copula_density = np.histogram2d(X_unif, Y_unif, bins, density=True)[0]
copula_freq = np.histogram2d(X_unif, Y_unif, bins, density=False)[0]
bin_area = (copula_freq / copula_density / len(X_unif))[0, 0]

since the last line looks like it backs out the bin_area from the same equation shown earlier, being a triple fraction also. I also don't know the importance of bin_area or what application it would be required for afterwards.

  • $\begingroup$ Probability integral transform (PIT) is used for transforming continuous random variables to uniform, and this is done when working with copulas. I wonder what the justification for other types of transformations is. $\endgroup$ Commented Aug 28, 2020 at 17:12
  • $\begingroup$ are you saying the binning approach above is different than integral transform (inverse CDF)? $\endgroup$
    – develarist
    Commented Aug 28, 2020 at 17:17
  • $\begingroup$ I don't know. If it were the same, why a different name? $\endgroup$ Commented Aug 28, 2020 at 17:32
  • $\begingroup$ By definition, the density is a value per unit area. Evidently, then, this estimate of the PDF is obtained by estimating probabilities as fractions of the sample count. $\endgroup$
    – whuber
    Commented Aug 28, 2020 at 18:24
  • $\begingroup$ "area" to me is length $\times$ width. how does this concept possibly match up with bin_area = copula_freq / copula_density / len(X_unif)? $\endgroup$
    – develarist
    Commented Aug 29, 2020 at 12:59


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.