RFC: First take on cleaning up EmpiricalUnivariateDistribution

[andreasnoack]

This one was pretty broken. I think I'll change the meanpdf function to utilize that the vector is sorted.

[juliohm]

Maybe parametrize the distribution with T<:Real instead of explicitly using Float64?

[andreasnoack]

Hm. It looks like we are assuming that DiscreteUnivariateDistributions can only be integer valued so, in general, an empirical distribution won't be a DiscreteUnivariateDistribution. Seems overly restrictive to me.

I'd also really like to rename this distribution to Empirical and since it was so broken, I considering if it can be done without deprecating it. Objections?

[juliohm]

I am in favor of new names, but ideally they would take into account continuous vs. discrete cases. I also agree that the implementation was quite broken to deserve a deprecation warning.

I am particularly interested in the continuous case as discussed here: https://discourse.julialang.org/t/empirical-distribution-type-for-continuous-variables/5676

In one of my spatial stats solvers I need to transform continuous empirical distributions to Normal, do the calculations, and transform back to the initial empirical distribution.

In that post, I wrote a simple transformation function that I am pasting here again:

"""
    transform(x, dist)

Transform the empirical distribution of samples `x` into `dist`.
"""
function transform(x::Vector, dist::ContinuousUnivariateDistribution)
    idx = sortperm(x)
    
    N = length(x)
    xcdf = [1:N-1; .99N] / N
   
    quantile(dist, xcdf[idx])
end

This function could be incorporated in Distributions.jl as well, it only depends on a Distribution type for which quantile is implemented.

[matbesancon]

@andreasnoack there seem to be conflicts to solve. Afterwards what's the status for this PR?

[andreasnoack]

It stalled when I realized that it is implicitly assumed that Discrete here means $\mathbb{N}_0$ which therefore doesn't cover the empirical distribution. I then thought about reworking the whole declared type tree in Distributions which I thought was (and still think is) both insufficient and unnecessarily complex but then I realized that I wouldn't have time for that. However, I still dream about redesigning the abstract types (and more out multivariate and matrix valued distributions.)

[matbesancon]

Agreed, this joins other issues, like #771
This is likely to need a complete re-work when we get the type hierarchy right.
I came to the same conclusion that we need discrete distributions as in over a discrete set when working on MixedDistributions:
https://github.com/matbesancon/MixedDistributions.jl/blob/master/src/simple_discrete.jl

[matbesancon]

Should we submit an issue documenting what the intent is and the requirements on the type hierarchy, then close this PR till the required changes are carried out?

[andreasnoack]

I think we keep this one open as it should be fine provided we get an appropriate distribution supertype for the distribution. I think #681 might cover, at least some of, what is needed.