Restructuring KernelDensity.jl

[jaksle]

The current capabilities of KernelDensity are quite limited, especially if we compare them to what is avaiable e.g. in SciPy, and this supposed to be is the main source of kernel estimation tools in Julia statistical ecosystem. The current interface and type system were made for calculating discretised pdf using Fourier transform in 1D or 2D and not for much else. This is a proposition of completely new type system and interface which is more flexible. See: #68, #110, #120, #126, #125, #129, #139.

Check exampleUse.jl to see how it works.

The main principles of this proposition are:

• The new KernelEstimate datatype contains full information about the fit, so a statistician can infer all they possibly want from it.
• KernelEstimate is a subtype of MixtureDistribution so it fits well the rest of JuliaStats, can be much more easily used in other tasks and has a lot of useful default methods already implemented in Distributions.
• It can be used for kernel estimation in any dimension.
• The discretised values of the pdf are distinct from KernelEstimate. However, they can be stored in it and if so, getting pdf from interpolation of it becomes avaiable. If it is not present, different methods for getting pdf can be used (e.g. from the naive sum formula, which for small samples is completely reasonable). It can be also added using precompute!.

The current approach does not allow for adaptive kde. You can allow for it with the same design, but for the sake of efficiency and code simplicity it might be better to add something like AdaptiveKernelEstimate if such a need arises.

I've currently implemented methods only for 1D. Adding 2D is only a matter of few simple additions, but in #110 there is code already for any dimension, it is debatable if it is worth to treat 2D case separately at all. But this is better to do that after getting some feedback.

Old interface may, or may not stay for backwards compability.

As it stands this proposition does not add any new numerical methods, but even simply linking it to Distributions adds a lot of features to this library, which for example Scipy or Sklearn do not have.