Consider implementing metalog size distributions

[smk78]

Following discussions at CanSAS-2023, @Kohlbrecher suggests SasView follow the lead of SASfit and implement its size distributions in terms of a Metalog Distribution:

https://en.wikipedia.org/wiki/Metalog_distribution
https://doi.org/10.1107/S1600576722009037, Section 5.1 & Fig 4

This very flexible distribution could overcome the limitations imposed on fit solutions by the shapes of singular (or composites of singular) distributions.

The Metalog Distribution is defined through its Quantile Probability Distribution (the inverse of the Cumulative Distribution Function).

Many 'conventional' distribution functions also have analytic quantile functions, for example:

https://en.wikipedia.org/wiki/Quantile_function#Simple_example
https://en.wikipedia.org/wiki/Log-normal_distribution, under 'Mode, Median, Quantiles'

It is also possible that the QPD approach could be more numerically stable because it is a proper integral (over 0 - 1) unlike the improper integrals (over 0 - inf) of traditional size distributions.

[lucas-wilkins]

I think the quantile approach will be good for 1D polydispersity, but it gets weird, complicated or both when it comes to multiple dimensions.

[Kohlbrecher]

I am not sure why you think it gets more complicated using multiple parameters with a distribution. The parameter having a distribution is simply replaced by its QDF. If you have multiple independent parameters each will be replaced by a separate QDF and the multiple integral has to be calculated over a hypercube [0,1]^n