Use estimated quantiles instead of std

[baggepinnen]

Similar to #26
Instead of reducing particles to uncertainty by calling std, we can make use of the quantiles of the sample, at least if number of particles is large. Example code:

MonteCarloMeasurements.jl/src/particles.jl

Line 385 in 5c4d138

≲(a::Real,p::AbstractParticles,lim=2) = (mean(p)-a)/std(p) > lim

[baggepinnen]

Asking @cscherrer for input here: I currently have definitions like the following to determine if particles differ significantly from each other or from a scalar

Base.:≈(a::Real,p::AbstractParticles, lim=2) = abs(mean(p)-a)/std(p) < lim
≲(p::AbstractParticles,a::AbstractParticles,lim=2) = (mean(p)-mean(a))/(2sqrt(std(p)^2 + std(a)^2)) > lim

but I guess it would make sense to change these to using sample quantiles instead, at least if the number of particles is reasonably large (might not make sense for sigmapoints etc.). E.g., the following definition

Base.:≈(a::Real,p::AbstractParticles, q=0.025) = quantile(p,q) < a < quantile(p,1-q)

I think this makes sense, but am open to hear your thoughts on it

[cscherrer]

Oh this is tricky - I think it really depends on what use case you're targeting.

If you're considering particles as proxies for random variables, I'd think the ordering matters (since permuted particles are no longer equal). So I'd probably go with "difference is not statistically significant", so maybe something like

function Base.:≈(a::AbstractParticles, b)
    lo,hi = quantile(a-b, (0.05,0.95))
    lo < 0.0 < hi
end

EDIT: I just realized this is the same as your example. Been a long day :)

If you want to consider them only as distributions, then I'd probably go with the Wasserstein or Earth mover distance

[baggepinnen]

Distribution free test statistic to determine if two samples come frome the same distribution.

A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Scholkopf, and A. Smola. ¨
A kernel two-sample test.