Add ZeroInflatedPoisson distribution

[emfeltham]

In the wake of a discussion over on StatisticalRethinkingTuring, I figured that it would be a good idea to put this together. It is used fairly extensively in the social sciences, and is probably not something that researchers should have to construct themselves. Thanks again.

[emfeltham]

Hi again, I ran and added tests, and changed the code according to the test criteria. It should pass now, at least it does locally. Apologies, and thanks again.

[mschauer]

Can’t we have the entire thing as actual MixtureDistribution of Dirac at 0 and Poisson, or make that work?

[codecov-commenter]

Codecov Report

Merging #1393 (2619337) into master (39f9899) will decrease coverage by 0.91%. The diff coverage is 5.12%.

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- Coverage   82.54%   81.63%   -0.92%     
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  Files         116      117       +1     
  Lines        6950     7001      +51     
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- Hits         5737     5715      -22     
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Impacted Files	Coverage Δ
src/univariates.jl	72.82% <ø> (ø)
src/univariate/discrete/zeroinflatedpoisson.jl	5.12% <5.12%> (ø)
src/univariate/discrete/discretenonparametric.jl	98.84% <0.00%> (-0.20%)	:arrow_down:
src/quantilealgs.jl	82.41% <0.00%> (ø)
src/mixtures/mixturemodel.jl	69.60% <0.00%> (+1.56%)	:arrow_up:

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[mschauer]

So let’s make it ZeroInflated{Poisson}? For the same amount of code we get a couple of related zero inflated distributions, e.g ZeroInflated{NegativeBinomial}.

[azev77]

Building on @mschauer we can generically allow creating all kinds of transformations (Zero-inflated/truncated/modified): (from my discourse post)

using Distributions
λ = 2.0; #Poisson parameter
zp = 0.4; #zero prob. 
dpoi = Poisson(λ)

dzip = MixtureModel([Dirac(0.0), dpoi], [zp, 1.0-zp])  #ZeroInflated ZIP
dztp = Truncated(dpoi, 1, Inf)                         #ZeroTruncated ZTP 
dzmp = MixtureModel([Dirac(0.0), dztp], [zp,1.0-zp])   #ZeroModified ZMP

# A function might look something like
ZeroInflated(d, zp) = MixtureModel([Dirac(0.0), d], [zp, 1.0-zp])
ZeroTruncated(d) = Truncated(d, 1, Inf)
ZeroModified(d, zp) = MixtureModel([Dirac(0.0), ZeroTruncated(d)], [zp,1.0-zp])

Btw, there are also one-inflated Binomial/Poisson/Beta etc which can be handled similarly. This really shows the amazing power of Julia & Distributions.jl!

# x_inflated/x_truncated might look something like
XInflated(d, xp, x) = MixtureModel([Dirac(x), d], [xp, 1.0-xp])
XTruncated(d, x) = Truncated(d, x+1, Inf) # note cts/discrete issue +1...
XModified(d, xp, x) = MixtureModel([Dirac(x), XTruncated(d, x)], [xp,1.0-xp])

# 
ZeroInflated(d, zp) = XInflated(d, zp, 0.0)
ZeroTruncated(d) = XTruncated(d, 0.0)
ZeroModified(d, zp) = XModified(d, zp, 0.0)

We can look at ZeroInflatedDistributions.jl & LRMoE.jl @jkbest2 & @sparktseung do you have any feedback on how to create Zero Inflated random variables?

[sparktseung]

@azev77

• Currently in LRMoE.jl v0.2.0, zero-inflated distributions are implemented as a separate object. e.g. there is PoissonExpert(λ) and there is ZIPoissonExpert(p0, λ). I don't this is a good way and I may need to change it up in the future.

• I think your example using the MixtureModel in Distributions.jl is much better and more maintainable.

• For discrete distributions with zero inflation & modification, I'd suggest adding some notes in the documentation, to explicitly specify what is the actual zero probability. For zero-inflated Poisson, it should be p0+(1-p0)*exp(-λ). A lot of people tend forget about the second term.

• For continuous distributions, some may have infinite density at zero, e.g. Gamma with shape<1. One should be careful about writing and interpreting pdf/logpdf in such case.