[Feature req] Convolutions of dependent random vectors.

[rand5]

I'm migrating a problem discussed here that involves computing the convolution of two dependent random variables where their joint distribution is defined by a copula. I believe the issue may stem from the fact that evaluating the pdf of the copula sometimes returns NaN when it should return 0.
Here is an example with marginal distributions that should have support over all reals:

X1 = Gumbel(15.,5.)
X2 = MixtureModel([Normal(-6.,0.1),X1],[0.9,0.1])

ρ = 0.4

C = GaussianCopula([1. ρ; ρ 1.])

D1 = SklarDist(C,(X1,X2))

pdf(D1,[-30., -30.]) #Returns NaN

The link above has an attempt at a brute force work-around, but it doesn't seem to be sufficient (see the plot in the link).

[lrnv]

Looks indeed like a bug, or rather a missed special case somewhere. By the way:

julia> pdf(D1,big.([-30., -30.])) # works with bigfloats
9.800107565129330824766557211886842343262822051986735333987174617133175015382767e-5024

julia> pdf(C, [0,0]) # here is where your nan is coming from
NaN

julia>

so yes, should definitely return 0. Let me take a look as soon as i have a moment.

---

ERR: Scratch that, the bug is comming from Distributions.jl :

julia> pdf(MvNormal(C.Σ),[-Inf,-Inf])
NaN

julia>

[rand5]

I tried a few other copulas (Clayton, Gumbel, Joe) and am getting similar domain errors. Would this need to be addressed for C::ArchimedeanCopula as well?

[lrnv]

For archimedeans the wanted pdf value at [0,0] ios not always 0, depends on the archimedean. But yes if some of them give domainerrors that is an issue.

Can you post your other examples ? Let's make tests cases out of them so that it stays fixed in the future.

[lrnv]

Hum... Taking another look at the gaussian case, the NaN actually comes from Distributions.jl :

julia> pdf(MvNormal(C.Σ),[-Inf,-Inf])
NaN

julia>

[rand5]

Here is what I was using to check the Archimedian Copulas

X1 = Normal()


d = 2
θ = 1.

Cs = [ClaytonCopula(d,θ), FrankCopula(d,θ), GumbelCopula(d,θ), JoeCopula(d,θ)] #Copulas to check. Make sure θ is appropriate

for k=1:length(Cs)
    D1 = SklarDist(Cs[k],(X1,X1))

    N = 100
    Zs = collect(range(-10.,10.,N))
    ConvD = map(Zs) do z
        quadgk(x->pdf(D1,[x,z-x]), -100., 100., rtol=1e-6)[1]
    end
end

It is not clear to me at which point the NaN is being produced since

X1 = Normal()

d = 2
θ = 1.

C = ClaytonCopula(d,θ)

 D1 = SklarDist(C,(X1,X1))

N = 100
Zs = collect(range(-10.,10.,N))
ConvD = map(Zs) do z
    quadgk(x->pdf(D1,[x,z-x]), -100., 100., rtol=1e-6)[1]
end

gives

DomainError with 0.0:
integrand produced NaN in the interval (-100.0, 100.0)

Stacktrace:
  [1] evalrule(f::var"#87#89"{Float64}, a::Float64, b::Float64, x::Vector{Float64}, w::Vector{Float64}, wg::Vector{Float64}, nrm::typeof(norm))
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\evalrule.jl:39
  [2] #8
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:54 [inlined]
  [3] ntuple
    @ .\ntuple.jl:48 [inlined]
  [4] do_quadgk(f::var"#87#89"{Float64}, s::Tuple{Float64, Float64}, n::Int64, atol::Nothing, rtol::Float64, maxevals::Int64, nrm::typeof(norm), _segbuf::Nothing, eval_segbuf::Nothing)
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:52
  [5] #50
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:83 [inlined]
  [6] handle_infinities(workfunc::QuadGK.var"#50#51"{Nothing, Float64, Int64, Int64, typeof(norm), Nothing, Nothing}, f::var"#87#89"{Float64}, s::Tuple{Float64, Float64})
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:189
  [7] #quadgk#49
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:82 [inlined]
  [8] quadgk
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:80 [inlined]
  [9] #86
    @ .\In[152]:14 [inlined]
 [10] iterate
    @ .\generator.jl:47 [inlined]
 [11] _collect(c::Vector{Float64}, itr::Base.Generator{Vector{Float64}, var"#86#88"}, ::Base.EltypeUnknown, isz::Base.HasShape{1})
    @ Base .\array.jl:854
 [12] collect_similar(cont::Vector{Float64}, itr::Base.Generator{Vector{Float64}, var"#86#88"})
    @ Base .\array.jl:763
 [13] map(f::Function, A::Vector{Float64})
    @ Base .\abstractarray.jl:3285
 [14] top-level scope
    @ In[152]:13

but

pdf(D1,[0.,0.]) # not NaN

[lrnv]

Okay so let's split up.

For the Gaussian, i reported upstream there : JuliaStats/Distributions.jl#1937

For the archimedeans, I'll come back when I know more

[lrnv]

@rand5 The archimedean ones should be fixed in the (currently registrating) v0.1.27 version of the copulas package. There was indeed an issue with the coputation of the pdf in the corners. On the new version, I have :

using Copulas, Distributions, QuadGK
X1 = Normal()


d = 2
θ = 1.

Cs = [
    ClaytonCopula(d,θ), 
    FrankCopula(d,θ), 
    GumbelCopula(d,θ), 
    JoeCopula(d,θ)
] #Copulas to check. Make sure θ is appropriate

N = 100
Zs = collect(range(-10.,10.,N))
ConvDs = zeros(length(Cs), length(Zs))
for k=1:length(Cs)
    D1 = SklarDist(Cs[k],(X1,X1))
    ConvDs[k,:] = map(Zs) do z
        quadgk(x->pdf(D1,[x,z-x]), -100., 100., rtol=1e-6)[1]
    end
end

p = plot()
for k in 1:length(Cs)
    plot!(p, Zs, ConvDs[k,:], label=k)
end
p

producing the folloiwng densities:

However, the Gaussian one is not fixed yet because the issues comes from Distributions.jl (and potentially form PDMats.jl directly), will take a bit more time as it is not up to me alone. :/

Apart the from gaussian, is this fix enough to cover what you need or do you have other questions?

[rand5]

@lrnv I'm using v0.1.27 and still getting domain errors:

X1 = Gumbel(15.,5.)
X2 = MixtureModel([Normal(-6.,0.1),X1],[0.9,0.1])

θ = 2.5
d= 2
C = GumbelCopula(d,θ)

D1 = SklarDist(C,(X1,X2))

N = 100
Zs = collect(range(-10.,10.,N))
ConvD = map(Zs) do z
    quadgk(x->pdf(D1,[x,z-x]), -100., 100., rtol=1e-6)[1]
end

which throws:

DomainError with 0.0:
integrand produced NaN in the interval (-100.0, 100.0)

Stacktrace:
  [1] evalrule(f::var"#2#4"{Float64}, a::Float64, b::Float64, x::Vector{Float64}, w::Vector{Float64}, wg::Vector{Float64}, nrm::typeof(norm))
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\evalrule.jl:39
  [2] #8
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:54 [inlined]
  [3] ntuple
    @ .\ntuple.jl:48 [inlined]
  [4] do_quadgk(f::var"#2#4"{Float64}, s::Tuple{Float64, Float64}, n::Int64, atol::Nothing, rtol::Float64, maxevals::Int64, nrm::typeof(norm), _segbuf::Nothing, eval_segbuf::Nothing)
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:52
  [5] #50
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:83 [inlined]
  [6] handle_infinities(workfunc::QuadGK.var"#50#51"{Nothing, Float64, Int64, Int64, typeof(norm), Nothing, Nothing}, f::var"#2#4"{Float64}, s::Tuple{Float64, Float64})
    @ QuadGK C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\adapt.jl:189
  [7] #quadgk#49
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:82 [inlined]
  [8] quadgk
    @ C:\Users\Monali\.julia\packages\QuadGK\BjmU0\src\api.jl:80 [inlined]
  [9] #1
    @ .\In[3]:4 [inlined]
 [10] iterate
    @ .\generator.jl:47 [inlined]
 [11] _collect(c::Vector{Float64}, itr::Base.Generator{Vector{Float64}, var"#1#3"}, ::Base.EltypeUnknown, isz::Base.HasShape{1})
    @ Base .\array.jl:854
 [12] collect_similar(cont::Vector{Float64}, itr::Base.Generator{Vector{Float64}, var"#1#3"})
    @ Base .\array.jl:763
 [13] map(f::Function, A::Vector{Float64})
    @ Base .\abstractarray.jl:3285
 [14] top-level scope
    @ In[3]:3

[rand5]

pdf(D1,[-100.,100.]) # = NaN

Edit: this error is being thrown despite the fact the test code above it works fine

[lrnv]

@rand5 You can replace the pdf call in your loop with

function verbose_pdf(D,x)
    u = cdf.(D.m, x)
    @show u, pdf(D.C, u), pdf.(D.m, x)
    return pdf(D, x)
end
ConvD = map(Zs) do z
    quadgk(x->verbose_pdf(D1,[x,z-x]), -100., 100., rtol=1e-6)[1]
end

With this, we see that in the Gumbel case the problem comes from values of the pdf on the boundary:

# These calls are currently NaN but should be 0:
pdf(GumbelCopula(2,2.5), [0.1,0.0])
pdf(GumbelCopula(2,2.5), [0.0,0.1])

# This one is truly undefined, but for simplicity well make it zero too.
pdf(GumbelCopula(2,2.5), [0.0,0.0])

This is fixed in #244 and now released in v0.1.28, if you have time to take a shot ?