Make fitting tests pass on all archimedeans

src/ArchimedeanCopulas/AMHCopula.jl

@@ -34,23 +34,39 @@ end
 
 τ(C::AMHCopula) = _amh_tau_f(C.θ) # no closed form inverse...
 
-_amh_tau_f(θ) = θ == 1 ? 1/3 : 1 - 2(θ+(1-θ)^2*log1p(-θ))/(3θ^2)
+function _amh_tau_f(θ)
 
-# if θ = -1, we obtain (5 -8*log(2))/3 
-
-function τ⁻¹(::Type{AMHCopula},τ)
-    if τ == zero(τ)
-        return τ
+    # unstable around zero, we instead cut its taylor expansion: 
+    if abs(θ) < 0.01
+        return 2/9  * θ
+            + 1/18  * θ^2 
+            + 1/45  * θ^3
+            + 1/90  * θ^4
+            + 2/315 * θ^5
+            + 1/252 * θ^6
+            + 1/378 * θ^7
+            + 1/540 * θ^8
+            + 2/1485 * θ^9
+            + 1/990 * θ^10
+    end
+    if iszero(θ)
+        return zero(θ)
     end
-    if τ > 1/3
+    u = isone(θ) ? θ : θ + (1-θ)^2 * log1p(-θ)
+    return 1 - (2/3)*u/θ^2
+end
+function τ⁻¹(::Type{AMHCopula},tau)
+    if tau == zero(tau)
+        return tau
+    elseif tau > 1/3
         @warn "AMHCopula cannot handle kendall tau's greater than 1/3. We capped it to 1/3."
         return one(τ)
-    end
-    if τ < (5 - 8*log(2))/3
+    elseif tau < (5 - 8*log(2))/3
         @warn "AMHCopula cannot handle kendall tau's smaller than (5- 8ln(2))/3 (approx -0.1817). We capped it to this value."
-        return -one(τ)
+        return -one(tau)
     end
-    return Roots.find_zero(θ -> _amh_tau_f(θ) - τ, (-one(τ), one(τ)))
+    search_range = tau > 0 ? (0,1) : (-1,0)
+    return Roots.find_zero(θ -> tau - _amh_tau_f(θ), search_range)
 end
 williamson_dist(C::AMHCopula{d,T}) where {d,T} = C.θ >= 0 ? WilliamsonFromFrailty(1 + Distributions.Geometric(1-C.θ),d) : WilliamsonTransforms.𝒲₋₁(t -> ϕ(C,t),d)
 

src/ArchimedeanCopulas/FrankCopula.jl

@@ -23,7 +23,7 @@ struct FrankCopula{d,T} <: ArchimedeanCopula{d}
     θ::T
     function FrankCopula(d,θ)
         if d > 2 && θ < 0
-            throw(ArgumentError("Negatively dependent Frank copulas cannot exists in dimensions > 2"))
+            throw(ArgumentError("Negatively dependent Frank copulas cannot exists in dimensions > 2. You passed θ = $θ"))
         end
         if θ == -Inf
             return WCopula(d)
@@ -47,22 +47,29 @@ end
 # D₁ = GSL.sf_debye_1 # sadly, this is C code.
 # could be replaced by : 
 # using QuadGK
-D₁(x) = QuadGK.quadgk(t -> t/(exp(t)-1), 0, x)[1]/x
+D₁(x) = QuadGK.quadgk(t -> t/expm1(t), 0, x)[1]/x
 # to make it more general. but once gain, it requires changing the integrator at each evlauation, 
 # which is problematic. 
 # Better option is to try to include this function into SpecialFunctions.jl. 
 
-
-τ(C::FrankCopula) = 1+4(D₁(C.θ)-1)/C.θ
-function τ⁻¹(::Type{FrankCopula},τ)
-    if τ == zero(τ)
-        return τ
+function _frank_tau_f(θ)
+    if abs(θ) < sqrt(eps(θ))
+        # return the taylor approx. 
+        return θ/9 * (1 - (θ/10)^2)
+    else
+        return 1+4(D₁(θ)-1)/θ
     end
-    if abs(τ==1)
-        return Inf * τ
+end
+τ(C::FrankCopula) = _frank_tau_f(C.θ)
+function τ⁻¹(::Type{FrankCopula},τ)
+    s,v = sign(τ),abs(τ)
+    if v == 0
+        return v
+    elseif v == 1
+        return s * Inf
+    else
+        return s*Roots.fzero(x -> _frank_tau_f(x)-v, 0, Inf)
     end
-    x₀ = (1-τ)/4
-    return Roots.fzero(x -> (1-D₁(x))/x - x₀, 1e-4, Inf)
 end
 
 williamson_dist(C::FrankCopula{d,T}) where {d,T} = C.θ > 0 ?  WilliamsonFromFrailty(Logarithmic(-C.θ), d) : WilliamsonTransforms.𝒲₋₁(t -> ϕ(C,t),d)

src/ArchimedeanCopulas/GumbelBarnettCopula.jl

@@ -43,8 +43,14 @@ function τ(C::GumbelBarnettCopula)
     return 1+4*result
 end
 function τ⁻¹(::Type{GumbelBarnettCopula}, tau)
-    if tau == zero(tau)
-        return tau
+    if tau == 0
+        return zero(tau)
+    elseif tau > 0 
+        @warn "GumbelBarnettCopula cannot handle positive kendall tau's, returning independence.."
+        return zero(tau)
+    elseif tau < τ(GumbelBarnettCopula(2,1))
+        @warn "GumbelBarnettCopula cannot handle negative kendall tau's smaller than  ≈ -0.3613, so we capped to that value."
+        return one(tau)
     end
 
     # Define an anonymous function that takes a value x and computes τ 

src/ArchimedeanCopulas/InvGaussianCopula.jl

@@ -28,38 +28,45 @@ struct InvGaussianCopula{d,T} <: ArchimedeanCopula{d}
             throw(ArgumentError("Theta must be non-negative."))
         elseif θ == 0
             return IndependentCopula(d)
-        elseif isinf(θ)
-            throw(ArgumentError("Theta cannot be infinite"))
+        # elseif isinf(θ)
+        #     throw(ArgumentError("Theta cannot be infinite"))
         else
             return new{d,typeof(θ)}(θ)
         end
     end
 end
 
-ϕ(  C::InvGaussianCopula,       t) = exp((1-sqrt(1+2*((C.θ)^(2))*t))/C.θ)
-ϕ⁻¹(C::InvGaussianCopula,       t) = ((1-C.θ*log(t))^(2)-1)/(2*(C.θ)^(2))
-function τ(C::InvGaussianCopula)
-
-    # Calculate the integral using an appropriate numerical integration method
-    result, _ = QuadGK.quadgk( x -> (x*((1-C.θ*log(x))^2-1))/(-2*C.θ*(1-C.θ*log(x))),0,1)
-
-    return 1+4*result
+ϕ(  C::InvGaussianCopula, t) = isinf(C.θ) ? exp(-sqrt(2*t)) : exp((1-sqrt(1+2*((C.θ)^(2))*t))/C.θ)
+ϕ⁻¹(C::InvGaussianCopula, t) = isinf(C.θ) ? ln(t)^2/2 : ((1-C.θ*log(t))^(2)-1)/(2*(C.θ)^(2))
+function _invg_tau_f(θ)
+    if θ == 0
+        return zero(θ)
+    elseif θ > 1e153 # should be Inf, but integrand has issues... 
+        return 1/2
+    elseif θ < sqrt(eps(θ))
+        return zero(θ)
+    end
+    function _integrand(x,θ) 
+        y = 1-θ*log(x)
+        ret = - x*(y^2-1)/(2θ*y)
+        return ret
+    end
+    rez, err = QuadGK.quadgk(x -> _integrand(x,θ),zero(θ),one(θ))
+    rez = 1+4*rez
+    return rez
 end
+τ(C::InvGaussianCopula) = _invg_tau_f(C.θ)
 function τ⁻¹(::Type{InvGaussianCopula}, tau)
     if tau == zero(tau)
         return tau
-    end
-    if tau < 0
+    elseif tau < 0
         @warn "InvGaussianCopula cannot handle negative dependencies, returning independence..."
-        return zero(τ)
+        return zero(tau)
+    elseif tau > 0.5
+        @warn "InvGaussianCopula cannot handle kendall tau greater than 0.5, using 0.5.."
+        return tau * Inf
     end
-
-    # Define an anonymous function that takes a value x and computes τ for an InvGaussianCopula copula with θ = x
-    τ_func(x) = τ(InvGaussianCopula(2,x))
-
-    # Set an initial value for x₀ (adjustable)
-    x = Roots.find_zero(x -> τ_func(x) - tau, (0.0, Inf))
-    return τ
+    return Roots.find_zero(x -> _invg_tau_f(x) - tau, (sqrt(eps(tau)), Inf))
 end
 
 williamson_dist(C::InvGaussianCopula{d,T}) where {d,T} = WilliamsonFromFrailty(Distributions.InverseGaussian(C.θ,1),d)

test/archimedean_tests.jl

@@ -7,16 +7,13 @@
     inv_works(T,tau) = Copulas.τ(T(2,Copulas.τ⁻¹(T,tau))) ≈ tau
     check_rnd(T,min,max,N) = all(inv_works(T,x) for x in min .+ (max-min) .* rand(rng,N))
 
-    # working: 
-    @test check_rnd(ClaytonCopula,-1,1,100)
-    @test check_rnd(GumbelCopula,0,1,100)
-    @test check_rnd(JoeCopula,0,1,100)
-
-    # not working: 
-    @test_broken check_rnd(AMHCopula,(5 - 8*log(2))/3,1/3,100)
-    @test_broken check_rnd(FrankCopula,-1,1,100)
-    @test_broken check_rnd(GumbelBarnettCopula,0,1,100)
-    @test_broken check_rnd(InvGaussianCopula,0,1,100)
+    @test check_rnd(ClaytonCopula,       -1,    1,   10)
+    @test check_rnd(GumbelCopula,         0,    1,   10)
+    @test check_rnd(JoeCopula,            0,    1,   10)
+    @test check_rnd(GumbelBarnettCopula, -0.35, 0,   10)
+    @test check_rnd(AMHCopula,           -0.18, 1/3, 10)
+    @test check_rnd(FrankCopula,         -1,    1,   10)
+    @test check_rnd(InvGaussianCopula,0,1/2,10)
 
 end
 
@@ -71,22 +68,22 @@ end
     data0 = rand(rng,C0,100)
     @test all(pdf(C0,data0) .>= 0)
     @test all(0 .<= cdf(C0,data0) .<= 1)
-    @test_broken fit(FrankCopula,data0)
+    fit(FrankCopula,data0)
 
     C1 = FrankCopula(2,log(rand(rng)))
     data1 = rand(rng,C1,100)
     @test all(pdf(C1,data1) .>= 0)
     @test all(0 .<= cdf(C1,data1) .<= 1)
-    @test_broken fit(FrankCopula,data1)
+    fit(FrankCopula,data1)
 
 
     for d in 2:10
-        for θ ∈ [0.0,rand(rng),1.0,-log(rand(rng)), Inf]
+        for θ ∈ [1.0,1-log(rand(rng)), Inf]
             C = FrankCopula(d,θ)
-            data = rand(rng,C,100)
+            data = rand(rng,C,10000)
             @test all(pdf(C,data) .>= 0)
             @test all(0 .<= cdf(C,data) .<= 1)
-            # fit(FrankCopula,data)
+            fit(FrankCopula,data)
         end
     end
     @test true
@@ -132,7 +129,7 @@ end
             data = rand(rng,C,100)
             @test all(pdf(C,data) .>= 0)
             @test all(0 .<= cdf(C,data) .<= 1)
-            @test_broken fit(GumbelBarnettCopula,data)
+            fit(GumbelBarnettCopula,data)
         end
     end
     @test true
@@ -148,7 +145,7 @@ end
             data = rand(rng,C,100)
             @test all(pdf(C,data) .>= 0)
             @test all(0 .<= cdf(C,data) .<= 1)
-            @test_broken fit(InvGaussianCopula,data)
+            fit(InvGaussianCopula,data)
         end
     end
     @test true