Fix Extreme Values Copulas (#226)

* Fix A function in AsymGalambosCopula to improve readability.

* refactor: Update AsymLogCopula struct with individual asymmetry parameters + Fix the A function

* refactor: Simplify calculation in `A` function in AsymMixedCopula.

* Fully rewrite BC2Copula parameters and methods

Changed the BC2Copula parameters to 'a' and 'b'. Updated the functions
to reflect the new parameters. Added functions τ and ρ for calculations.

* refactor: Rename function `ρₛ` to `ρ` in CuadrasAugeCopula.

* refactor(GalambosCopula.jl): Improve handling of large parameter values

* refactor: Simplify calculation functions in HuslerReissCopula

* refactor: Update specific A function of LogCopula

* refactor: Rename parameters in MOCopula and update related functions

* refactor: Improve spacing in MixedCopula calculation formula

* refactor: Remove unused _pdf function in tEVCopula

* refactor: Simplify _pdf function and add τ and ρ functions to EVCs

src/ExtremeValueCopula.jl

@@ -90,14 +90,6 @@ function D_B_ℓ(C::ExtremeValueCopula, t::Vector{Float64}, B::Vector{Int})
 end
 
 # Función PDF para ExtremeValueCopula usando ℓ
-function _pdf(C::ExtremeValueCopula, u::AbstractArray{<:Real})
-    t = -log.(u)
-    c = exp(-ℓ(C, t))
-    D1 = D_B_ℓ(C, t, [1])
-    D2 = D_B_ℓ(C, t, [2])
-    D12 = D_B_ℓ(C, t, [1, 2])
-    return c * (-D12 + D1 * D2) / (u[1] * u[2])
-end
 function Distributions._logpdf(C::ExtremeValueCopula, u::AbstractArray{<:Real})
     t = -log.(u)
     c = exp(-ℓ(C, t))
@@ -106,26 +98,12 @@ function Distributions._logpdf(C::ExtremeValueCopula, u::AbstractArray{<:Real})
     D12 = D_B_ℓ(C, t, [1, 2])
     return log(c) + log(-D12 + D1 * D2) - log(u[1] * u[2])
 end
-# Definir la función para calcular τ
-function τ(C::ExtremeValueCopula)
-    integrand(x) = begin
-        a = A(C, x)
-        da = dA(C, x)
-        return (x * (1 - x) / a) * da
-    end
-
-    integrate, _ = QuadGK.quadgk(integrand, 0.0, 1.0)
-    return integrate
-end
+# Definir la función para calcular τ and ρ
+# Warning: the τ function can be veeeery unstable... it is actually very hard to compute correctly. 
+# In some case, it simply fails by a lot. 
+τ(C::ExtremeValueCopula) = QuadGK.quadgk(x -> d²A(C, x) * x * (1 - x) / A(C, x), 0.0, 1.0)[1]
+ρ(C::ExtremeValueCopula) = 12 *  QuadGK.quadgk(x -> 1 / (1 + A(C, x))^2, 0, 1)[1] - 3
 
-function ρₛ(C::ExtremeValueCopula)
-    integrand(x) = 1 / (1 + A(C, x))^2
-
-    integral, _ = QuadGK.quadgk(integrand, 0, 1)
-
-    ρs = 12 * integral - 3
-    return ρs
-end
 # Función para calcular el coeficiente de dependencia en el límite superior
 function λᵤ(C::ExtremeValueCopula)
     return 2(1 - A(C, 0.5))
@@ -150,9 +128,6 @@ function Distributions._rand!(rng::Distributions.AbstractRNG, C::ExtremeValueCop
     u1, u2 = rand(rng, Distributions.Uniform(0,1), 2)
     z = rand(rng, ExtremeDist(C))
     p = probability_z(C, z)
-    if p < -eps() || p > eps()
-        p = 0
-    end
     c = rand(rng, Distributions.Bernoulli(p))
     w = 0
     if c == 1

src/ExtremeValueCopulas/AsymGalambosCopula.jl

@@ -46,14 +46,7 @@ struct AsymGalambosCopula{P} <: ExtremeValueCopula{P}
 end
 
 function A(C::AsymGalambosCopula, t::Real)
-    α = C.α
-    θ = C.θ
-
-    term1 = (θ[1] * t)^(-α)
-    term2 = (θ[2] * (1 - t))^(-α)
-
-    inner_term = term1 + term2
-
-    result = 1 - inner_term^(-1 / α)
-    return result
+    x₁ = - C.α * log(C.θ[1] * t)
+    x₂ = - C.α * log(C.θ[2] * (1-t))
+    return -expm1(-LogExpFunctions.logaddexp(x₁,x₂) / C.α)
 end

src/ExtremeValueCopulas/AsymLogCopula.jl

@@ -21,7 +21,8 @@ References:
 """
 struct AsymLogCopula{P} <: ExtremeValueCopula{P}
     α::P  # Dependence Parameter
-    θ::Vector{P}  # Asymmetry parameters (size 2)
+    θ₁::P
+    θ₂::P
     function AsymLogCopula(α::P, θ::Vector{P}) where {P}
         if length(θ) != 2
             throw(ArgumentError("The vector θ must have 2 elements for the bivariate case"))
@@ -30,15 +31,8 @@ struct AsymLogCopula{P} <: ExtremeValueCopula{P}
         elseif  !(0 <= θ[1] <= 1)  || !(0 <= θ[2] <= 1)  
             throw(ArgumentError("All parameters θ must be in the interval [0, 1]"))
         else
-            return new{P}(α, θ)
+            return new{P}(α, θ[1],θ[2])
         end
     end
 end
-
-function A(C::AsymLogCopula, t::Real)
-    α = C.α
-    θ = C.θ
-
-    A = ((θ[1]^α)*(1-t)^α + (θ[2]^α)*(t^α))^(1/α)+(θ[1]- θ[2])*t + 1 -θ[1]  
-    return A
-end
+A(C::AsymLogCopula, t::Real) = ((C.θ₁^C.α)*(1-t)^C.α + (C.θ₂^C.α)*(t^C.α))^(1/C.α)+(C.θ₁- C.θ₂)*t + 1 - C.θ₁

src/ExtremeValueCopulas/AsymMixedCopula.jl

@@ -52,8 +52,6 @@ struct AsymMixedCopula{P} <: ExtremeValueCopula{P}
 end
 
 function A(C::AsymMixedCopula, t::Real)
-    θ = C.θ
-
-    a = θ[2]*t^3 + θ[1]*t^2-(θ[1]+θ[2])*t+1
-    return a
+    θ₁, θ₂ = C.θ[1], C.θ[2]
+    return θ₂*t^3 + θ₁*t^2 - (θ₁+θ₂)*t + 1
 end

src/ExtremeValueCopulas/BC2Copula.jl

@@ -3,75 +3,63 @@
 
 Fields:
 
-    - θ1::Real - parameter
-    - θ1::Real - parameter
+    - a::Real - parameter
+    - a::Real - parameter
     
 Constructor
 
-    BC2Copula(θ1, θ2)
+    BC2Copula(a, b)
 
-The bivariate BC₂ copula is parameterized by two parameters ``\\theta_{i} \\in [0,1], i=1,2``. It is an Extreme value copula with Pickands dependence function: 
+The bivariate BC2 copula is parameterized by two parameters ``a,b \\in [0,1]``. It is an Extreme value copula with Pickands dependence function: 
 
 ```math
-A(t) = \\max\\{\\theta_1 t, \\theta_2(1-t) \\} + \\max\\{(1-\\theta_1)t, (1-\\theta_2)(1-t)\\}
+A(t) = \\max\\{a t, b (1-t) \\} + \\max\\{(1-a)t, (1-b)(1-t)\\}
 ```
 
 References:
 * [mai2011bivariate](@cite) Mai, J. F., & Scherer, M. (2011). Bivariate extreme-value copulas with discrete Pickands dependence measure. Extremes, 14, 311-324. Springer, 2011.
 """
 struct BC2Copula{P} <: ExtremeValueCopula{P}
-    θ1::P
-    θ2::P
-
-    function BC2Copula(θ::Vararg{Real})
-        if length(θ) !== 2
-            throw(ArgumentError("BC2Copula requires only 2 arguments."))
-        end
-        T = promote_type(typeof(θ[1]), typeof(θ[2]))
-        θ1, θ2 = T(θ[1]), T(θ[2])
-
-        if !(0 <= θ1 <= 1) || !(0 <= θ2 <= 1) 
-            throw(ArgumentError("All θ parameters must be in [0,1]"))
+    a::P
+    b::P
+    function BC2Copula(a,b)
+        T = promote_type(typeof(a), typeof(b))
+        if !(0 <= a <= 1) || !(0 <= b <= 1) 
+            throw(ArgumentError("Both parameters a and b must be in [0,1]"))
         end
-        return new{T}(θ1, θ2)
+        return new{T}(T(a), T(b))
     end
 end
 
 function ℓ(C::BC2Copula, t::Vector)
-    θ1, θ2 = C.θ1, C.θ2
+    a, b = C.a, C.b
     t₁, t₂ = t
-    return max(θ1*t₁, θ2*t₂) + max((1-θ1)*t₁, (1-θ2)*t₂)
+    return max(a*t₁, b*t₂) + max((1-a)*t₁, (1-b)*t₂)
 end
 
 function A(C::BC2Copula, t::Real)
-    θ1, θ2 = C.θ1, C.θ2
-    return max(θ1*t, θ2*(1-t)) + max((1-θ1)*t, (1-θ2)*(1-t))
+    a, b = C.a, C.b
+    return max(a*t, b*(1-t)) + max((1-a)*t, (1-b)*(1-t))
 end
 
 function dA(C::BC2Copula, t::Float64)
-    θ1, θ2 = C.θ1, C.θ2
-
-    # Conditions for the derivative of the first part
-    if θ1*t >= θ2*(1-t)
-        f1_derivative = θ1
-    else
-        f1_derivative = -θ2
-    end
-
-    # Conditions for the derivative of the second part
-    if (1-θ1)*t >= (1-θ2)*(1-t)
-        f2_derivative = 1-θ1
-    else
-        f2_derivative = -(1-θ2)
-    end
-
-    return f1_derivative + f2_derivative
+    a, b = C.a, C.b
+    der1 =  a*t >= b*(1-t) ? a : -b
+    der2 = (1-a)*t >= (1-b)*(1-t) ? -a : -(1-b)
+    return der1 + der2
 end
 
 function Distributions._rand!(rng::Distributions.AbstractRNG, C::BC2Copula, u::AbstractVector{T}) where {T<:Real}
-    θ1, θ2 = C.θ1, C.θ2
+    a, b = C.a, C.b
     v1, v2 = rand(rng, Distributions.Uniform(0,1), 2)
-    u[1] = max(v1^(1/θ1),v2^(1/(1-θ1)))
-    u[2] = max(v1^(1/θ2),v2^(1/(1-θ2)))
+    u[1] = max(v1^(1/a), v2^(1/(1-a)))
+    u[2] = max(v1^(1/b), v2^(1/(1-b)))
     return u
+end
+
+τ(C::BC2Copula) = 1 - abs(C.a - C.b)
+
+function ρ(C::BC2Copula)
+    a,b = C.a, C.b
+    return 2 * (a + b + a*b + max(a,b) - 2a^2 - 2b^2) / (3 - a - b - min(a,b)) / (a + b + max(a,b))
 end

src/ExtremeValueCopulas/CuadrasAugeCopula.jl

@@ -39,7 +39,7 @@ dA(C::CuadrasAugeCopula, t::Real) = t <= 0.5 ? -C.θ : C.θ
 
 τ(C::CuadrasAugeCopula) = C.θ/(2-C.θ)
 
-ρₛ(C::CuadrasAugeCopula) = (3*C.θ)/(4-C.θ)
+ρ(C::CuadrasAugeCopula) = (3*C.θ)/(4-C.θ)
 
 # specific ℓ específica of Cuadras-Augé Copula
 function ℓ(C::CuadrasAugeCopula, t::Vector)

src/ExtremeValueCopulas/GalambosCopula.jl

@@ -27,7 +27,7 @@ struct GalambosCopula{P} <: ExtremeValueCopula{P}
     θ::P  # Copula parameter
     function GalambosCopula(θ)
         if θ > 19.5
-            @warn "The parameter θ = $(θ) is large, which may lead to numerical instability in any functions. Consider regularizing the input."
+            @info "GalambosCopula(θ=$(θ)): θ is large, which may lead to numerical issues."
         end
         if θ < 0
             throw(ArgumentError("Theta must be >= 0"))
@@ -41,7 +41,7 @@ struct GalambosCopula{P} <: ExtremeValueCopula{P}
     end
 end
 
-A(C::GalambosCopula, t::Real) = 1 - (t^(-C.θ) + (1 - t)^(-C.θ))^(-1/C.θ)
+A(C::GalambosCopula, t::Real) = -expm1(-LogExpFunctions.logaddexp(-C.θ*log(t),-C.θ*log(1-t))/C.θ)
 # This auxiliary function helps determine if we need binary search or not in the generation of random samples
 function needs_binary_search(C::GalambosCopula)
     return C.θ > 19.5

src/ExtremeValueCopulas/HuslerReissCopula.jl

@@ -50,11 +50,7 @@ function A(H::HuslerReissCopula, t::Real)
     θ = H.θ
     term1 = t * Distributions.cdf(Distributions.Normal(), θ^(-1) + 0.5 * θ * log(t / (1 - t)))
     term2 = (1 - t) * Distributions.cdf(Distributions.Normal(), θ^(-1) + 0.5 * θ * log((1 - t) / t))
-
-    a = term1 + term2
-
-
-    return a
+    return term1 + term2
 end
 
 function dA(H::HuslerReissCopula, t::Real)
@@ -66,7 +62,5 @@ function dA(H::HuslerReissCopula, t::Real)
     dA_term2 = -Distributions.cdf(Distributions.Normal(), θ^(-1) + 0.5 * θ * log((1 - t) / t)) + 
                   (1 - t) * Distributions.pdf(Distributions.Normal(), θ^(-1) + 0.5 * θ * log((1 - t) / t)) * (0.5 * θ * (-1 / t - 1 / (1 - t)))
 
-    da = dA_term1 + dA_term2
-
-    return da
+    return dA_term1 + dA_term2
 end

src/ExtremeValueCopulas/LogCopula.jl

@@ -42,8 +42,5 @@ function ℓ(G::LogCopula, t::Vector)
     t₁, t₂ = t
     return (t₁^θ + t₂^θ)^(1/θ)
 end
-# #  specific A funcion of HuslerReissCopula
-function A(C::LogCopula, t::Real)
-    θ = C.θ
-    return (t^θ + (1 - t)^θ)^(1/θ)
-end
+# #  specific A funcion of LogCopula
+A(C::LogCopula, t::Real) = exp(LogExpFunctions.logaddexp(C.θ*log(t),C.θ*log(1-t))/C.θ)

src/ExtremeValueCopulas/MOCopula.jl

@@ -3,9 +3,9 @@
 
 Fields:
 
-    - λ1::Real - parameter
-    - λ2::Real - parameter
-    - λ12::Real - parameter
+    - λ₁::Real - parameter
+    - λ₂::Real - parameter
+    - λ₁₂::Real - parameter
     
 Constructor
 
@@ -21,49 +21,22 @@ References:
 * [mai2012simulating](@cite) Mai, J. F., & Scherer, M. (2012). Simulating copulas: stochastic models, sampling algorithms, and applications (Vol. 4). World Scientific.
 """
 struct MOCopula{P} <: ExtremeValueCopula{P}
-    λ1::P
-    λ2::P
-    λ12::P
-
-    function MOCopula(λ::Vararg{Real})
-        if length(λ) !== 3
-            throw(ArgumentError("MOCopula requires only 3 arguments."))
-        end
-
-        T = promote_type(typeof(λ[1]), typeof(λ[2]), typeof(λ[3]))
-        λ1, λ2, λ12 = T(λ[1]), T(λ[2]), T(λ[3])
-
-        if λ1 < 0 || λ2 < 0 || λ12 < 0
+    a::P
+    b::P
+    function MOCopula(λ₁,λ₂,λ₁₂)
+        if λ₁ < 0 || λ₂ < 0 || λ₁₂ < 0
             throw(ArgumentError("All λ parameters must be >= 0"))
         end
-
-        return new{T}(λ1, λ2, λ12)
+        a, b = λ₁ / (λ₁ + λ₁₂), λ₂ / (λ₂ + λ₁₂)
+        return new{typeof(a)}(a,b)
     end
 end
-
-function A(C::MOCopula, t::Real)
-    λ1, λ2, λ12 = C.λ1, C.λ2, C.λ12
-    return ((λ1 * (1 - t))/(λ1 + λ12)) + ((λ2 * t)/(λ2 + λ12)) + λ12 * max((1-t)/(λ1 + λ12), t/(λ2 + λ12))
-end
-
-function ℓ(C::MOCopula, t::Vector)
-    λ1, λ2, λ12 = C.λ1, C.λ2, C.λ12
-    t₁, t₂ = t
-    return ((λ1 * t₂)/(λ1 + λ12)) + ((λ2 * t₁)/(λ2 + λ12)) + λ12 * max(t₂/(λ1 + λ12), t₁/(λ2 + λ12))
-end
-
-function _cdf(C::MOCopula, u::AbstractArray{<:Real})
-    λ1, λ2, λ12 = C.λ1, C.λ2, C.λ12
-    u₁, u₂ = u
-    return min(u₂*u₁^(1 - λ12/(λ1 + λ12)), u₁ * u₂^(1 - λ12/(λ2 + λ12)))
-end
-
+A(C::MOCopula, t::Real) = max(t + (1-t)*C.b, (1-t)+C.a*t)
+_cdf(C::MOCopula, u::AbstractArray{<:Real}) = min(u[1]^C.a * u[2], u[1] * u[2]^C.b)
 function Distributions._rand!(rng::Distributions.AbstractRNG, C::MOCopula, u::AbstractVector{T}) where {T<:Real}
-    λ1, λ2, λ12 = C.λ1, C.λ2, C.λ12
-    r, s, t = rand(rng, Distributions.Uniform(0,1),3)
-    x = min(-log(r)/λ1, -log(t)/λ12)
-    y = min(-log(s)/λ2, -log(t)/λ12)
-    u[1] = exp(-(λ1+λ12)*x)
-    u[2] = exp(-(λ2+λ12)*y)
+    r, s, t = -log.(rand(rng,3)) # Exponentials(1)
+    u[1] = exp(-min(r/(1-C.a), t/C.a))
+    u[2] = exp(-min(s/(1-C.b), t/C.b))
     return u
 end
+τ(C::MOCopula) = C.a*C.b/(C.a+C.b-C.a*C.b)

src/ExtremeValueCopulas/MixedCopula.jl

@@ -34,4 +34,4 @@ struct MixedCopula{P} <: ExtremeValueCopula{P}
     end
 end
 
-A(C::MixedCopula, t::Real) = C.θ*t^2 - C.θ*t + 1
+A(C::MixedCopula, t::Real) = C.θ * t^2 - C.θ * t + 1

src/ExtremeValueCopulas/tEVCopula.jl

@@ -93,16 +93,6 @@ function d²A(C::tEVCopula, t::Real)
     d2A = (dA_plus - dA_minus) / (2 * h)
     return d2A
 end
-
-# PDF function for ExtremeValueCopula using ℓ
-function _pdf(C::tEVCopula, u::AbstractArray{<:Real})
-    t = -log.(u)
-    c = exp(-ℓ(C, t))
-    D1 = D_B_ℓ(C, t, [1])
-    D2 = D_B_ℓ(C, t, [2])
-    D12 = D_B_ℓ(C, t, [1, 2])
-    return c * (-D12 + D1 * D2) / (u[1] * u[2])
-end
 function Distributions._logpdf(C::tEVCopula, u::AbstractArray{<:Real})
     t = -log.(u)
     c = exp(-ℓ(C, t))