# ZKSI/DatagenCopulaBased.jl

copula based data generator
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# DatagenCopulaBased.jl

Copula based data generator. Returns data in a form of a matrix U: size(U) = (t,n) - being t realisations of n-variate random variable. Be default each marginal, i.e. U[:,i], is uniformly distributed on [0,1]. Interdependence between marginals is modelled by appropriate n-variate copula function, see e.g.: U. Cherubini, E. Luciano, W. Vecchiato, 'Copula Methods in Finance', Wiley 2004.

This module support following copula families:

• Elliptical copulas (Gaussian, t-Student),
• Archimedean copulas (Clayton, Frank, Gumbel, Ali-Mikhail-Haq), including nested ones,
• Frechet familly copulas (maximal, minimal, independent),
• Marshall-Olkin copulas.

## Installation

Within Julia, run

to install the files Julia 0.7 or higher is required.

## Elliptical copulas

We use elliptical multivariate distribution (such as Gaussian or t-Student) to construct a copula. Suppose F(x₁, ..., xₙ) is a cumulative density function (cdf.) of such multivariate distribution, and Fᵢ(xᵢ) is univariate cdf. of its i th marginal. Hence uᵢ = Fᵢ(xᵢ) is from the uniform distribution on [0,1], and the elliptical copula is: C(u₁, ..., uₙ) = F(F₁⁻¹(u₁), ..., Fₙ⁻¹(uₙ)).

### Gaussian copula

julia> gausscopulagen(t::Int, Σ::Matrix{Float64} = [1. 0.5; 0.5 1.])

The function returns U: size(U) = (t,n) - t realisations of n-variate random variable, each marginal, i.e. U[:,i], is uniformly distributed on [0,1] and a cross-correlation is modelled by a Gaussian copula, parametrised by the symmetric positively defined correlation matrix Σ with ones on diagonals Σᵢᵢ=1 and all elements -1 ≤ Σᵢⱼ ≤ 1 . Number of marginal variables is n = size(Σ, 1) = size(Σ, 2). If the symmetric covariance matrix without ones on a diagonals is imputed, it will be converted into a correlation matrix automatically.

julia> using Random

julia> Random.seed!(43);

julia> gausscopulagen(10)
10×2 Array{Float64,2}:
0.589188  0.815308
0.708285  0.924962
0.747341  0.156994
0.227634  0.183116
0.227575  0.957376
0.271558  0.364803
0.445691  0.52792
0.585362  0.23135
0.498593  0.48266
0.190283  0.594451

### t-Student copula

julia> tstudentcopulagen(t::Int, Σ::Matrix{Float64} = [1. 0.5; 0.5 1.], ν::Int=10)

The function returns U: size(U) = (t,n) - t realisations of n-variate random variable, each marginal, i.e. U[:,i], is uniformly distributed on [0,1] and a cross-correlation is modelled by a t-Student copula parametrised by the symmetric matrix Σ (with ones on diagonals as in a Gaussian case) and by a numver ν ∈ N.

julia> Random.seed!(43);

julia> tstudentcopulagen(10)
10×2 Array{Float64,2}:
0.658199  0.937148
0.718244  0.92602
0.809521  0.0980325
0.263068  0.222589
0.187187  0.971109
0.245373  0.346428
0.452336  0.524498
0.57113   0.272525
0.498443  0.48082
0.113788  0.633349

## Archimedean copulas

Archimedean one parameter bivariate copula C(u₁,u₂) = φ⁻¹(φ(u₁)+φ(u₂)) is defined by using the continuous strictly decreasing generator parametrised by θ, such that φ(t): [0,1] → [0, ∞) and φ⁻¹(s) is the pseudo-inverse.

We define similarly n-variate Archimedean copula C(u₁,..., uₙ) = φ⁻¹(φ(u₁)+...+φ(uₙ)). Here constrains for theθ parameter are more strict, see: M. Hofert, 'Sampling Archimedean copulas', Computational Statistics & Data Analysis, 52 (2008), 5163-5174.

• Clayton copula - keyword = "clayton": θ ∈ (0, ∞) for n > 2 and θ ∈ [-1, 0) ∪ (0, ∞) for n = 2,
• Frank copula - keyword = "frank": θ ∈ (0, ∞) for n > 2 and θ ∈ (-∞, 0) ∪ (0, ∞) for n = 2,
• Gumbel copula - keyword = "gumbel": θ ∈ [1, ∞),
• Ali-Mikhail-Haq copula - keyword = "amh": θ ∈ (0, 1) for n > 2 and θ ∈ [-1, 1] for n = 2.

For2-dimensional copula generate algorithms see P. Kumar, Probability Distributions and Estimation of Ali-Mikhail-Haq Copula, Applied Mathematical Sciences, Vol. 4, 2010, no. 14, 657 - 666, and R. Nelsen 'An Introduction to Copulas', Springer Science & Business Media, 1999 - 216.

To generate t realisations of n-variate data from Archimedean copula with parameter θ run

julia> archcopulagen(t::Int, n::Int, θ::Union{Float64, Int}, copula::String; rev::Bool = false, cor::String = "")

The function returns U: size(U) = (t,n) - t realisations of n-variate random variable, each marginal, i.e. U[:,i], is uniformly distributed on [0,1] and a cross-correlation is modelled by corresponding Archimedean copula.

julia> Random.seed!(43);

julia> archcopulagen(10, 2, 1, "clayton")
10×2 Array{Float64,2}:
0.770331  0.932834
0.472847  0.0806845
0.970749  0.653029
0.622159  0.0518025
0.402461  0.228549
0.946375  0.842883
0.809076  0.129038
0.747983  0.433829
0.374341  0.437269
0.973066  0.910103
• If cor = Kendall, uses Kendall's τ correlation coefficients θ.
• If cor = Spearman, uses Spearman ρ correlation coefficient instead of θ.
• If reversed = true returns data from reversed copula.

To generate data from reversed copula:

• Generated data from corresponding copula [u₁, ..., uᵢ, ..., uₙ],
• Perform transformation ∀ᵢ uᵢ → 1-uᵢ.

For modelling justification see: K. Domino, T. Błachowicz, M. Ciupak, 'The use of copula functions for predictive analysis of correlations between extreme storm tides', Physica A: Statistical Mechanics and its Applications 413, 489-497, 2014, and K. Domino, T. Błachowicz, 'The use of copula functions for modeling the risk of investment in shares traded on the Warsaw Stock Exchange', Physica A: Statistical Mechanics and its Applications 413, 77-85, 2014.

### Nested Archimedean copulas

To generate t realisations of ∑ᵢ nᵢ + m variate data from nested Archimedean copulas, McNeil, A.J., 2008. 'Sampling nested Archimedean copulas'. Journal of Statistical Computation and Simulation 78, 567–581, run:

julia> nestedarchcopulagen(t::Int, n::Vector{Int}, ϕ::Vector{Float64}, θ::Float64, copula::String, m::Int = 0)

Here n is a vector of number of variates of child copulas, ϕ are their parameters, θ is a parameter of parents copula. Here last m variates are generated using parents copula only. Only such nesting that child and parents copulas are from the same family is supported. Nesting condition requires 0 < θ ≤ minimum(ϕ).

julia> Random.seed!(43);

julia> nestedarchcopulagen(10, [2,2], [2., 3.], 1.1, "clayton", 1)
10×5 Array{Float64,2}:
0.333487  0.584206   0.970471  0.352363  0.793386
0.249313  0.0802689  0.298697  0.46432   0.78807
0.765832  0.272857   0.461754  0.125465  0.661781
0.897061  0.346811   0.745457  0.899775  0.477065
0.387096  0.268233   0.533175  0.42922   0.294137
0.42065   0.247676   0.641627  0.538728  0.507315
0.598049  0.138186   0.659411  0.876095  0.19471
0.125968  0.0643853  0.824152  0.601356  0.662842
0.57524   0.625373   0.688956  0.57825   0.545043
0.96839   0.899199   0.827176  0.544107  0.862751

#### If copula == "gumbel" further nesting is supported.

To generate t realisations of length(θ)+1 variate data from hierarchically nested Gumbel copula: C_θₙ(... C_θ₂(C_θ₁(u₁, u₂), u₃)..., uₙ) run:

julia>   nestedarchcopulagen(t::Int, θ::Vector{Float64}, copula::String = "gumbel")

Nesting condition 1 ≤ θ_{i+1} ≤ θᵢ

julia> Random.seed!(43);

julia> x = nestedarchcopulagen(5, [4., 3., 2.], "gumbel")
5×4 Array{Float64,2}:
0.832902  0.915821   0.852532  0.903184
0.266333  0.293338   0.307899  0.0346497
0.152431  0.0432532  0.319465  0.42015
0.812182  0.685689   0.721783  0.554992
0.252867  0.521345   0.406719  0.511759

To generate t realisations of ∑ᵢ ∑ⱼ nᵢⱼ variate data from double nested gumbel copula: C_θ(C_ϕ₁(C_Ψ₁₁(u,...), ..., C_C_Ψ₁,ₗ₁(u...)), ..., C_ϕₖ(C_Ψₖ₁(u,...), ..., C_Ψₖ,ₗₖ(u,...))) where lᵢ = length(n[i]) run:

julia> nestedarchcopulagen::Int, n::Vector{Vector{Int}}, Ψ::Vector{Vector{Float64}}, ϕ::Vector{Float64}, θ₀::Float64, copula::String = "gumbel")
julia> Random.seed!(43);

julia> x = nestedarchcopulagen(5, [[2,2],[2]], [[3., 2.], [4.]], [1.5, 2.1], 1.2, "gumbel")
5×6 Array{Float64,2}:
0.464403  0.711722   0.883035   0.896706   0.888614   0.826514
0.750596  0.768193   0.0659561  0.0252472  0.996014   0.989127
0.825211  0.712079   0.581356   0.507739   0.882675   0.84959
0.276326  0.0827071  0.240836   0.434629   0.0184611  0.031363
0.208422  0.504727   0.27561    0.639089   0.481855   0.573715

### Chain of bivariate Archimedean copulas

To generate t realisations of length(θ)+1 variate data, using a chain of one parameter bivariate Archimedean copulas parametrised by θᵢ for - i'th and i+1'th marginal:

julia> chaincopulagen(t::Int, θ::Union{Vector{Float64}, Vector{Int}}, copula::Vector{String}; rev::Bool = false, cor::String = "")

In other words ∀i∈[1, length(θ)] data are generated form Archimedean copula C_{θᵢ}(uᵢ, u_{i+1}). Due to features of bivariate copulas, each marginal uᵢ is uniformly distributed on [0,1], hence we got a multivariate copula, defined by subsequent bivariate sub-copulas. The cross-corelation between marginals i and j: i ≠ j+1 are introduced by a chain of bivariate copulas.

Following families are supported: "clayton", "frank" and "amh" - Ali-Mikhail-Haq. Conditions for θᵢ parameters ranges such as in corresponding bivariate copula case.

julia> Random.seed!(43);

julia> chaincopulagen(10, [4., 11.], ["frank", "frank"])
10×3 Array{Float64,2}:
0.180975  0.386303   0.879254
0.775377  0.247895   0.144803
0.888934  0.426854   0.772457
0.924876  0.395564   0.223155
0.408278  0.139002   0.142997
0.912603  0.901252   0.949828
0.828727  0.0295759  0.0897796
0.400537  0.0337673  0.27872
0.429437  0.462771   0.425435
0.955881  0.953623   0.969038

## Marshall-Olkin copula

To generate t realisations of n-variate data from Marshall-Olkin copula with parameter series λ with non-negative elements λₛ, run:

julia> marshallolkincopulagen(t::Int, λ::Vector{Float64}; reverse::Bool = false)

Number of marginals is n = ceil(Int, log(2, length(λ)-1)). Parameters are ordered as follow: λ = [λ₁, λ₂, ..., λₙ, λ₁₂, λ₁₃, ..., λ₁ₙ, λ₂₃, ..., λₙ₋₁ₙ, λ₁₂₃, ..., λ₁₂...ₙ] If reversed = true, returns data from reversed Marshall-Olkin copula , i.e. generates data [u₁, ..., uᵢ, ..., uₙ] from given Marshall-Olkin copula and perform transformation ∀ᵢ uᵢ → 1-uᵢ

julia> Random.seed!(43);

julia> marshallolkincopulagen(10, [0.2, 1.2, 1.6])
10×2 Array{Float64,2}:
0.99636   0.994344
0.167268  0.0619408
0.977418  0.965093
0.495167  0.0247053
0.410336  0.250159
0.778989  0.678064
0.50927   0.350059
0.925875  0.887095
0.353646  0.219006
0.782477  0.686799

To generate data from the Marshall-Olkin copula we use algorithm presented P. Embrechts, F. Lindskog, A McNeil 'Modelling Dependence with Copulas and Applications to Risk Management', 2001 ∗∗

## Frechet family copulas

To generate t realisation of n variate one parameter Frechet copula Cf = α C_{max} + (1-α) C_{⟂}, where 0 ≤ α ≤ 1 run:

julia> frechetcopulagen(t::Int, n::Int, α::Union{Int, Float64})
julia> Random.seed!(43);

julia> frechetcopulagen(10, 2, 0.5)
10×2 Array{Float64,2}:
0.180975  0.661781
0.775377  0.775377
0.888934  0.125437
0.924876  0.924876
0.408278  0.408278
0.912603  0.740184
0.828727  0.00463791
0.400537  0.0288987
0.429437  0.429437
0.955881  0.851275

Two parameters Frechet copula, C = α C_{max} + β C_{min} + (1- α - β) C_{⟂} is supported only for n == 2:

julia> frechetcopulagen(t::Int, n::Int, α::Union{Int, Float64}, β::Union{Int, Float64})

Here where 0 ≤ α , where 0 ≤ β and α + β ≤ 1

julia> Random.seed!(43);

julia> frechetcopulagen(10, 2, 0.4, 0.2)
10×2 Array{Float64,2}:
0.180975  0.661781
0.775377  0.775377
0.888934  0.125437
0.924876  0.924876
0.408278  0.591722
0.912603  0.740184
0.828727  0.171273
0.400537  0.0288987
0.429437  0.429437
0.955881  0.851275

### Chain of bivariate Frechet copulas

To generate t realisations of length(α)+1 multivariate data using a chain two parameter bivariate Frechet copulas with parameter αᵢ and βᵢ for each neighbour (i'th and i+1'th) marginals run:

julia> chainfrechetcopulagen(t::Int, α::Vector{Float64}, β::Vector{Float64} = zeros(α))

In other words ∀i∈[1, length(θ)] data are generated from following Frechet copula C_{αᵢ,βᵢ}(uᵢ, u_{i+1}). Due to features of bivariate copulas, each marginal uᵢ is uniformly distributed on [0,1], hence we got a multivariate copula, defined by subsequent bivariate sub-copulas. The relation between marginals i and j: i ≠ j+1 are defined by a sequence of bivariate copulas.

julia> srand(43)

julia> chainfrechetcopulagen(10, [0.6, 0.4], [0.3, 0.5])
10×3 Array{Float64,2}:
0.996764  0.996764  0.996764
0.204033  0.795967  0.204033
0.979901  0.979901  0.0200985
0.120669  0.879331  0.120669
0.453027  0.453027  0.453027
0.800909  0.199091  0.800909
0.54892   0.54892   0.54892
0.933832  0.933832  0.0661679
0.396943  0.396943  0.396943
0.804096  0.851275  0.955881

## Correlation matrix generation

We supply a few methods to generate a n x n correlation matrix Σ.

### Fully random cases

to generate randomly a correlation matrix run

julia> cormatgen(n::Int)

or

julia> cormatgen_rand(n::Int)

for different methods we have different outputs:

julia> Random.seed!(43);

julia> cormatgen(4)
4×4 Array{Float64,2}:
1.0       0.396865  0.339354  0.193335
0.396865  1.0       0.887028  0.51934
0.339354  0.887028  1.0       0.551519
0.193335  0.51934   0.551519  1.0

julia> cormatgen_rand(4)
4×4 Array{Float64,2}:
1.0       0.659183  0.916879  0.486979
0.659183  1.0       0.676167  0.808264
0.916879  0.676167  1.0       0.731206
0.486979  0.808264  0.731206  1.0

### Deterministic cases

To generate a correlation matrix with constant elements run:

julia> cormatgen_constant(n::Int, α::Float64)

parameter α should satisfy 0 <= α <= 1

julia> cormatgen_constant(4, 0.4)
4×4 Array{Float64,2}:
1.0  0.4  0.4  0.4
0.4  1.0  0.4  0.4
0.4  0.4  1.0  0.4
0.4  0.4  0.4  1.0

the generalisation is

julia> cormatgen_two_constant(n::Int, α::Float64, β::Float64)

parameters should satisfy 0 <= α <= 1 and α > β.

julia> cormatgen_two_constant(4, 0.5, 0.2)
4×4 Array{Float64,2}:
1.0  0.5  0.2  0.2
0.5  1.0  0.2  0.2
0.2  0.2  1.0  0.2
0.2  0.2  0.2  1.0

to generate Toeplitz matrix with parameter 0 <= ρ <= 1 run:

julia> cormatgen_toeplitz(n::Int, ρ::Float64)

julia> cormatgen_toeplitz(4, 0.5)
4×4 Array{Float64,2}:
1.0    0.5   0.25  0.125
0.5    1.0   0.5   0.25
0.25   0.5   1.0   0.5
0.125  0.25  0.5   1.0

### Partially random and partially deterministic cases

To generate constant matrix with noise run:

julia> cormatgen_constant_noised(n::Int, α::Float64; ϵ::Float64 = (1.-α)/2.)

where the parameter ϵ must satisfy 0 <= ϵ <= 1-α

julia> Random.seed!(43);

julia> cormatgen_constant_noised(4, 0.5)
4×4 Array{Float64,2}:
1.0       0.314724  0.590368  0.346992
0.314724  1.0       0.314256  0.512183
0.590368  0.314256  1.0       0.538089
0.346992  0.512183  0.538089  1.0

Analogically generate noised two constants matrix run

julia> Random.seed!(43);

julia> cormatgen_two_constant_noised(4, 0.5, 0.2)
4×4 Array{Float64,2}:
1.0        0.314724  0.290368  0.0469922
0.314724   1.0       0.014256  0.212183
0.290368   0.014256  1.0       0.238089
0.0469922  0.212183  0.238089  1.0

Finally to generate noised Toeplitz matrix run:

julia> cormatgen_toeplitz_noised(n::Int, ρ::Float64; ϵ=(1-ρ)/(1+ρ)/2)

where the parameter ϵ must satisfy 0 <= ϵ <= (1-ρ)/(1+ρ)

julia> Random.seed!(43);

julia> cormatgen_toeplitz_noised(4, 0.5)
4×4 Array{Float64,2}:
1.0        0.376483  0.310246  0.0229948
0.376483   1.0       0.376171  0.258122
0.310246   0.376171  1.0       0.525393
0.0229948  0.258122  0.525393  1.0

## Changes the subset of marginals of multivariate Gaussian distributed data

To change a chosen marginals subset ind of multivariate Gaussian distributed data x by means of t-Student sub-copula with a parameter ν run:

julia> gcop2tstudent(x::Matrix{Float64}, ind::Vector{Int}, ν::Int; naive::Bool = false)

all univariate marginal distributions are Gaussian and unaffected by a transformation.

julia> Σ = [1. 0.5 0.5; 0.5 1. 0.5; 0.5 0.5 1];

julia> Random.seed!(42);

julia> x = Array(rand(MvNormal(Σ), 6)')
6×3 Array{Float64,2}:
-0.556027  -0.662861   -0.384124
-0.299484   1.38993    -0.571326
-0.468606  -0.0990787  -2.3464
1.00331    1.43902     0.966819
0.518149   1.55065     0.989712
-0.886205   0.149748   -1.54419

julia> gcop2tstudent(x, [1,2], 6)
6×3 Array{Float64,2}:
-0.519458  -0.498377   -0.384124
-0.37937    1.66806    -0.571326
-0.432902  -0.0178933  -2.3464
1.01216    1.50814     0.966819
0.226484   1.12436     0.989712
-0.727203   0.238701   -1.54419

To change a chosen marginals subset inds[i][2] of multivariate Gaussian distributed data x by means of Archimedean sub-copula of family inds[i][1] run:

julia> gcop2arch(x::Matrix{Float64}, inds::Vector{Pair{String,Vector{Int64}}}; naive::Bool = false, notnested::Bool = false)

many disjoint subsets numbered by i with different Archimedean sub-copulas are possible. As before all univariate marginal distributions are Gaussian and unaffected by a transformation. Named parameter naive indicates a use of a naive algorithm of data substitution. Named parameter notnested means the use of one parameter Archimedean copula instead of a nested one.

julia> Σ = [1. 0.5 0.5; 0.5 1. 0.5; 0.5 0.5 1];

julia> Random.seed!(42)

julia> x = Array(rand(MvNormal(Σ), 6)')
6×3 Array{Float64,2}:
-0.556027  -0.662861   -0.384124
-0.299484   1.38993    -0.571326
-0.468606  -0.0990787  -2.3464
1.00331    1.43902     0.966819
0.518149   1.55065     0.989712
-0.886205   0.149748   -1.54419

julia> gcop2arch(x, ["clayton" => [1,2]])
6×3 Array{Float64,2}:
-0.742443   0.424851  -0.384124
0.211894   0.195774  -0.571326
-0.989417  -0.299369  -2.3464
0.157683   1.47768    0.966819
0.154893   0.893253   0.989712
-0.657297  -0.339814  -1.54419

To change a chosen marginals subset ind of multivariate Gaussian distributed data x by means of Frechet maximal sub-copula:

julia> gcop2frechet(x::Matrix{Float64}, ind::Vector{Int}; naive::Bool = false)

all univariate marginal distributions are Gaussian and unaffected by a transformation.

julia> Σ = [1. 0.5 0.5; 0.5 1. 0.5; 0.5 0.5 1];

julia> Random.seed!(42)

julia> x = Array(rand(MvNormal(Σ), 6)')
6×3 Array{Float64,2}:
-0.556027  -0.662861   -0.384124
-0.299484   1.38993    -0.571326
-0.468606  -0.0990787  -2.3464
1.00331    1.43902     0.966819
0.518149   1.55065     0.989712
-0.886205   0.149748   -1.54419

julia> gcop2frechet(x, [1,2])
6×3 Array{Float64,2}:
-0.875777   -0.374723   -0.384124
0.0960334   0.905703   -0.571326
-0.599792   -0.0110945  -2.3464
0.813717    1.8513      0.966819
0.599255    1.56873     0.989712
-0.7223     -0.172507   -1.54419

To change a chosen marginals subset ind of multivariate Gaussian distributed data x by means of bivariate Marshall-Olkin copula:

julia> gcop2marshallolkin(x::Matrix{Float64}, ind::Vector{Int}, λ1::Float64 = 1., λ2::Float64 = 1.5; naive::Bool = false)

all univariate marginal distributions are Gaussian and unaffected by a transformation. We require length(ind) = 2 λ1 ≧ 0 and λ2 ≧ 0. The parameter λ12 is computed from expected correlation between both changed marginals.

julia> Σ = [1. 0.5 0.5; 0.5 1. 0.5; 0.5 0.5 1];

julia> Random.seed!(42);

julia> x = Array(rand(MvNormal(Σ), 6)')
6×3 Array{Float64,2}:
-0.556027  -0.662861   -0.384124
-0.299484   1.38993    -0.571326
-0.468606  -0.0990787  -2.3464
1.00331    1.43902     0.966819
0.518149   1.55065     0.989712
-0.886205   0.149748   -1.54419

julia> gcop2marshallolkin(x, [1,2])
6×3 Array{Float64,2}:
-0.790756   0.784371  -0.384124
-0.28088    0.338086  -0.571326
-0.90688   -0.509684  -2.3464
0.738628   1.71026    0.966819
0.353654   1.19357    0.989712
-0.867606  -0.589929  -1.54419

## Helpers

### Converting marginals

Takes matrix X: size(X) = (t, n) ie t realisations of n-dimensional random variable, with all uniform marginal univariate distributions ∀ᵢ X[:,i] ∼ Uniform(0,1), and convert those marginals to common distribution d with parameters p[i]

julia> convertmarg!(U::Matrix{T}, d::UnionAll, p::Union{Vector{Vector{Int64}}, Vector{Vector{Float64}}}; testunif::Bool = true)

If testunif = true each marginal is tested for uniformity.

julia> using Distributions

julia> Random.seed!(43);

julia> U = gausscopulagen(10);

julia> convertmarg!(U, Normal, [[0, 1],[0, 10]])

julia> U
10×2 Array{Float64,2}:
0.225457      8.97627
0.548381     14.3926
0.666147    -10.0689
-0.746662     -9.03553
-0.746857     17.2101
-0.608109     -3.45649
-0.136555      0.700419
0.215631     -7.34409
-0.00352701   -0.434793
-0.876853      2.39009

To convert i th marginal to univariate distribution d with parameters array p run

julia> using Distributions

julia> quantile.(d(p...), U[:,i])
julia> Random.seed!(43);

julia> U = gausscopulagen(10);

julia> quantile.(Levy(0, 1), U[:,2])
10-element Array{Float64,1}:
18.327904335047272
112.72788160148863
0.4992650891811052
0.5642861403809334
350.0676959136128
1.2175971128674394
2.510078079677982
0.6980591543550244
2.0290242635860944
3.527994542141473

To convert all marginals to the same d with the same parameters p run

julia> using Distributions

julia> quantile.(d(p...), U)
julia> julia> quantile.(Levy(0, 1), U)
10×2 Array{Float64,2}:
3.42919    18.3279
7.14305   112.728
9.6359      0.499265
0.687009    0.564286
0.686835  350.068
0.827224    1.2176
1.71944     2.51008
3.3597      0.698059
2.18374     2.02902
0.582946    3.52799

# Citing this work

This project was partially financed by the National Science Centre, Poland – project number 2014/15/B/ST6/05204.

• while reffering to gcop2arch(), gcop2frechet(), and gcop2marshallolkin() - cite K. Domino, A. Glos: 'Introducing higher order correlations to marginals' subset of multivariate data by means of Archimedean copulas', [arXiv:1803.07813] (https://arxiv.org/abs/1803.07813).

• while reffering to gcop2tstudent() - cite K. Domino: ' The use of the Higher Order Singular Value Decomposition of the 4-cumulant's tensors in features selection and outlier detection', [arXiv:1804.00541] (https://arxiv.org/abs/1804.00541).