BivariateCopulas.jl

An implementation of bivariate copulas and bivariate distributions in Julia
Author AnderGray
Github
Popularity
11 Stars
Updated Last
1 Year Ago
Started In
March 2021

DOI

2-copulas julia

An implementation of bivariate copulas and bivariate distributions in Julia.

This module has the following capabilities:

  • Construction of continous bivariate distributions with Distributions.jl
  • Evaluation of cdf and density
  • Conditioning (cdfs)
  • Sampling
  • Bivariate scatter plots, cdf and density plots and contour plots
  • Common copula families:
    • π (independence)
    • M (maximal correlation)
    • W (maximal negative correlation)
    • Gaussian (correlation coefficient r; r = -1 for W, r = 0 for π, r = 1 for M)
    • Frank (s is real; -inf for W, 0 for π, inf for M)
    • Clayton (t>=-1; -1 for W, 0 for π and inf for M)

Any possible multivariate dependence can be encoded in a copula. Copulas are joint cdfs with standard uniform marginals, and are a way to model dependency independently of marginal distributions. Most probabilistic dependence problems can be reduced to an analysis of copulas.

Authors

  • Ander Gray, Institute for Risk and Uncertainty, University of Liverpool
  • Liam T. Henry, Clarus Financial Technology, Belfast

Installation

This is a registered julia package:

julia> ]
pkg> add BivariateCopulas

or the most up to date version:

julia> ]
pkg> add https://github.com/AnderGray/BivariateCopulas.jl#master

Sklar's theorem

A bivariate copula (2-copula) C is a function C:[0,1]2 -> [0,1] with the following properties:

Three important 2-copulas are:

with W and M being bounds on all 2-copulas: W ≤ C ≤ M

Copulas are mainly used in dependence modelling, and can be used to construct any continous multivariate distribution function (df) given their univariate marginals. This is enabled by a theorem from Sklar:

Where FX and FY are the cumulative distribution functions (cdf) of X and Y. Moreover, X and Y are stochastically independent if and only if CXY = π, maximally correlated iff CXY = M, and minimally correlated iff CXY = W.

Usage

Constructing copulas

julia> a = Pi()
Copula ~ π()

julia> b = M()
Copula ~ M()

julia> c = W()
Copula ~ W()

julia> d = Gaussian(0.8)
Copula ~ Gau(r=0.8)

julia> e = Frank(10)
Copula ~ F(s=10.0)

julia> g = Clayton(-0.2)
Copula ~ Cla(t=-0.2)

Sklar's theorem

Use any two continous distribution from Distributions.jl may be used to make a bivariate distribution with C

julia> C1 = Gaussian(0.8);

julia> j1 = C1(Normal(0, 1), Normal(0, 1)) # Correlated normal
Joint ~ Gau( r=0.8, Normal{Float64}(μ=0.0, σ=1.0), Normal{Float64}(μ=0.0, σ=1.0) )

julia> C2 = M();

julia> j2 = C2(Beta(2, 4), Beta(4, 2)) # Maximally correlated betas
Joint ~ M( Beta{Float64}(α=2.0, β=4.0), Beta{Float64}(α=4.0, β=2.0) )

julia> j3 = W(Normal(-1, 20), Beta(4, 2)) # Minimally correlated normal and beta
Joint ~ W( Normal{Float64}(μ=-1.0, σ=20.0), Beta{Float64}(α=4.0, β=2.0) )

CDF and Density for C and Joints

julia> cop = Gaussian(0.8);

julia> cop(0.2, 0.4) # cdf eval
0-dimensional Array{Float64,0}:
0.17890296062944128

julia> cdf(cop, 0.2, 0.4) # same as above 
0-dimensional Array{Float64,0}:
0.17890296062944128

julia> cop(0.2:0.1:0.8, 0.2:0.1:0.8) # ask for matrix
7×7 Array{Float64,2}:
 0.129034  0.16003   0.178903  0.189859  0.195758  0.198579  0.199683
 0.16003   0.21118   0.247245  0.271358  0.286343  0.294693  0.298579
 0.178903  0.247245  0.300933  0.340944  0.368789  0.386343  0.395758
 0.189859  0.271358  0.340944  0.397584  0.440944  0.471358  0.489859
 0.195758  0.286343  0.368789  0.440944  0.500933  0.547245  0.578903
 0.198579  0.294693  0.386343  0.471358  0.547245  0.61118   0.66003
 0.199683  0.298579  0.395758  0.489859  0.578903  0.66003   0.729034

julia> density(cop, 0.2, 0.4) # density eval at a point
1.347233996821345

julia> density(cop, 0.2:0.1:0.8, 0.2:0.1:0.8) # ask for matrix
7×7 Array{Float64,2}:
 2.28311    1.8543    1.34723   0.888161  0.522393  0.26094   0.0981165
 1.8543     1.88328   1.65627   1.30532   0.917827  0.554958  0.26094
 1.34723    1.65627   1.71488   1.57426   1.28937   0.917827  0.522393
 0.888161   1.30532   1.57426   1.66667   1.57426   1.30532   0.888161
 0.522393   0.917827  1.28937   1.57426   1.71488   1.65627   1.34723
 0.26094    0.554958  0.917827  1.30532   1.65627   1.88328   1.8543
 0.0981165  0.26094   0.522393  0.888161  1.34723   1.8543    2.28311

julia> j1 = cop(Beta(2, 4), Normal(5, 3.4));

julia> j1(0.1:0.2:0.8, 5.1:0.2:5.8) # cdf
4×4 Array{Float64,2}:
 0.0806019  0.0807922  0.0809446  0.0810654
 0.388167   0.398055   0.407184   0.415547
 0.5023     0.523831   0.544962   0.565593
 0.511647   0.535043   0.558309   0.581364

julia> density(j1, 0.1:0.2:0.8, 5.1:0.2:5.8) # density
4×4 Array{Float64,2}:
 0.0461003   0.0380492   0.0311039   0.0251831
 0.398346    0.390902    0.379929    0.365733
 0.128563    0.142985    0.157505    0.171839
 0.00374385  0.00473385  0.00592841  0.00735337

Sampling

julia> rand(cop, 10^4) # sample a copula
10000×2 Array{Float64,2}:
 0.342484   0.293974
 0.970305   0.892417
 0.344887   0.670971
 0.177868   0.091284
 ⋮          
 0.577286   0.439561
 0.560594   0.405275
 0.259065   0.0325417
 0.424095   0.389324

julia> rand(j1, 10^4) # sample a bivariate distribution
10000×2 Array{Float64,2}:
 0.0737024  -0.791622
 0.330924    4.26197
 0.14506     5.36363
 0.353489    7.28413
 0.530191   10.199
 ⋮          
 0.366434    5.22849
 0.109036    1.30304
 0.555351    6.06244
 0.653661   10.6358
 0.242393    4.85661

Plots

PyPlot.jl may be used for this packages plotting features. If so, it must be explicitally stated:

Scatter plots

julia> using PyPlot
julia> samps = rand(cop, 10^4);

julia> scatter(samps)

julia> samps2 = rand(j1, 10^4);

julia> scatter(samps2)

CDF plots

julia> using3D()
julia> plot(cop)

julia> plot(j1)

Density plots

julia> c1 = Frank(0.8);

julia> plotDen(c1)

julia> j1 = c1(Beta(2, 5), Beta(2, 5));

julia> plotDen(j1)

Contour plots

julia> plotContourCdf(j1)

julia> plotContourDen(j1)

Bibliography

Required Packages

Used By Packages