diff --git a/dev/archimedeans/index.html b/dev/archimedeans/index.html index 5f879208..1582f9ad 100644 --- a/dev/archimedeans/index.html +++ b/dev/archimedeans/index.html @@ -6,4 +6,4 @@ ϕ⁻¹(C::MyAC{d},x) τ(C::MyAC{d}) τ⁻¹(::MyAC{d},τ) -radial_dist(C::MyAC{d})

Available Archimedean copulas

Independence

Copulas.IndependentCopula — Type

IndependentCopula{d,T}

Constructor

IndependentCopula(d, θ)

The Independant Copula in dimension $d$ is the simplest copula, that has the form :

\[C(\mathbf{x}) = \prod_{i=1}^d x_i.\]

It happends to be an Archimedean Copula, with generator :

\[\phi(t) = \exp{-t}\]

source

Clayton

Copulas.ClaytonCopula — Type
ClaytonCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

ClaytonCopula(d, θ)

The Clayton copula in dimension $d$ is parameterized by $\theta \in [-1,\infty)$ when $d=2$ and $\theta \in [0,\infty)$ if $d>2$. It is an Archimedean copula with generator :

\[\phi(t) = \left(1+\mathrm{sign}(\theta)*t\right)^{-1\frac{1}{\theta}}\]

source

Frank

Copulas.FrankCopula — Type
FrankCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

FrankCopula(d, θ)

The Frank copula in dimension $d$ is parameterized by $\theta \in [0,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = -\frac{\log\left(1+e^{-t}(e^{-\theta-1})\right)}{ heta}\]

source

Gumbel

Copulas.GumbelCopula — Type
GumbelCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

GumbelCopula(d, θ)

The Gumbel copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = \exp{-t^{\frac{1}{θ}}}\]

source

Ali-Mikhail-Haq

Copulas.AMHCopula — Type
AMHCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

AMHCopula(d, θ)

The AMH copula in dimension $d$ is parameterized by $\theta \in [0,1)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \frac{1-\theta}{e^{-t}-\theta}\]

source

Joe

Copulas.JoeCopula — Type
JoeCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

JoeCopula(d, θ)

The Joe copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \left(1 - e^{-t}\right)^{\frac{1}{\theta}}\]

source
+radial_dist(C::MyAC{d})

Available Archimedean copulas

Independence

Copulas.IndependentCopula — Type

IndependentCopula{d,T}

Constructor

IndependentCopula(d, θ)

The Independant Copula in dimension $d$ is the simplest copula, that has the form :

\[C(\mathbf{x}) = \prod_{i=1}^d x_i.\]

It happends to be an Archimedean Copula, with generator :

\[\phi(t) = \exp{-t}\]

source

Clayton

Copulas.ClaytonCopula — Type
ClaytonCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

ClaytonCopula(d, θ)

The Clayton copula in dimension $d$ is parameterized by $\theta \in [-1,\infty)$ when $d=2$ and $\theta \in [0,\infty)$ if $d>2$. It is an Archimedean copula with generator :

\[\phi(t) = \left(1+\mathrm{sign}(\theta)*t\right)^{-1\frac{1}{\theta}}\]

source

Frank

Copulas.FrankCopula — Type
FrankCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

FrankCopula(d, θ)

The Frank copula in dimension $d$ is parameterized by $\theta \in [0,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = -\frac{\log\left(1+e^{-t}(e^{-\theta-1})\right)}{ heta}\]

source

Gumbel

Copulas.GumbelCopula — Type
GumbelCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

GumbelCopula(d, θ)

The Gumbel copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = \exp{-t^{\frac{1}{θ}}}\]

source

Ali-Mikhail-Haq

Copulas.AMHCopula — Type
AMHCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

AMHCopula(d, θ)

The AMH copula in dimension $d$ is parameterized by $\theta \in [0,1)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \frac{1-\theta}{e^{-t}-\theta}\]

source

Joe

Copulas.JoeCopula — Type
JoeCopula{d,T}

Fields:

  • θ::Real - parameter

Constructor

JoeCopula(d, θ)

The Joe copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \left(1 - e^{-t}\right)^{\frac{1}{\theta}}\]

source
diff --git a/dev/ellipticals/index.html b/dev/ellipticals/index.html index bddf8ba9..61159060 100644 --- a/dev/ellipticals/index.html +++ b/dev/ellipticals/index.html @@ -1,6 +1,6 @@ Elliptical Copulas · Copulas.jl
Edit on GitHub

Elliptical Copulas

Details about the elliptical Copulas

Generic Elliptical copulas

explain how we could define a generic elliptical copulas, and methods that are related to them.

Missing docstring.

Missing docstring for EllipticalCopula. Check Documenter's build log for details.

Available elliptical copulas

Gaussian

Copulas.GaussianCopula — Type
GaussianCopula{d,MT}

Fields:

  • Σ::MT - covariance matrix

Constructor

GaussianCopula(Σ)

The Gaussian Copula is the copula of a Multivariate normal distribution. It is constructed as :

\[C(\mathbf{x}; \boldsymbol{\Sigma}) = F_{\Sigma}(F_{\Sigma,i}^{-1}(x_i),i\in 1,...d)\]

where $F_{\Sigma}$ is a cdf of a gaussina random vector and F_{\Sigma,i} is the ith marignal cdf, while `$\Sigma$ is the covariance matrix.

It can be constructed in Julia via: 

C = GaussianCopula(Σ)

The random number generation works as expected:

rand(C,1000)
 # or
-Random.rand!(C,u)

And yo can fit the distribution via : 

fit(GaussianCopula,data)
source

Student's T

Copulas.TCopula — Type
TCopula{d,MT}

Fields:

  • df::Int - number of degree of freedom
  • Σ::MT - covariance matrix

Constructor

TCopula(df,Σ)

The Student's T Copula is the copula of a Multivariate Student distribution. It is constructed as :

\[C(\mathbf{x}; \boldsymbol{n,\Sigma}) = F_{n,\Sigma}(F_{n,\Sigma,i}^{-1}(x_i),i\in 1,...d)\]

where $F_{n,\Sigma}$ is a cdf of a multivariate student random vector with covariance matrix $\Sigma$ and $n$ degrees of freedom. and F_{n,\Sigma,i} is the ith marignal cdf.

It can be constructed in Julia via: 

C = TCopula(n,Σ)

The random number generation works as expected:

rand(C,1000)
+Random.rand!(C,u)

And yo can fit the distribution via : 

fit(GaussianCopula,data)
source

Student's T

Copulas.TCopula — Type
TCopula{d,MT}

Fields:

  • df::Int - number of degree of freedom
  • Σ::MT - covariance matrix

Constructor

TCopula(df,Σ)

The Student's T Copula is the copula of a Multivariate Student distribution. It is constructed as :

\[C(\mathbf{x}; \boldsymbol{n,\Sigma}) = F_{n,\Sigma}(F_{n,\Sigma,i}^{-1}(x_i),i\in 1,...d)\]

where $F_{n,\Sigma}$ is a cdf of a multivariate student random vector with covariance matrix $\Sigma$ and $n$ degrees of freedom. and F_{n,\Sigma,i} is the ith marignal cdf.

It can be constructed in Julia via: 

C = TCopula(n,Σ)

The random number generation works as expected:

rand(C,1000)
 # or
-Random.rand!(C,u)

And yo can fit the distribution via : 

fit(TCopula,data)

Except that currently it does not work since fit(Distributions.MvTDist,data) does not dispatch.

source
+Random.rand!(C,u)

And yo can fit the distribution via : 

fit(TCopula,data)

Except that currently it does not work since fit(Distributions.MvTDist,data) does not dispatch.

source diff --git a/dev/index.html b/dev/index.html index becc7497..9b96552d 100644 --- a/dev/index.html +++ b/dev/index.html @@ -19,4 +19,4 @@ ϕ⁻¹(C::ClaytonCopula,t) = sign(C.θ)*(t^(-C.θ)-1) # Inverse Generator τ(C::ClaytonCopula) = C.θ/(C.θ+2) # θ -> τ τ⁻¹(::Type{ClaytonCopula},τ) = 2τ/(1-τ) # τ -> θ -radial_dist(C::ClaytonCopula) = Distributions.Gamma(1/C.θ,1) # Radial distribution

The Archimedean API is modular:

Dev Roadmap

Urgent things

Next

Maybe later

Contributions are welcome

Do not hesitate to open an issue to discuss :)

source
Copulas.EmpiricalCopula — Type
EmpiricalCopula{d,MT}

Fields:

  • u::MT - the matrix of observations.

Constructor

EmpiricalCopula(u;pseudos=true)

The EmpiricalCopula in dimension $d$ is parameterized by a pseudo-data matrix wich should have shape (d,N). Its expression is given as :

\[C(\mathbf x) = \frac{1}{N}\sum_{i=1}^n \mathbf 1_{\mathbf u_i \le \mathbf x}\]

This function is very practical, be be aware that this is not a true copula (since $\mathbf u$ are only pseudo-observations). The constructor allows you to pass dirctly pseudo-observations (the default) or will compute them for you. You can then compute the cdf of the copula, and sample it through the standard interface.

source
Base.rand — Method

Generate independent stable random numbers.

:param α: characteristic exponent (0.1 to 2.0) :param β: skew (-1 to +1) :param scale: scale parameter :param loc: location parameter (mean for α > 1, median/mode when β=0)

This implementation is based on the method in J.M. Chambers, C.L. Mallows and B.W. Stuck, "A Method for Simulating Stable Random Variables," JASA 71 (1976): 340-4. McCulloch's MATLAB implementation (1996) served as a reference in developing this code.

source
+radial_dist(C::ClaytonCopula) = Distributions.Gamma(1/C.θ,1) # Radial distribution

The Archimedean API is modular:

Dev Roadmap

Urgent things

Next

Maybe later

Contributions are welcome

Do not hesitate to open an issue to discuss :)

source
Copulas.EmpiricalCopula — Type
EmpiricalCopula{d,MT}

Fields:

  • u::MT - the matrix of observations.

Constructor

EmpiricalCopula(u;pseudos=true)

The EmpiricalCopula in dimension $d$ is parameterized by a pseudo-data matrix wich should have shape (d,N). Its expression is given as :

\[C(\mathbf x) = \frac{1}{N}\sum_{i=1}^n \mathbf 1_{\mathbf u_i \le \mathbf x}\]

This function is very practical, be be aware that this is not a true copula (since $\mathbf u$ are only pseudo-observations). The constructor allows you to pass dirctly pseudo-observations (the default) or will compute them for you. You can then compute the cdf of the copula, and sample it through the standard interface.

source
Base.rand — Method

Generate independent stable random numbers.

:param α: characteristic exponent (0.1 to 2.0) :param β: skew (-1 to +1) :param scale: scale parameter :param loc: location parameter (mean for α > 1, median/mode when β=0)

This implementation is based on the method in J.M. Chambers, C.L. Mallows and B.W. Stuck, "A Method for Simulating Stable Random Variables," JASA 71 (1976): 340-4. McCulloch's MATLAB implementation (1996) served as a reference in developing this code.

source
diff --git a/dev/search/index.html b/dev/search/index.html index 23b2a253..5587c8de 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · Copulas.jl

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    +Search · Copulas.jl

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