copulas.bivariate.gumbel module

Gumbel module.

class copulas.bivariate.gumbel.Gumbel(*args, **kwargs)

Bases: Bivariate

Class for clayton copula model.

compute_theta()

Compute theta parameter using Kendall’s tau.

On Gumbel copula \(\tau\) is defined as \(τ = \frac{θ−1}{θ}\) that we solve as \(θ = \frac{1}{1-τ}\)

copula_type = 2

cumulative_distribution(X)

Compute the cumulative distribution function for the Gumbel copula.

The cumulative density(cdf), or distribution function for the Gumbel family of copulas correspond to the formula:

\[C(u,v) = e^{-((-\ln u)^{\theta} + (-\ln v)^{\theta})^{\frac{1}{\theta}}}\]

Parameters:

X (np.ndarray)

Returns:

cumulative probability for the given datapoints, cdf(X).

Return type:

np.ndarray

generator(t)

Return the generator function.

invalid_thetas = []

partial_derivative(X)

Compute partial derivative of cumulative distribution.

The partial derivative of the copula(CDF) is the conditional CDF.

\[F(v|u) = \frac{\partial C(u,v)}{\partial u} = C(u,v)\frac{((-\ln u)^{\theta} + (-\ln v)^{\theta})^{\frac{1}{\theta} - 1}} {\theta(- \ln u)^{1 -\theta}}\]

Parameters:

• X (np.ndarray)

• y (float)

Returns:

numpy.ndarray

percent_point(y, V)

Compute the inverse of conditional cumulative distribution \(C(u|v)^{-1}\).

Parameters:

• y (np.ndarray) – value of \(C(u|v)\).

• v (np.ndarray) – given value of v.

probability_density(X)

Compute probability density function for given copula family.

The probability density(PDF) for the Gumbel family of copulas correspond to the formula:

\[\begin{align} c(U,V) &= \frac{\partial^2 C(u,v)}{\partial v \partial u} &= \frac{C(u,v)}{uv} \frac{((-\ln u)^{\theta} # noqa: JS101 + (-\ln v)^{\theta})^{\frac{2} # noqa: JS101 {\theta} - 2 }}{(\ln u \ln v)^{1 - \theta}} # noqa: JS101 ( 1 + (\theta-1) \big((-\ln u)^\theta + (-\ln v)^\theta\big)^{-1/\theta}) \end{align}\]

Parameters:

X (numpy.ndarray)

Returns:

numpy.ndarray

theta_interval = [1, inf]