copulas.bivariate.gumbel module¶
Gumbel module.
-
class
copulas.bivariate.gumbel.
Gumbel
(copula_type=None, random_state=None)[source]¶ Bases:
copulas.bivariate.base.Bivariate
Class for clayton copula model.
-
compute_theta
()[source]¶ 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)[source]¶ 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
-
invalid_thetas
= []¶
-
partial_derivative
(X)[source]¶ 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)[source]¶ 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)[source]¶ 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]¶
-