copulas.bivariate.clayton module

Clayton module.

class copulas.bivariate.clayton.Clayton(*args, **kwargs)

Bases: Bivariate

Class for clayton copula model.

compute_theta()

Compute theta parameter using Kendall’s tau.

On Clayton copula this is

\[τ = θ/(θ + 2) \implies θ = 2τ/(1-τ)\]

\[θ ∈ (0, ∞)\]

On the corner case of \(τ = 1\), return infinite.

copula_type = 0

cumulative_distribution(X)

Compute the cumulative distribution function for the clayton copula.

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

\[C(u,v) = (u^{-θ} + v^{-θ} - 1)^{-1/θ}\]

Parameters:

X (numpy.ndarray)

Returns:

cumulative probability.

Return type:

numpy.ndarray

generator(t)

Compute the generator function for Clayton copula family.

The generator is a function \(\psi: [0,1]\times\Theta \rightarrow [0, \infty)\) # noqa: JS101

that given an Archimedian copula fulfills: .. math:: C(u,v) = psi^{-1}(psi(u) + psi(v))

Parameters:

t (numpy.ndarray)

Returns:

numpy.ndarray

invalid_thetas = []

partial_derivative(X)

Compute partial derivative of cumulative distribution.

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

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

Parameters:

• X (np.ndarray)

• y (float)

Returns:

Derivatives

Return type:

numpy.ndarray

percent_point(y, V)

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

Parameters:

• y (numpy.ndarray) – Value of \(C(u|v)\).

• v (numpy.ndarray) – given value of v.

probability_density(X)

Compute probability density function for given copula family.

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

\[c(U,V) = \frac{\partial^2}{\partial v \partial u}C(u,v) = (\theta + 1)(uv)^{-\theta-1}(u^{-\theta} + v^{-\theta} - 1)^{-\frac{2\theta + 1}{\theta}}\]

Parameters:

X (numpy.ndarray)

Returns:

Probability density for the input values.

Return type:

numpy.ndarray

theta_interval = [0, inf]