copulas.bivariate.clayton module¶
Clayton module.
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class
copulas.bivariate.clayton.
Clayton
(copula_type=None, random_state=None)[source]¶ Bases:
copulas.bivariate.base.Bivariate
Class for clayton copula model.
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compute_theta
()[source]¶ 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.
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copula_type
= 0¶
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cumulative_distribution
(X)[source]¶ 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
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generator
(t)[source]¶ 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
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invalid_thetas
= []¶
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partial_derivative
(X)[source]¶ Compute partial derivative of cumulative distribution.
The partial derivative of the copula(CDF) is the conditional CDF.
\[F(v|u) = \frac{\partial C(u,v)}{\partial u} = u^{- \theta - 1}(u^{-\theta} + v^{-\theta} - 1)^{-\frac{\theta+1}{\theta}}\]- Parameters
X (np.ndarray) –
y (float) –
- Returns
Derivatives
- Return type
numpy.ndarray
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percent_point
(y, V)[source]¶ 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.
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probability_density
(X)[source]¶ 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
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theta_interval
= [0, inf]¶
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