copulas.bivariate.frank module¶
Frank module.
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class
copulas.bivariate.frank.
Frank
(copula_type=None, random_state=None)[source]¶ Bases:
copulas.bivariate.base.Bivariate
Class for Frank copula model.
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compute_theta
()[source]¶ Compute theta parameter using Kendall’s tau.
On Frank copula, the relationship between tau and theta is defined by:
\[\tau = 1 − \frac{4}{\theta} + \frac{4}{\theta^2}\int_0^\theta \! \frac{t}{e^t -1} \mathrm{d}t.\]In order to solve it, we can simplify it as
\[0 = 1 + \frac{4}{\theta}(D_1(\theta) - 1) - \tau\]where the function D is the Debye function of first order, defined as:
\[D_1(x) = \frac{1}{x}\int_0^x\frac{t}{e^t -1} \mathrm{d}t.\]
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copula_type
= 1¶
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cumulative_distribution
(X)[source]¶ Compute the cumulative distribution function for the Frank copula.
The cumulative density(cdf), or distribution function for the Frank family of copulas correspond to the formula:
\[C(u,v) = −\frac{\ln({\frac{1 + g(u) g(v)}{g(1)}})}{\theta}\]- Parameters
X – np.ndarray
- Returns
cumulative distribution
- Return type
np.array
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invalid_thetas
= [0]¶
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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.
\[F(v|u) = \frac{\partial}{\partial u}C(u,v) = \frac{g(u)g(v) + g(v)}{g(u)g(v) + g(1)}\]- Parameters
X (np.ndarray) –
y (float) –
- Returns
np.ndarray
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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.
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probability_density
(X)[source]¶ Compute probability density function for given copula family.
The probability density(PDF) for the Frank family of copulas correspond to the formula:
\[c(U,V) = \frac{\partial^2 C(u,v)}{\partial v \partial u} = \frac{-\theta g(1)(1 + g(u + v))}{(g(u) g(v) + g(1)) ^ 2}\]Where the g function is defined by:
\[g(x) = e^{-\theta x} - 1\]- Parameters
X – np.ndarray
- Returns
probability density
- Return type
np.array
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theta_interval
= [-inf, inf]¶
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