copul.family.extreme_value package

Submodules

copul.family.extreme_value.bb5 module

class copul.family.extreme_value.bb5.BB5(*args, **kwargs)

Bases: BivExtremeValueCopula

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

The symbolic CDF of the BB5 copula contains deeply nested logarithms and powers that are prohibitively slow to evaluate through SymPy’s evalf. This override routes all concrete (u, v) evaluations to the fast numpy path.

delta = delta

dim: int

intervals: dict = {'delta': Interval.open(0, oo), 'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [theta, delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

theta = theta

copul.family.extreme_value.biv_extreme_value_copula module

class copul.family.extreme_value.biv_extreme_value_copula.BivExtremeValueCopula(*args, **kwargs)

Bases: MultivariateExtremeValueCopula, BivCoreCopula

Bivariate extreme value copula.

Specialization of MultivariateExtremeValueCopula to the 2-dimensional case. Provides additional methods specific to bivariate extreme value theory using Pickands dependence functions.

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta\) for extreme value copulas.

Since \(C(\tfrac12,\tfrac12) = (\tfrac14)^{A(1/2)}\):

\[\beta = 4 \cdot (1/4)^{A(1/2)} - 1\]

Return type:

float

property cdf

Cumulative distribution function of the copula.

Returns:

Wrapper around the symbolic CDF expression.

Return type:

CDFWrapper

cdf_vectorized(u, v)

Vectorized cumulative distribution function.

Parameters:

• u (array_like) – Uniform marginals in [0,1].

• v (array_like) – Uniform marginals in [0,1].

Returns:

Values of \(C(u,v)\).

Return type:

numpy.ndarray

Notes

Implements

\[C(u,v) = (uv)^{A(\log v / \log(uv))}.\]

cond_distr_1(u=None, v=None)

First conditional distribution \(\partial C/\partial u\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial u\).

Return type:

float

cond_distr_2(u=None, v=None)

Second conditional distribution \(\partial C/\partial v\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial v\).

Return type:

float

deriv_pickand_at_0()

Derivative of the Pickands function at \(t=0\).

Returns:

Value of \(A'(0)\).

Return type:

float or sympy.Expr

classmethod from_pickands(pickands, params=None)

Construct a new copula from a Pickands dependence function.

Parameters:

• pickands (str or sympy.Expr) – Pickands dependence function. May contain t or another symbol.

• params (list[str|sympy.Symbol] | str | None) – Parameter names/symbols. If None, symbols are detected automatically.

Returns:

Instance with the specified Pickands function.

Return type:

BivExtremeValueCopula

gini_gamma(*args, **kwargs)

Gini’s \(\gamma\) for extreme value copulas.

\[\gamma = 4\!\left[ \int_0^1 t^{A(1/2)}\,dt + \int_0^1 \bigl(t(1-t)\bigr)^{A\!\left(\frac{\ln(1-t)} {\ln(t(1-t))}\right)}\,dt \right] - 2\]

Computed numerically via the vectorized CDF.

Return type:

float

intervals: dict = {}

property is_ci

Whether the copula is conditionally increasing.

Returns:

Always True for extreme value copulas.

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Spearman’s \(\rho\).

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

lambda_L()

Lower tail dependence coefficient for extreme value copulas.

All bivariate extreme value copulas have \(\lambda_L = 0\).

Returns:

Always 0.

Return type:

float

lambda_U()

Upper tail dependence coefficient for extreme value copulas.

\[\lambda_U = 2\bigl(1 - A(\tfrac12)\bigr)\]

where \(A\) is the Pickands dependence function.

Return type:

float

References

Gudendorf & Segers (2010), Extreme-Value Copulas.

params: list = []

property pdf

Probability density function of the copula.

Returns:

Wrapper around the symbolic or numerical PDF.

Return type:

SymPyFuncWrapper

property pickands

Return the Pickands dependence function with parameter values substituted.

Returns:

A wrapper that supports evaluation at \(t\) values and symbolic use.

Return type:

PickandsWrapper

plot_pickands(subs=None, **kwargs)

Plot the Pickands dependence function.

Parameters:

• subs (dict, optional) – Parameter substitutions.

• **kwargs – Additional substitutions.

spearmans_rho(*args, **kwargs)

Spearman’s \(\rho\) for the extreme value copula.

Parameters:

• *args – Copula parameters.

• **kwargs – Copula parameters.

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

t = t

tail_dependence_function(t, lower=True)

Evaluate the tail dependence function at \(t \in [0,1]\).

For bivariate extreme value copulas the upper TDF has the closed form

\[b_U(t) = 1 - A(t)\]

The lower TDF is identically 0 (since \(\lambda_L = 0\)).

Parameters:

• t (float or array_like) – Point(s) in \([0,1]\).

• lower (bool) – If True, return the lower TDF (always 0). If False, return the upper TDF.

Return type:

float or numpy.ndarray

tail_order()

Tail order \(\kappa\) for extreme value copulas.

For any bivariate extreme value copula the lower tail order is \(\kappa_L = 1 / A(1/2)\) and the upper tail order is 1 whenever \(\lambda_U > 0\), or the rate at which \(A(t) \to 1\) near the endpoints.

Returns:

{"lower": kappa_L, "upper": kappa_U}

Return type:

dict

u = u

v = v

copul.family.extreme_value.biv_extreme_value_copula.from_pickands(pickands, params=None)

Construct a bivariate extreme value copula from a Pickands function.

Parameters:

• pickands (str or sympy.Expr) – Pickands dependence function. May contain t or another symbol.

• params (list or str, optional) – Parameter names. If None, symbols are detected automatically.

Returns:

Instance with the specified Pickands function.

Return type:

BivExtremeValueCopula

copul.family.extreme_value.cuadras_auge module

class copul.family.extreme_value.cuadras_auge.CuadrasAuge(*args, **kwargs)

Bases: BivExtremeValueCopula

Cuadras-Auge copula, special case of the Marshall-Olkin copula.

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta\) for the Cuadras-Augé copula.

\[\beta = 4\,C(\tfrac12,\tfrac12) - 1 = 4\cdot 2^{-\delta}\cdot 2^{-(2-2\delta)} - 1 = 2^{2-\delta} \cdot 2^{-2+2\delta}\cdot 4 - 1 \;=\; 2^{\delta} - 1\]

chatterjees_xi(*args, **kwargs)

Compute Chatterjee’s xi correlation measure.

This method sets the parameters, computes intermediate integrals, and returns the simplified expression for xi.

Returns:

A wrapper around the symbolic expression for Chatterjee’s xi.

Return type:

SymPyFuncWrapper

cond_distr_1(u=None, v=None)

First conditional distribution \(\partial C/\partial u\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial u\).

Return type:

float

delta = delta

dim: int

gini_gamma(*args, **kwargs)

Gini’s \(\gamma\) for the Cuadras-Augé copula.

\[\gamma = 4\!\left[\int_0^1 C(t,t)\,dt + \int_0^1 C(t,1{-}t)\,dt\right] - 2\]

For the CA copula, \(C(t,t) = t^{2-\delta}\) (since \(\min(t,t)=t\)), so the diagonal integral is \(1/(3-\delta)\). The anti-diagonal \(C(t,1-t)\) requires splitting at \(t=1/2\).

hoeffdings_d(*args, **kwargs)

Hoeffding’s \(D\) for the Cuadras-Augé copula.

Expanding the double integral \(90\iint(C-uv)^2\,du\,dv\) over the two regions \(\{v \le u\}\) and \(\{u < v\}\) (using symmetry) gives:

\[D(\delta) = \frac{10\,\delta^{2}} {18 - 9\,\delta + \delta^{2}}\]

which satisfies \(D(0)=0\) and \(D(1)=1\).

intervals: dict = {'delta': Interval(0, 1)}

property is_absolutely_continuous

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Spearman’s \(\rho\).

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

params: list = [delta]

property pdf

Probability density function of the copula.

Returns:

Wrapper around the symbolic or numerical PDF.

Return type:

SymPyFuncWrapper

schweizer_wolff_sigma(*args, **kwargs)

Schweizer–Wolff \(\sigma\) for the Cuadras-Augé copula.

The CA copula is PQD for all \(\delta \in (0,1]\), so \(\sigma = \rho_S = 3\delta/(4-\delta)\).

spearman_footrule(*args, **kwargs)

Spearman’s footrule \(\psi\) for the Cuadras-Augé copula.

\[\psi = 6\int_0^1 C(t,t)\,dt - 2 = \frac{6}{3-\delta} - 2 = \frac{2\delta}{3-\delta}\]

spearmans_rho(*args, **kwargs)

Spearman’s \(\rho\) for the extreme value copula.

Parameters:

• *args – Copula parameters.

• **kwargs – Copula parameters.

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

copul.family.extreme_value.galambos module

class copul.family.extreme_value.galambos.Galambos(*args, **kwargs)

Bases: BivExtremeValueCopula

Galambos extreme value copula with parameter \(\delta > 0\).

CDF:

C(u,v) = u v exp( ((-log u)^(-delta) + (-log v)^(-delta))^(-1/delta) )

for (u,v) in (0,1]^2, with the usual boundary extensions.

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

delta = delta

dim: int

intervals: dict = {'delta': Interval.open(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [delta]

property pdf

Numerical PDF via finite differences on cdf_vectorized.

copul.family.extreme_value.gumbel_hougaard module

class copul.family.extreme_value.gumbel_hougaard.GumbelHougaardEV(*args, **kwargs)

Bases: MultivariateGumbelHougaard, BivExtremeValueCopula

Bivariate Gumbel-Hougaard Extreme Value Copula.

A specialized version of the multivariate Gumbel-Hougaard copula for the bivariate case. This copula combines features of both the bivariate extreme value copula and the multivariate Gumbel-Hougaard copula.

The CDF is given by: C(u,v) = exp(-(((-ln u)^θ + (-ln v)^θ)^(1/θ)))

Special cases: - θ = 1: Independence copula - θ → ∞: Comonotonicity copula (perfect positive dependence)

Parameters:

theta (float, optional) – Dependence parameter (default is None). Must be greater than or equal to 1.

dim: int

intervals: dict = {'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

The Gumbel-Hougaard copula is absolutely continuous.

Returns:

True

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

The Gumbel-Hougaard copula is symmetric.

Returns:

True

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Kendall’s tau for the Gumbel-Hougaard copula.

For the Gumbel-Hougaard copula, Kendall’s tau has the closed form (θ-1)/θ.

Parameters:

• *args – Parameters for the copula.

• **kwargs – Parameters for the copula.

Returns:

Kendall’s tau value or expression.

Return type:

float or sympy.Expr

params: list = [theta]

theta = theta

copul.family.extreme_value.huesler_reiss module

class copul.family.extreme_value.huesler_reiss.HueslerReiss(*args, **kwargs)

Bases: BivExtremeValueCopula

Hüsler–Reiss extreme value copula with parameter \(\delta \ge 0\). When \(\delta=0\), it reduces to the independence copula.

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

cdf_vectorized(u, v)

Vectorized implementation of the Hüsler–Reiss copula CDF:

C(u,v) = (u * v)^A(t), where t = ln(v) / ln(u*v), A(t) = (1 - t)*Φ(z(1 - t)) + t*Φ(z(t)), z(x) = 1/delta + (delta/2)*ln(x/(1 - x)).

delta = delta

dim: int

intervals: dict = {'delta': Interval(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

copul.family.extreme_value.joeev module

class copul.family.extreme_value.joeev.JoeEV(*args, **kwargs)

Bases: BivExtremeValueCopula

alpha_1 = alpha_1

alpha_2 = alpha_2

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

The symbolic CDF of the JoeEV copula contains deeply nested logarithms and powers that are slow to evaluate through SymPy’s evalf. This override routes all concrete (u, v) evaluations to the fast numpy path.

delta = delta

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1), 'delta': Interval.open(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [alpha_1, alpha_2, delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

copul.family.extreme_value.marshall_olkin module

class copul.family.extreme_value.marshall_olkin.MarshallOlkin(*args, **kwargs)

Bases: BivExtremeValueCopula

property alpha_1

property alpha_2

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta\) for the Marshall-Olkin copula.

\[\beta = 4\,C(\tfrac12,\tfrac12) - 1 = 4 \cdot 2^{-\max(\alpha_1,\alpha_2)} \cdot (\tfrac14)^{1-?} - 1\]

Uses the general \(4C(1/2,1/2)-1\) formula directly.

chatterjees_xi(*args, **kwargs)

Compute Chatterjee’s xi correlation measure.

This method sets the parameters, computes intermediate integrals, and returns the simplified expression for xi.

Returns:

A wrapper around the symbolic expression for Chatterjee’s xi.

Return type:

SymPyFuncWrapper

cond_distr_1(u=None, v=None)

First conditional distribution \(\partial C/\partial u\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial u\).

Return type:

float

cond_distr_2(u=None, v=None)

Second conditional distribution \(\partial C/\partial v\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial v\).

Return type:

float

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1)}

property is_absolutely_continuous

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Spearman’s \(\rho\).

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

params: list = [alpha_1, alpha_2]

property pdf

Probability density function of the copula.

Returns:

Wrapper around the symbolic or numerical PDF.

Return type:

SymPyFuncWrapper

schweizer_wolff_sigma(*args, **kwargs)

Schweizer–Wolff \(\sigma\) for the Marshall-Olkin copula.

The MO copula is PQD for all \(\alpha_1, \alpha_2 \ge 0\), so

\[\sigma = \rho_S = \frac{3\,\alpha_1\,\alpha_2} {2\alpha_1 + 2\alpha_2 - \alpha_1\alpha_2}\]

spearmans_footrule(*args, **kwargs)

spearmans_rho(*args, **kwargs)

Spearman’s \(\rho\) for the extreme value copula.

Parameters:

• *args – Copula parameters.

• **kwargs – Copula parameters.

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

copul.family.extreme_value.marshall_olkin.MarshallOlkinDiag()

Creates a Marshall-Olkin copula with alpha_1 = alpha_2

copul.family.extreme_value.multivar_ev_independence_copula module

class copul.family.extreme_value.multivar_ev_independence_copula.MultivariateExtremeIndependenceCopula(dimension=2, **kwargs)

Bases: MultivariateExtremeValueCopula

Multivariate Independence Copula as an Extreme Value Copula.

This class represents the independence copula C(u₁, u₂, …, uₙ) = u₁ × u₂ × … × uₙ as a special case of an extreme value copula.

The independence copula is the only copula that is both an extreme value copula and an Archimedean copula. In the extreme value context, it corresponds to having complete tail independence between variables.

Parameters:

dimension (int, optional) – Dimension of the copula (number of variables). Default is 2.

property cdf

Compute the cumulative distribution function (CDF) of the independence copula.

For the independence copula, the CDF is the product of all marginals: C(u₁, u₂, …, uₙ) = u₁ × u₂ × … × uₙ

Returns:

A wrapper around the CDF function.

Return type:

CallableCDFWrapper

cdf_vectorized(*args)

Vectorized implementation of the CDF for the independence copula.

Parameters:

*args (array_like) – Arrays of dimension values to evaluate the CDF at.

Returns:

Array of CDF values.

Return type:

numpy.ndarray

intervals = {}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

The independence copula is absolutely continuous.

Returns:

True

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

The independence copula is symmetric in all its arguments.

Returns:

True

Return type:

bool

kendalls_tau()

Compute Kendall’s tau for the independence copula.

For the independence copula, Kendall’s tau is 0 as there is no dependence.

Returns:

0

Return type:

float

lambda_L()

Compute the lower tail dependence coefficient.

For the independence copula, there is no tail dependence.

Returns:

0

Return type:

float

lambda_U()

Compute the upper tail dependence coefficient.

For the independence copula, there is no tail dependence.

Returns:

0

Return type:

float

params = []

property pdf

Compute the probability density function (PDF) of the independence copula.

For the independence copula, the PDF is constant 1 on the unit cube.

Returns:

A wrapper around the PDF function.

Return type:

SymPyFuncWrapper

pdf_vectorized(*args)

Vectorized implementation of the PDF for the independence copula.

Parameters:

*args (array_like) – Arrays of dimension values to evaluate the PDF at.

Returns:

Array of PDF values (all 1s).

Return type:

numpy.ndarray

rvs(n=1, random_state=None)

Generate random variates from the independence copula.

For the independence copula, this simply generates independent uniform random variables.

Parameters:

• n (int, optional) – Number of samples to generate (default is 1).

• random_state (int or None, optional) – Seed for the random number generator.

Returns:

Array of shape (n, dim) containing independent uniform samples.

Return type:

numpy.ndarray

spearmans_rho()

Compute Spearman’s rho for the independence copula.

For the independence copula, Spearman’s rho is 0 as there is no dependence.

Returns:

0

Return type:

float

copul.family.extreme_value.multivariate_extreme_value_copula module

class copul.family.extreme_value.multivariate_extreme_value_copula.CallableCDFWrapper(callable_func)

Bases: object

A wrapper for Python callable functions to be used as CDF functions.

This is different from CDFWrapper and SymPyFuncWrapper which expect sympy expressions. This wrapper accepts Python callables and provides a compatible interface.

evalf()

Placeholder for the evalf method to maintain compatibility.

Returns:

This instance, unchanged.

Return type:

self

subs(*args, **kwargs)

Placeholder for the subs method to maintain compatibility.

Returns:

This instance, unchanged.

Return type:

self

class copul.family.extreme_value.multivariate_extreme_value_copula.MultivariateExtremeValueCopula(dimension, *args, **kwargs)

Bases: Copula

Multivariate Extreme Value Copula.

An extension of the Copula class designed for multivariate extreme value distributions. Extreme value copulas arise as the limiting distributions of component-wise maxima of random vectors.

property cdf

Compute the cumulative distribution function (CDF) of the extreme value copula.

Returns:

A wrapper around the CDF function.

Return type:

CallableCDFWrapper

cdf_vectorized(*args)

Vectorized implementation of the CDF function for improved performance with arrays.

Parameters:

*args (array_like) – Arrays of dimension values to evaluate the CDF at.

Returns:

Array of CDF values.

Return type:

numpy.ndarray

intervals = {}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params = []

sample_parameters(n=1)

Sample random parameter values within the defined intervals.

Parameters:

n (int, optional) – Number of parameter sets to sample (default is 1).

Returns:

Dictionary of parameter name-value pairs.

Return type:

dict

suppress_warnings()

Context manager to temporarily suppress warnings.

Yields:

None

copul.family.extreme_value.multivariate_gumbel_hougaard module

class copul.family.extreme_value.multivariate_gumbel_hougaard.MultivariateGumbelHougaard(dimension=2, *args, **kwargs)

Bases: MultivariateExtremeValueCopula

Multivariate Gumbel-Hougaard Extreme Value Copula.

A specialized extreme value copula with the form: C(u₁, u₂, …, uₙ) = exp(-((-ln u₁)^θ + (-ln u₂)^θ + … + (-ln uₙ)^θ)^(1/θ))

Special cases: - θ = 1: Independence copula - θ → ∞: Comonotonicity copula (perfect positive dependence)

Parameters:

• dimension (int) – Dimension of the copula (number of variables).

• theta (float, optional) – Dependence parameter (default is None). Must be greater than or equal to 1.

property cdf

C(u1,…,ud) = exp(-((sum_j (-log uj)^theta)^(1/theta))) With boundary handling:

• if any u_j == 0 -> 0

• if all u_j == 1 -> 1

• else -> interior expression

cdf_vectorized(*args)

Vectorized CDF with correct boundary handling.

intervals = {'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

The Gumbel-Hougaard copula is absolutely continuous.

Returns:

True

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

The Gumbel-Hougaard copula is symmetric in all its arguments.

Returns:

True

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Kendall’s tau for the multivariate Gumbel-Hougaard copula.

For the bivariate case, Kendall’s tau is (θ-1)/θ. For higher dimensions, the pairwise Kendall’s tau is the same for any pair.

Parameters:

• *args – Parameters for the copula.

• **kwargs – Parameters for the copula.

Returns:

Kendall’s tau value or expression.

Return type:

float or sympy.Expr

params = [theta]

theta = theta

copul.family.extreme_value.multivariate_gumbel_hougaard.MultivariateGumbelHougaardEV

alias of MultivariateGumbelHougaard

copul.family.extreme_value.t_ev module

class copul.family.extreme_value.t_ev.tEV(*args, **kwargs)

Bases: BivExtremeValueCopula

Student-t Extreme Value Copula.

Parameters:

• nu (float) – Degrees of freedom, nu > 0

• rho (float) – Correlation parameter, -1 < rho < 1

cdf(u=None, v=None, **kwargs)

Evaluate the tEV CDF numerically via cdf_vectorized.

cdf_vectorized(u, v)

Optimized vectorized implementation of the CDF.

dim: int

intervals: dict = {'nu': Interval.open(0, oo), 'rho': Interval.open(-1, 1)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

nu = nu

params: list = [nu, rho]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

property pickands

Return the Pickands dependence function with parameter values substituted.

Returns:

A wrapper that supports evaluation at \(t\) values and symbolic use.

Return type:

PickandsWrapper

rho = rho

copul.family.extreme_value.tawn module

class copul.family.extreme_value.tawn.Tawn(*args, **kwargs)

Bases: BivExtremeValueCopula

alpha_1 = alpha_1

alpha_2 = alpha_2

property cdf

Cumulative distribution function of the copula.

Returns:

Wrapper around the symbolic CDF expression.

Return type:

CDFWrapper

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1), 'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [alpha_1, alpha_2, theta]

theta = theta

Module contents

class copul.family.extreme_value.BB5(*args, **kwargs)

Bases: BivExtremeValueCopula

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

The symbolic CDF of the BB5 copula contains deeply nested logarithms and powers that are prohibitively slow to evaluate through SymPy’s evalf. This override routes all concrete (u, v) evaluations to the fast numpy path.

delta = delta

dim: int

intervals: dict = {'delta': Interval.open(0, oo), 'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [theta, delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

theta = theta

class copul.family.extreme_value.CuadrasAuge(*args, **kwargs)

Bases: BivExtremeValueCopula

Cuadras-Auge copula, special case of the Marshall-Olkin copula.

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta\) for the Cuadras-Augé copula.

\[\beta = 4\,C(\tfrac12,\tfrac12) - 1 = 4\cdot 2^{-\delta}\cdot 2^{-(2-2\delta)} - 1 = 2^{2-\delta} \cdot 2^{-2+2\delta}\cdot 4 - 1 \;=\; 2^{\delta} - 1\]

chatterjees_xi(*args, **kwargs)

Compute Chatterjee’s xi correlation measure.

This method sets the parameters, computes intermediate integrals, and returns the simplified expression for xi.

Returns:

A wrapper around the symbolic expression for Chatterjee’s xi.

Return type:

SymPyFuncWrapper

cond_distr_1(u=None, v=None)

First conditional distribution \(\partial C/\partial u\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial u\).

Return type:

float

delta = delta

dim: int

gini_gamma(*args, **kwargs)

Gini’s \(\gamma\) for the Cuadras-Augé copula.

\[\gamma = 4\!\left[\int_0^1 C(t,t)\,dt + \int_0^1 C(t,1{-}t)\,dt\right] - 2\]

For the CA copula, \(C(t,t) = t^{2-\delta}\) (since \(\min(t,t)=t\)), so the diagonal integral is \(1/(3-\delta)\). The anti-diagonal \(C(t,1-t)\) requires splitting at \(t=1/2\).

hoeffdings_d(*args, **kwargs)

Hoeffding’s \(D\) for the Cuadras-Augé copula.

Expanding the double integral \(90\iint(C-uv)^2\,du\,dv\) over the two regions \(\{v \le u\}\) and \(\{u < v\}\) (using symmetry) gives:

\[D(\delta) = \frac{10\,\delta^{2}} {18 - 9\,\delta + \delta^{2}}\]

which satisfies \(D(0)=0\) and \(D(1)=1\).

intervals: dict = {'delta': Interval(0, 1)}

property is_absolutely_continuous

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Spearman’s \(\rho\).

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

params: list = [delta]

property pdf

Probability density function of the copula.

Returns:

Wrapper around the symbolic or numerical PDF.

Return type:

SymPyFuncWrapper

schweizer_wolff_sigma(*args, **kwargs)

Schweizer–Wolff \(\sigma\) for the Cuadras-Augé copula.

The CA copula is PQD for all \(\delta \in (0,1]\), so \(\sigma = \rho_S = 3\delta/(4-\delta)\).

spearman_footrule(*args, **kwargs)

Spearman’s footrule \(\psi\) for the Cuadras-Augé copula.

\[\psi = 6\int_0^1 C(t,t)\,dt - 2 = \frac{6}{3-\delta} - 2 = \frac{2\delta}{3-\delta}\]

spearmans_rho(*args, **kwargs)

Spearman’s \(\rho\) for the extreme value copula.

Parameters:

• *args – Copula parameters.

• **kwargs – Copula parameters.

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

class copul.family.extreme_value.Galambos(*args, **kwargs)

Bases: BivExtremeValueCopula

Galambos extreme value copula with parameter \(\delta > 0\).

CDF:

C(u,v) = u v exp( ((-log u)^(-delta) + (-log v)^(-delta))^(-1/delta) )

for (u,v) in (0,1]^2, with the usual boundary extensions.

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

delta = delta

dim: int

intervals: dict = {'delta': Interval.open(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [delta]

property pdf

Numerical PDF via finite differences on cdf_vectorized.

class copul.family.extreme_value.GumbelHougaardEV(*args, **kwargs)

Bases: MultivariateGumbelHougaard, BivExtremeValueCopula

Bivariate Gumbel-Hougaard Extreme Value Copula.

A specialized version of the multivariate Gumbel-Hougaard copula for the bivariate case. This copula combines features of both the bivariate extreme value copula and the multivariate Gumbel-Hougaard copula.

The CDF is given by: C(u,v) = exp(-(((-ln u)^θ + (-ln v)^θ)^(1/θ)))

Special cases: - θ = 1: Independence copula - θ → ∞: Comonotonicity copula (perfect positive dependence)

Parameters:

theta (float, optional) – Dependence parameter (default is None). Must be greater than or equal to 1.

dim: int

intervals: dict = {'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

The Gumbel-Hougaard copula is absolutely continuous.

Returns:

True

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

The Gumbel-Hougaard copula is symmetric.

Returns:

True

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Kendall’s tau for the Gumbel-Hougaard copula.

For the Gumbel-Hougaard copula, Kendall’s tau has the closed form (θ-1)/θ.

Parameters:

• *args – Parameters for the copula.

• **kwargs – Parameters for the copula.

Returns:

Kendall’s tau value or expression.

Return type:

float or sympy.Expr

params: list = [theta]

theta = theta

class copul.family.extreme_value.HueslerReiss(*args, **kwargs)

Bases: BivExtremeValueCopula

Hüsler–Reiss extreme value copula with parameter \(\delta \ge 0\). When \(\delta=0\), it reduces to the independence copula.

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

cdf_vectorized(u, v)

Vectorized implementation of the Hüsler–Reiss copula CDF:

C(u,v) = (u * v)^A(t), where t = ln(v) / ln(u*v), A(t) = (1 - t)*Φ(z(1 - t)) + t*Φ(z(t)), z(x) = 1/delta + (delta/2)*ln(x/(1 - x)).

delta = delta

dim: int

intervals: dict = {'delta': Interval(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

class copul.family.extreme_value.JoeEV(*args, **kwargs)

Bases: BivExtremeValueCopula

alpha_1 = alpha_1

alpha_2 = alpha_2

cdf(u=None, v=None, **kwargs)

Evaluate the CDF numerically via cdf_vectorized.

The symbolic CDF of the JoeEV copula contains deeply nested logarithms and powers that are slow to evaluate through SymPy’s evalf. This override routes all concrete (u, v) evaluations to the fast numpy path.

delta = delta

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1), 'delta': Interval.open(0, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [alpha_1, alpha_2, delta]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

class copul.family.extreme_value.MarshallOlkin(*args, **kwargs)

Bases: BivExtremeValueCopula

property alpha_1

property alpha_2

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta\) for the Marshall-Olkin copula.

\[\beta = 4\,C(\tfrac12,\tfrac12) - 1 = 4 \cdot 2^{-\max(\alpha_1,\alpha_2)} \cdot (\tfrac14)^{1-?} - 1\]

Uses the general \(4C(1/2,1/2)-1\) formula directly.

chatterjees_xi(*args, **kwargs)

Compute Chatterjee’s xi correlation measure.

This method sets the parameters, computes intermediate integrals, and returns the simplified expression for xi.

Returns:

A wrapper around the symbolic expression for Chatterjee’s xi.

Return type:

SymPyFuncWrapper

cond_distr_1(u=None, v=None)

First conditional distribution \(\partial C/\partial u\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial u\).

Return type:

float

cond_distr_2(u=None, v=None)

Second conditional distribution \(\partial C/\partial v\).

Uses exact boundary values where possible, otherwise numerical finite differences on cdf_vectorized().

Parameters:

• u (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

• v (float or tuple) – Evaluation point in \([0,1]^2\). If u is a tuple/list (u_val, v_val) and v is None, the values are unpacked automatically.

Returns:

Value of \(\partial C(u,v)/\partial v\).

Return type:

float

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1)}

property is_absolutely_continuous

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

kendalls_tau(*args, **kwargs)

Compute Spearman’s \(\rho\).

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

params: list = [alpha_1, alpha_2]

property pdf

Probability density function of the copula.

Returns:

Wrapper around the symbolic or numerical PDF.

Return type:

SymPyFuncWrapper

schweizer_wolff_sigma(*args, **kwargs)

Schweizer–Wolff \(\sigma\) for the Marshall-Olkin copula.

The MO copula is PQD for all \(\alpha_1, \alpha_2 \ge 0\), so

\[\sigma = \rho_S = \frac{3\,\alpha_1\,\alpha_2} {2\alpha_1 + 2\alpha_2 - \alpha_1\alpha_2}\]

spearmans_footrule(*args, **kwargs)

spearmans_rho(*args, **kwargs)

Spearman’s \(\rho\) for the extreme value copula.

Parameters:

• *args – Copula parameters.

• **kwargs – Copula parameters.

Returns:

Symbolic expression of Spearman’s \(\rho\).

Return type:

sympy.Expr

copul.family.extreme_value.MarshallOlkinDiag()

Creates a Marshall-Olkin copula with alpha_1 = alpha_2

class copul.family.extreme_value.Tawn(*args, **kwargs)

Bases: BivExtremeValueCopula

alpha_1 = alpha_1

alpha_2 = alpha_2

property cdf

Cumulative distribution function of the copula.

Returns:

Wrapper around the symbolic CDF expression.

Return type:

CDFWrapper

dim: int

intervals: dict = {'alpha_1': Interval(0, 1), 'alpha_2': Interval(0, 1), 'theta': Interval(1, oo)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

params: list = [alpha_1, alpha_2, theta]

theta = theta

class copul.family.extreme_value.tEV(*args, **kwargs)

Bases: BivExtremeValueCopula

Student-t Extreme Value Copula.

Parameters:

• nu (float) – Degrees of freedom, nu > 0

• rho (float) – Correlation parameter, -1 < rho < 1

cdf(u=None, v=None, **kwargs)

Evaluate the tEV CDF numerically via cdf_vectorized.

cdf_vectorized(u, v)

Optimized vectorized implementation of the CDF.

dim: int

intervals: dict = {'nu': Interval.open(0, oo), 'rho': Interval.open(-1, 1)}

property is_absolutely_continuous: bool

Check if the copula is absolutely continuous.

Returns:

Must be implemented by subclasses.

Return type:

bool

property is_symmetric: bool

Check if the copula is symmetric.

Returns:

Must be implemented by subclasses.

Return type:

bool

nu = nu

params: list = [nu, rho]

property pdf

Numerical PDF via finite-difference on cdf_vectorized.

property pickands

Return the Pickands dependence function with parameter values substituted.

Returns:

A wrapper that supports evaluation at \(t\) values and symbolic use.

Return type:

PickandsWrapper

rho = rho