copul.family.frechet package

Submodules

copul.family.frechet.biv_independence_copula module

class copul.family.frechet.biv_independence_copula.BivIndependenceCopula(*args, **kwargs)

Bases: Frechet, BivArchimedeanCopula

Bivariate Independence Copula implementation.

The independence copula represents statistical independence between random variables: C(u,v) = u*v

This is a special case of both the Frechet family (with alpha=beta=0) and the Archimedean family (with generator -log(t)).

property alpha

property beta

property cdf

C(u,v) = u*v

Type:

CDF of the independence copula

cond_distr_1(u=None, v=None)

Conditional distribution: C_2(v|u) = v

For an independence copula, the conditional distribution of v given u is just v.

cond_distr_2(u=None, v=None)

Conditional distribution: C_1(u|v) = u

For an independence copula, the conditional distribution of u given v is just u.

lambda_L()

Lower tail dependence coefficient (= 0 for independence).

lambda_U()

Upper tail dependence coefficient (= 0 for independence).

property pdf

PDF of the independence copula is constant 1 on the unit square.

property pickands

copul.family.frechet.frechet module

class copul.family.frechet.frechet.Frechet(*args, **kwargs)

Bases: BivCopula

Bivariate Fréchet copula (a convex combination of the upper/lower Fréchet bounds and independence).

Parameters:

• alpha (\(\alpha \in [0,1]\)) – Weight of the upper Fréchet bound \(\min(u,v)\).

• beta (\(\beta \in [0,1]\)) – Weight of the lower Fréchet bound \(\max(u+v-1,0)\).

Notes

The CDF is

\[C(u,v) \;=\; \alpha\,\min(u,v) \;+\; (1-\alpha-\beta)\,u\,v \;+\; \beta\,\max(u+v-1,0).\]

The parameter domain must satisfy \(\alpha\ge 0\), \(\beta\ge 0\) and \(\alpha+\beta \le 1\).

The copula is absolutely continuous iff \(\alpha=\beta=0\).

property alpha

property beta

blests_nu(*args, **kwargs)

Blest’s measure of rank correlation ν. For the Fréchet copula: ν = α − β.

property cdf

Cumulative distribution function

\[C(u,v) \;=\; \alpha\,\min(u,v) \;+\; (1-\alpha-\beta)\,u\,v \;+\; \beta\,\max(u+v-1,0).\]

cdf_vectorized(u, v)

Vectorized CDF on many points.

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

Uses NumPy broadcasting; implements the same formula as above.

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)

\(F_{U_{-1}\mid U_1}(u_{-1}\mid u_1)\).

Parameters:

• *args – See cond_distr().

• **kwargs – See cond_distr().

cond_distr_2(u=None, v=None)

\(F_{U_{-2}\mid U_2}(u_{-2}\mid u_2)\).

Parameters:

• *args – See cond_distr().

• **kwargs – See cond_distr().

dim: int

ginis_gamma(*args, **kwargs)

Gini’s gamma \(\gamma\).

For the Fréchet copula,

\[\gamma \;=\; \alpha \;-\; \beta,\]

which coincides with Spearman’s \(\rho_S\) for this family.

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

property is_absolutely_continuous: bool

Whether the copula is absolutely continuous.

Returns:

True if the copula has a density a.e. on \([0,1]^d\), otherwise False.

Return type:

bool

Notes

Subclasses must override this property.

property is_symmetric: bool

Whether \(C(u,v)=C(v,u)\) (always True for this family).

kendalls_tau(*args, **kwargs)

Kendall’s \(\tau\)

\[\tau \;=\; \frac{(\alpha-\beta)\,\bigl(2+\alpha+\beta\bigr)}{3}.\]

property lambda_L

Compute the lower tail dependence coefficient.

Returns:

The symbolic expression for the lower tail dependence.

Return type:

sympy.Expr

property lambda_U

Compute the upper tail dependence coefficient.

Returns:

The simplified symbolic expression for the upper tail dependence.

Return type:

sympy.Expr

params: list = [alpha, beta]

property pdf

Evaluate (or partially evaluate) the PDF: ∂^d C / ∂u1 … ∂ud

spearmans_footrule(*args, **kwargs)

Spearman’s footrule \(\psi \;=\; \mathbb{E}\,\lvert U-V\rvert\).

Closed form:

\[\psi \;=\; \alpha \;-\; \tfrac{1}{2}\,\beta.\]

spearmans_rho(*args, **kwargs)

Spearman’s rank correlation

\[\rho_S \;=\; \alpha \;-\; \beta.\]

copul.family.frechet.frechet_multi module

class copul.family.frechet.frechet_multi.MVFrechet(dimension, *args, **kwargs)

Bases: _FrechetMultiMixin, Copula

Multivariate Fréchet family for arbitrary dimension d >= 2.

Usage:

C = MVFrechet(dimension=3, alpha=0.4) # beta auto-forced to 0 C = MVFrechet(dimension=2, alpha=0.3, beta=0.2)

Notes

• Intervals for alpha/beta are auto-tightened based on (dimension, other parameter).

• In d≥3, attempting to set beta!=0 raises ValueError.

copul.family.frechet.lower_frechet module

class copul.family.frechet.lower_frechet.LowerFrechet(*args, **kwargs)

Bases: Frechet

property alpha

property beta

copul.family.frechet.mardia module

class copul.family.frechet.mardia.Mardia(*args, **kwargs)

Bases: BivCopula

Mardia Copula.

A convex mixture of the Fréchet bounds and the independence copula.

C(u,v) = theta^2 * (1 + theta) / 2 * min(u,v) +

(1 - theta^2) * u*v + theta^2 * (1 - theta) / 2 * max(u+v-1, 0)

Parameters:

thetafloat, -1 ≤ theta ≤ 1

Dependence parameter

property cdf

Cumulative distribution function of the copula.

C(u,v) = theta^2 * (1 + theta) / 2 * min(u,v) +

(1 - theta^2) * u*v + theta^2 * (1 - theta) / 2 * max(u+v-1, 0)

chatterjees_xi(*args, **kwargs)

Calculate Chatterjee’s xi for the Mardia copula.

For Mardia, xi = theta^4 * (3*theta^2 + 1) / 4

dim: int

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

property is_absolutely_continuous: bool

Whether the copula is absolutely continuous.

Returns:

True if the copula has a density a.e. on \([0,1]^d\), otherwise False.

Return type:

bool

Notes

Subclasses must override this property.

property is_symmetric: bool

Whether the copula is exchangeable (symmetric under coordinate permutations).

Returns:

True if \(C(u_{\pi(1)},\ldots,u_{\pi(d)}) = C(u_1,\ldots,u_d)\) for all permutations \(\pi\), otherwise False.

Return type:

bool

Notes

Subclasses must override this property.

kendalls_tau(*args, **kwargs)

Calculate Kendall’s tau for the Mardia copula.

For Mardia, tau = theta^3 * (theta^2 + 2) / 3

property lambda_L

Lower tail dependence coefficient.

For Mardia, lambda_L = theta^2 * (1 + theta) / 2

When theta = 1, this equals 1 When theta = -1, this equals 0

property lambda_U

Upper tail dependence coefficient.

For Mardia, lambda_U = theta^2 * (1 + theta) / 2

When theta = 1, this equals 1 When theta = -1, this equals 0

params: list = [theta]

property pdf

Probability density function of the copula.

The Mardia copula does not have a PDF due to its singular components.

spearmans_rho(*args, **kwargs)

Calculate Spearman’s rho for the Mardia copula.

For Mardia, rho = theta^3

theta = theta

copul.family.frechet.rho_d_lower_boundary module

class copul.family.frechet.rho_d_lower_boundary.RhoDLowerBoundary(*args, **kwargs)

Bases: Frechet

One-parameter family attaining the lower boundary of the (rho, D) region:

C = (1 - |rho|) * Pi + |rho| * B_rho,

where B_rho = M if rho >= 0 and B_rho = W if rho < 0.

This can be encoded as the Frechet mixture with:

alpha(rho) = max(rho, 0), beta(rho) = max(-rho, 0), independence weight = 1 - |rho|.

Parameters:

rho (sympy symbol or float in [-1, 1]) – Target Spearman’s rho. For this family, rho(C) = rho.

Notes

We expose only the single parameter ‘rho’ and compute alpha, beta piecewise via sympy.Max to keep expressions symbolic when needed.

property alpha

property beta

property cdf

C(u,v) = alpha * min(u,v) + (1 - alpha - beta) * u v + beta * max(u+v-1, 0), where 1 - alpha - beta = 1 - |rho|.

intervals: dict = {'rho': Interval(-1, 1)}

params: list = [rho]

property rho

copul.family.frechet.rho_d_upper_boundary module

class copul.family.frechet.rho_d_upper_boundary.RhoDUpperBoundary(*args, **kwargs)

Bases: Frechet

One-parameter family attaining the upper boundary of the (rho, D) region:

C = alpha * M + (1 - alpha) * W, with alpha = (1 + rho)/2.

Hence the independence weight is zero and beta = 1 - alpha = (1 - rho)/2. This family satisfies Spearman’s rho(C) = rho.

Parameters:

rho (sympy symbol or float in [-1, 1]) – Target Spearman’s rho.

Notes

We implement this as a Frechet subclass but expose only the single parameter ‘rho’. Internally:

alpha(rho) = (1 + rho)/2, beta(rho) = (1 - rho)/2.

property alpha

property beta

property cdf

C(u,v) = alpha * min(u,v) + (1 - alpha - beta) * u v + beta * max(u+v-1, 0) = alpha * M + beta * W (since 1 - alpha - beta = 0 in this family).

intervals: dict = {'rho': Interval(-1, 1)}

params: list = [rho]

property rho

copul.family.frechet.upper_frechet module

class copul.family.frechet.upper_frechet.UpperFrechet(*args, **kwargs)

Bases: Frechet

property alpha

property beta

dim: int

property pickands

t = t

Module contents