copul.family.frechet package

Submodules

copul.family.frechet.biv_independence_copula module

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

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)

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.

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta = \alpha - \beta\) (affine in the weights).

blum_kiefer_rosenblatt(*args, **kwargs)

Blum-Kiefer-Rosenblatt coefficient for the Fr'echet copula:

\[B = \tfrac13\bigl(\alpha^2(1+2\alpha)+\beta^2(1+2\beta)\bigr) - \tfrac12\,\alpha\beta(\alpha+\beta) - \tfrac{7}{12}\,\alpha\beta\,,\]

with \(B(M)=B(W)=1\), obtained by integrating the squared deviation against the mixture measure \(\alpha\,\mathrm{d}M+(1-\alpha-\beta)\,\mathrm{d}\Pi+\beta\,\mathrm{d}W\).

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

gini_gamma(*args, **kwargs)

Compute Gini’s gamma concordance coefficient.

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

Range: \([-1, 1]\). \(\gamma = 1\) for the upper Fréchet bound (comonotonic), \(\gamma = -1\) for the lower Fréchet bound (countermonotonic), \(\gamma = 0\) for independence.

Returns:

The symbolic expression for Gini’s gamma.

Return type:

sympy.Expr

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.

hoeffdings_d(*args, **kwargs)

Hoeffding’s dependence index for the Fr'echet copula:

\[\Phi^2 = \alpha^2 + \beta^2 - \tfrac{7}{4}\,\alpha\beta\,,\]

using \(90\!\iint(M-\Pi)^2 = 90\!\iint(W-\Pi)^2 = 1\) and \(90\!\iint(M-\Pi)(W-\Pi) = -\tfrac{7}{8}\).

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

spearman_footrule(*args, **kwargs)

Compute Spearman’s footrule coefficient \(\psi\).

\[\psi(C) = 6 \int_0^1 C(t,t)\,dt - 2\]

Equivalently, \(\psi(C) = 6\,\mathbb{E}[\min(U,V)] - 2\) where \((U,V) \sim C\).

Range: \([-\tfrac{1}{2}, 1]\). \(\psi = 1\) for the upper Fréchet bound, \(\psi = -\tfrac{1}{2}\) for the lower Fréchet bound, \(\psi = 0\) for independence.

Returns:

The symbolic expression for Spearman’s footrule coefficient.

Return type:

sympy.Expr

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)

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)

property alpha

property beta

copul.family.frechet.mardia module

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

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

blests_nu(*args, **kwargs)

Blest’s \(\nu = \theta^3\); the Mardia copula is radially symmetric, so \(\nu\) coincides with Spearman’s \(\rho\).

blomqvists_beta(*args, **kwargs)

Blomqvist’s \(\beta = \theta^3\) for the Mardia copula.

blum_kiefer_rosenblatt(*args, **kwargs)

Blum-Kiefer-Rosenblatt coefficient \(B = \theta^4(2\theta^2+1)(15\theta^2+1)/48\) for the Mardia copula, obtained from the Fr'echet-copula formula by substituting the Mardia weights.

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

gini_gamma(*args, **kwargs)

Gini’s \(\gamma = \theta^3\) for the Mardia copula.

hoeffdings_d(*args, **kwargs)

Hoeffding’s dependence index \(\Phi^2 = \theta^4(1+15\theta^2)/16\), obtained from the Fr'echet-copula formula by substituting the Mardia weights.

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.

spearman_footrule(*args, **kwargs)

Spearman’s footrule \(\psi = \theta^2(1+3\theta)/4\).

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)

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

Symbolic CDF of the lower-boundary family:

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)

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)

property alpha

property beta

dim: int

property pickands

t = t

Module contents