copulas.bivariate package

Submodules

• copulas.bivariate.base module
• copulas.bivariate.clayton module
• copulas.bivariate.frank module
• copulas.bivariate.gumbel module
• copulas.bivariate.independence module
• copulas.bivariate.utils module

Module contents

Bivariate copulas.

class copulas.bivariate.Bivariate(copula_type=None, random_state=None)

Bases: object

Base class for bivariate copulas.

This class allows to instantiate all its subclasses and serves as a unique entry point for the bivariate copulas classes.

>>> Bivariate(copula_type=CopulaTypes.FRANK).__class__
copulas.bivariate.frank.Frank

>>> Bivariate(copula_type='frank').__class__
copulas.bivariate.frank.Frank

Parameters

• copula_type (Union[CopulaType, str]) – Subtype of the copula.

• random_state (Union[int, np.random.RandomState, None]) – Seed or RandomState for the random generator.

copula_type

Family of the copula a subclass belongs to.

Type

CopulaTypes

_subclasses

List of declared subclasses.

Type

list[type]

theta_interval

Interval of valid thetas for the given copula family.

Type

list[float]

invalid_thetas

Values that, even though they belong to theta_interval, shouldn’t be considered valid.

Type

list[float]

tau

Kendall’s tau for the data given at fit().

Type

float

theta

Parameter for the copula.

Type

float

cdf(X)

Shortcut to cumulative_distribution().

check_fit()

Assert that the model is fit and the computed theta is valid.

Raises

• NotFittedError – if the model is not fitted.

• ValueError – if the computed theta is invalid.

check_marginal(u)

Check that the marginals are uniformly distributed.

Parameters

u (np.ndarray) – Array of datapoints with shape (n,).

Raises

ValueError – If the data does not appear uniformly distributed.

check_theta()

Validate the computed theta against the copula specification.

This method is used to assert the computed theta is in the valid range for the copula.

Raises

ValueError – If theta is not in theta_interval or is in invalid_thetas,

compute_theta()

Compute theta parameter using Kendall’s tau.

copula_type = None

cumulative_distribution(X)

Compute the cumulative distribution function for the copula, \(C(u, v)\).

Parameters

X (np.ndarray) –

Returns

cumulative probability

Return type

numpy.array

fit(X)

Fit a model to the data updating the parameters.

Parameters

X (np.ndarray) – Array of datapoints with shape (n,2).

Returns

None

classmethod from_dict(copula_dict)

Create a new instance from the given parameters.

Parameters

copula_dict – dict with the parameters to replicate the copula. Like the output of Bivariate.to_dict

Returns

Instance of the copula defined on the parameters.

Return type

Bivariate

generator(t)

Compute the generator function for Archimedian copulas.

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))

In a more generic way:

\[C(u_1, u_2, ..., u_n;\theta) = \psi^-1(\sum_0^n{\psi(u_i;\theta)}; \theta)\]

infer(X)

Take in subset of values and predicts the rest.

invalid_thetas = []

classmethod load(copula_path)

Create a new instance from a file.

Parameters

copula_path (str) – Path to file with the serialized copula.

Returns

Instance with the parameters stored in the file.

Return type

Bivariate

log_probability_density(X)

Return log probability density of model.

The log probability should be overridden with numerically stable variants whenever possible.

Parameters

X – np.ndarray of shape (n, 1).

Returns

np.ndarray

partial_derivative(X)

Compute partial derivative of cumulative distribution.

The partial derivative of the copula(CDF) is the conditional CDF.

\[F(v|u) = \frac{\partial C(u,v)}{\partial u}\]

The base class provides a finite difference approximation of the partial derivative of the CDF with respect to u.

Parameters

• X (np.ndarray) –

• y (float) –

Returns

np.ndarray

partial_derivative_scalar(U, V)

Compute partial derivative \(C(u|v)\) of cumulative density of single values.

pdf(X)

Shortcut to probability_density().

percent_point(y, V)

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.

ppf(y, V)

Shortcut to percent_point().

probability_density(X)

Compute probability density function for given copula family.

The probability density(pdf) for a given copula is defined as:

\[c(U,V) = \frac{\partial^2 C(u,v)}{\partial v \partial u}\]

Parameters

X (np.ndarray) – Shape (n, 2).Datapoints to compute pdf.

Returns

Probability density for the input values.

Return type

np.array

sample(*args, **kwargs)

save(filename)

Save the internal state of a copula in the specified filename.

Parameters

filename (str) – Path to save.

Returns

None

classmethod select_copula(X)

Select best copula function based on likelihood.

Given out candidate copulas the procedure proposed for selecting the one that best fit to a dataset of pairs \(\{(u_j, v_j )\}, j=1,2,...n\) , is as follows:

1. Estimate the most likely parameter \(\theta\) of each copula candidate for the given dataset.

2. Construct \(R(z|\theta)\). Calculate the area under the tail for each of the copula candidates.

3. Compare the areas: \(a_u\) achieved using empirical copula against the ones achieved for the copula candidates. Score the outcome of the comparison from 3 (best) down to 1 (worst).

4. Proceed as in steps 2- 3 with the lower tail and function \(L\).

5. Finally the sum of empirical upper and lower tail functions is compared against \(R + L\). Scores of the three comparisons are summed and the candidate with the highest value is selected.

Parameters

X (np.ndarray) – Matrix of shape (n,2).

Returns

Best copula that fits for it.

Return type

copula

set_random_state(random_state)

Set the random state.

Parameters

random_state (int, np.random.RandomState, or None) – Seed or RandomState for the random generator.

classmethod subclasses()

Return a list of subclasses for the current class object.

Returns

Subclasses for given class.

Return type

list[Bivariate]

tau = None

theta = None

theta_interval = []

to_dict()

Return a dict with the parameters to replicate this object.

Returns

Parameters of the copula.

Return type

dict

class copulas.bivariate.Clayton(copula_type=None, random_state=None)

Bases: copulas.bivariate.base.Bivariate

Class for clayton copula model.

compute_theta()

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.

copula_type = 0

cumulative_distribution(X)

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

generator(t)

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

invalid_thetas = []

partial_derivative(X)

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

percent_point(y, V)

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.

probability_density(X)

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

theta_interval = [0, inf]

class copulas.bivariate.CopulaTypes

Bases: enum.Enum

Available copula families.

CLAYTON = 0

FRANK = 1

GUMBEL = 2

INDEPENDENCE = 3

class copulas.bivariate.Frank(copula_type=None, random_state=None)

Bases: copulas.bivariate.base.Bivariate

Class for Frank copula model.

compute_theta()

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:

\[D_1(x) = \frac{1}{x}\int_0^x\frac{t}{e^t -1} \mathrm{d}t.\]

copula_type = 1

cumulative_distribution(X)

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

generator(t)

Return the generator function.

invalid_thetas = [0]

partial_derivative(X)

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

percent_point(y, V)

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.

probability_density(X)

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

theta_interval = [-inf, inf]

class copulas.bivariate.Gumbel(copula_type=None, random_state=None)

Bases: copulas.bivariate.base.Bivariate

Class for clayton copula model.

compute_theta()

Compute theta parameter using Kendall’s tau.

On Gumbel copula \(\tau\) is defined as \(τ = \frac{θ−1}{θ}\) that we solve as \(θ = \frac{1}{1-τ}\)

copula_type = 2

cumulative_distribution(X)

Compute the cumulative distribution function for the Gumbel copula.

The cumulative density(cdf), or distribution function for the Gumbel family of copulas correspond to the formula:

\[C(u,v) = e^{-((-\ln u)^{\theta} + (-\ln v)^{\theta})^{\frac{1}{\theta}}}\]

Parameters

X (np.ndarray) –

Returns

cumulative probability for the given datapoints, cdf(X).

Return type

np.ndarray

generator(t)

Return the generator function.

invalid_thetas = []

partial_derivative(X)

Compute partial derivative of cumulative distribution.

The partial derivative of the copula(CDF) is the conditional CDF.

\[F(v|u) = \frac{\partial C(u,v)}{\partial u} = C(u,v)\frac{((-\ln u)^{\theta} + (-\ln v)^{\theta})^{\frac{1}{\theta} - 1}} {\theta(- \ln u)^{1 -\theta}}\]

Parameters

• X (np.ndarray) –

• y (float) –

Returns

numpy.ndarray

percent_point(y, V)

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.

probability_density(X)

Compute probability density function for given copula family.

The probability density(PDF) for the Gumbel family of copulas correspond to the formula:

\[\begin{align} c(U,V) &= \frac{\partial^2 C(u,v)}{\partial v \partial u} &= \frac{C(u,v)}{uv} \frac{((-\ln u)^{\theta} # noqa: JS101 + (-\ln v)^{\theta})^{\frac{2} # noqa: JS101 {\theta} - 2 }}{(\ln u \ln v)^{1 - \theta}} # noqa: JS101 ( 1 + (\theta-1) \big((-\ln u)^\theta + (-\ln v)^\theta\big)^{-1/\theta}) \end{align}\]

Parameters

X (numpy.ndarray) –

Returns

numpy.ndarray

theta_interval = [1, inf]