climb.tool.impl.data_suite.third_party.copulas.bivariate package

Submodules

climb.tool.impl.data_suite.third_party.copulas.bivariate.base module

This module contains a base class for bivariate copulas.

class climb.tool.impl.data_suite.third_party.copulas.bivariate.base.Bivariate(*args, **kwargs)

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_seed (Union[int, None]) – Seed 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)

The marginals are supposed to be 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 \(\psi: [0,1]\times\Theta \rightarrow [0, \infty)\) that given an Archimedian copula fulills:

\[C(u,v) = \psi^{-1}(\psi(u) + \psi(v))\]

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. It 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

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 climb.tool.impl.data_suite.third_party.copulas.bivariate.base.CopulaTypes(value)

Bases: Enum

Available copula families.

CLAYTON = 0

FRANK = 1

GUMBEL = 2

INDEPENDENCE = 3

climb.tool.impl.data_suite.third_party.copulas.bivariate.clayton module

class climb.tool.impl.data_suite.third_party.copulas.bivariate.clayton.Clayton(*args, **kwargs)

Bases: 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 \(\psi: [0,1]\times\Theta \rightarrow [0, \infty)\) that given an Archimedian copula fulills:

\[C(u,v) = \psi^{-1}(\psi(u) + \psi(v))\]

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]

climb.tool.impl.data_suite.third_party.copulas.bivariate.frank module

class climb.tool.impl.data_suite.third_party.copulas.bivariate.frank.Frank(*args, **kwargs)

Bases: 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]

climb.tool.impl.data_suite.third_party.copulas.bivariate.gumbel module

class climb.tool.impl.data_suite.third_party.copulas.bivariate.gumbel.Gumbel(*args, **kwargs)

Bases: 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:

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

Parameters:

X (numpy.ndarray)

Returns:

numpy.ndarray

theta_interval = [1, inf]

climb.tool.impl.data_suite.third_party.copulas.bivariate.independence module

class climb.tool.impl.data_suite.third_party.copulas.bivariate.independence.Independence(*args, **kwargs)

Bases: Bivariate

This class represent the copula for two independent variables.

copula_type = 3

cumulative_distribution(X)

Compute the cumulative distribution of the independence bivariate copula is the product.

Parameters:

X (numpy.array) – Matrix of shape (n,2), whose values are in [0, 1]

Returns:

Cumulative distribution values of given input.

Return type:

numpy.array

fit(X)

Fit the copula to the given data.

Parameters:

X (numpy.array) – Probabilites in a matrix shaped (n, 2)

Returns:

None

generator(t)

Compute the generator function for the Copula.

The generator function is a function f(t), such that an archimedian copula can be defined as

C(u1, …, uN) = f(f^-1(u1), …, f^-1(uN)).

Parameters:

t (numpy.array)

Returns:

np.array

partial_derivative(X)

Compute the conditional probability of one event conditiones to the other.

In the case of the independence copula, due to C(u,v) = u*v, we have that F(u|v) = dC/du = v.

Parameters:

X()

percent_point(y, V)

Compute the inverse of conditional cumulative distribution \(F(u|v)^-1\).

Parameters:

• y – np.ndarray value of \(F(u|v)\).

• v – np.ndarray given value of v.

probability_density(X)

Compute the probability density for the independence copula.

climb.tool.impl.data_suite.third_party.copulas.bivariate.utils module

climb.tool.impl.data_suite.third_party.copulas.bivariate.utils.split_matrix(X)

Split an (n,2) numpy.array into two vectors.

Parameters:

X (numpy.array) – Matrix of shape (n,2)

Returns:

Both of shape (n,)

Return type:

tuple[numpy.array]

Module contents

class climb.tool.impl.data_suite.third_party.copulas.bivariate.Bivariate(*args, **kwargs)

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_seed (Union[int, None]) – Seed 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)

The marginals are supposed to be 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 \(\psi: [0,1]\times\Theta \rightarrow [0, \infty)\) that given an Archimedian copula fulills:

\[C(u,v) = \psi^{-1}(\psi(u) + \psi(v))\]

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. It 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

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 climb.tool.impl.data_suite.third_party.copulas.bivariate.Clayton(*args, **kwargs)

Bases: 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 \(\psi: [0,1]\times\Theta \rightarrow [0, \infty)\) that given an Archimedian copula fulills:

\[C(u,v) = \psi^{-1}(\psi(u) + \psi(v))\]

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 climb.tool.impl.data_suite.third_party.copulas.bivariate.CopulaTypes(value)

Bases: Enum

Available copula families.

CLAYTON = 0

FRANK = 1

GUMBEL = 2

INDEPENDENCE = 3

class climb.tool.impl.data_suite.third_party.copulas.bivariate.Frank(*args, **kwargs)

Bases: 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 climb.tool.impl.data_suite.third_party.copulas.bivariate.Gumbel(*args, **kwargs)

Bases: 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:

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

Parameters:

X (numpy.ndarray)

Returns:

numpy.ndarray

theta_interval = [1, inf]