from typing import Union
import numpy
import numpy as np
from beartype import beartype
from UQpy.utilities.ValidationTypes import Numpy2DFloatArray
from UQpy.distributions.baseclass import Copula
from numpy import log, exp
[docs]class Gumbel(Copula):
@beartype
def __init__(self, theta: Union[None, float]):
"""
:param theta: Parameter of the Gumbel copula, real number in :math:`[1, +\infty)`.
"""
super().__init__(theta=theta)
[docs] def evaluate_cdf(self, unit_uniform_samples: Numpy2DFloatArray) -> np.ndarray:
"""
Compute the copula cdf :math:`C(u_1, u_2, ..., u_d)` for a `d`-variate uniform distribution.
For a generic multivariate distribution with marginal cdfs :math:`F_1, ..., F_d` the joint cdf is computed as:
:math:`F(x_1, ..., x_d) = C(u_1, u_2, ..., u_d)`
where :math:`u_i = F_i(x_i)` is uniformly distributed. This computation is performed in the
:meth:`.JointCopula.cdf` method.
:param unit_uniform_samples: Points (uniformly distributed) at which to evaluate the copula cdf, must be of
shape :code:`(npoints, dimension)`.
:return: Values of the cdf.
"""
if unit_uniform_samples.shape[1] > 2:
raise ValueError("Maximum dimension for the Gumbel Copula is 2")
theta, u, v = self.extract_data(unit_uniform_samples)
cdf_val = exp(-(((-log(u)) ** theta + (-log(v)) ** theta) ** (1 / theta)))
return cdf_val
[docs] def evaluate_pdf(self, unit_uniform_samples: Numpy2DFloatArray) -> numpy.ndarray:
"""
Compute the copula pdf :math:`c(u_1, u_2, ..., u_d)` for a `d`-variate uniform distribution.
For a generic multivariate distribution with marginals pdfs :math:`f_1, ..., f_d` and marginals cdfs
:math:`F_1, ..., F_d`, the joint pdf is computed as:
:math:`f(x_1, ..., x_d) = c(u_1, u_2, ..., u_d) f_1(x_1) ... f_d(x_d)`
where :math:`u_i = F_i(x_i)` is uniformly distributed. This computation is performed in the
:meth:`.JointCopula.pdf` method.
:param unit_uniform_samples: Points (uniformly distributed) at which to evaluate the copula pdf, must be of
shape :code:`(npoints, dimension)`.
:return: Values of the copula pdf term.
"""
theta, u, v = self.extract_data(unit_uniform_samples)
c = exp(-(((-log(u)) ** theta + (-log(v)) ** theta) ** (1 / theta)))
pdf_val = (c * 1 / u * 1 / v
* ((-log(u)) ** theta + (-log(v)) ** theta) ** (-2 + 2 / theta)
* (log(u) * log(v)) ** (theta - 1)
* (1 + (theta - 1) * ((-log(u)) ** theta + (-log(v)) ** theta) ** (-1 / theta)))
return pdf_val
def extract_data(self, unit_uniform_samples):
u = unit_uniform_samples[:, 0]
v = unit_uniform_samples[:, 1]
theta = self.parameters["theta"]
return theta, u, v