def partial_derivative(self, X):
r"""Compute partial derivative of cumulative distribution.
The partial derivative of the copula(CDF) is the conditional CDF.
.. math:: 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}}
Args:
X (np.ndarray)
y (float)
Returns:
numpy.ndarray
"""
self.check_fit()
U, V = split_matrix(X)
if self.theta == 1:
return V
else:
t1 = np.power(-np.log(U), self.theta)
t2 = np.power(-np.log(V), self.theta)
p1 = self.cumulative_distribution(X)
p2 = np.power(t1 + t2, -1 + 1.0 / self.theta)
p3 = np.power(-np.log(V), self.theta - 1)
return np.divide(np.multiply(np.multiply(p1, p2), p3), V)
def probability_density(self, X):
r"""Compute probability density function for given copula family.
The probability density(PDF) for the Frank family of copulas correspond to the formula:
.. math:: 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:
.. math:: g(x) = e^{-\theta x} - 1
Args:
X: `np.ndarray`
Returns:
np.array: probability density
"""
self.check_fit()
U, V = split_matrix(X)
if self.theta == 0:
return np.multiply(U, V)
else:
num = np.multiply(np.multiply(-self.theta, self._g(1)),
1 + self._g(np.add(U, V)))
aux = np.multiply(self._g(U), self._g(V)) + self._g(1)
den = np.power(aux, 2)
return num / den
def cumulative_distribution(self, X):
r"""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:
.. math:: C(u,v) = e^{-((-\ln u)^{\theta} + (-\ln v)^{\theta})^{\frac{1}{\theta}}}
Args:
X (np.ndarray)
Returns:
np.ndarray: cumulative probability for the given datapoints, cdf(X).
"""
self.check_fit()
U, V = split_matrix(X)
if self.theta == 1:
return np.multiply(U, V)
else:
h = np.power(-np.log(U), self.theta) + np.power(-np.log(V), self.theta)
h = -np.power(h, 1.0 / self.theta)
cdfs = np.exp(h)
return cdfs
def cumulative_distribution(self, X):
r"""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:
.. math:: C(u,v) = −\frac{\ln({\frac{1 + g(u) g(v)}{g(1)}})}{\theta}
Args:
X: `np.ndarray`
Returns:
np.array: cumulative distribution
"""
self.check_fit()
U, V = split_matrix(X)
num = np.multiply(
np.exp(np.multiply(-self.theta, U)) - 1,
np.exp(np.multiply(-self.theta, V)) - 1)
den = np.exp(-self.theta) - 1
return -1.0 / self.theta * np.log(1 + num / den)
def cumulative_distribution(self, 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:
.. math:: C(u,v) = max[(u^{-θ} + v^{-θ} - 1),0]^{-1/θ}
Args:
X (numpy.ndarray)
Returns:
numpy.ndarray: cumulative probability.
"""
self.check_fit()
U, V = split_matrix(X)
if (V == 0).all() or (U == 0).all():
return np.zeros(V.shape[0])
else:
cdfs = [
np.power(
np.power(U[i], -self.theta) + np.power(V[i], -self.theta) - 1,
-1.0 / self.theta
)
if (U[i] > 0 and V[i] > 0) else 0
for i in range(len(U))
]
return np.array([max(x, 0) for x in cdfs])
def partial_derivative(self, X):
r"""Compute partial derivative of cumulative distribution.
The partial derivative of the copula(CDF) is the conditional CDF.
.. math:: F(v|u) = \frac{\partial}{\partial u}C(u,v) =
\frac{g(u)g(v) + g(v)}{g(u)g(v) + g(1)}
Args:
X (np.ndarray)
y (float)
Returns:
np.ndarray
"""
self.check_fit()
U, V = split_matrix(X)
if self.theta == 0:
return V
else:
num = np.multiply(self._g(U), self._g(V)) + self._g(U)
den = np.multiply(self._g(U), self._g(V)) + self._g(1)
return num / den
def partial_derivative(self, X, y=0):
r"""Compute partial derivative of cumulative distribution.
The partial derivative of the copula(CDF) is the value of the conditional probability.
.. math:: F(v|u) = \frac{\partial C(u,v)}{\partial u} =
u^{- \theta - 1}(u^{-\theta} + v^{-\theta} - 1)^{-\frac{\theta+1}{\theta}}
Args:
X (np.ndarray)
y (float)
Returns:
numpy.ndarray: Derivatives
"""
self.check_fit()
U, V = split_matrix(X)
A = np.power(V, -self.theta - 1)
# If theta tends to inf, A tends to inf
# And the next partial_derivative tends to 0
if (A == np.inf).any():
return np.zeros(len(V))
B = np.power(V, -self.theta) + np.power(U, -self.theta) - 1
h = np.power(B, (-1 - self.theta) / self.theta)
return np.multiply(A, h) - y
def probability_density(self, X):
r"""Compute probability density function for given copula family.
The probability density(PDF) for the Gumbel family of copulas correspond to the formula:
.. math:: 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})
Args:
X (numpy.ndarray)
Returns:
numpy.ndarray
"""
self.check_fit()
U, V = split_matrix(X)
if self.theta == 1:
return np.multiply(U, V)
else:
a = np.power(np.multiply(U, V), -1)
tmp = np.power(-np.log(U), self.theta) + np.power(
-np.log(V), self.theta)
b = np.power(tmp, -2 + 2.0 / self.theta)
c = np.power(np.multiply(np.log(U), np.log(V)), self.theta - 1)
d = 1 + (self.theta - 1) * np.power(tmp, -1.0 / self.theta)
return self.cumulative_distribution(X) * a * b * c * d
def _compute_empirical(X):
"""Compute empirical distribution.
Args:
X(numpy.array): Shape (n,2); Datapoints to compute the empirical(frequentist) copula.
Return:
tuple(list):
"""
z_left = []
z_right = []
L = []
R = []
U, V = split_matrix(X)
N = len(U)
base = np.linspace(EPSILON, 1.0 - EPSILON, COMPUTE_EMPIRICAL_STEPS)
# See https://github.com/DAI-Lab/Copulas/issues/45
for k in range(COMPUTE_EMPIRICAL_STEPS):
left = sum(np.logical_and(U <= base[k], V <= base[k])) / N
right = sum(np.logical_and(U >= base[k], V >= base[k])) / N
if left > 0:
z_left.append(base[k])
L.append(left / base[k]**2)
if right > 0:
z_right.append(base[k])
R.append(right / (1 - z_right[k])**2)
return z_left, L, z_right, R
def fit(self, X):
"""Fit a model to the data updating the parameters.
Args:
X(np.ndarray): Array of datapoints with shape (n,2).
Return:
None
"""
U, V = split_matrix(X)
self.tau = stats.kendalltau(U, V)[0]
self._compute_theta()
def partial_derivative(self, 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.
Args:
X()
"""
_, V = split_matrix(X)
return V
def cumulative_distribution(self, X):
"""Compute the cumulative distribution of the independence bivariate copula is the product.
Args:
X(numpy.array): Matrix of shape (n,2), whose values are in [0, 1]
Returns:
numpy.array: Cumulative distribution values of given input.
"""
U, V = split_matrix(X)
return np.multiply(U, V)
def fit(self, X):
"""Fit a model to the data updating the parameters.
Args:
X(np.ndarray): Array of datapoints with shape (n,2).
Return:
None
"""
U, V = split_matrix(X)
self.check_marginal(U)
self.check_marginal(V)
self.tau = stats.kendalltau(U, V)[0]
if np.isnan(self.tau):
if len(np.unique(U)) == 1 or len(np.unique(V)) == 1:
raise ValueError("Constant column.")
raise ValueError("Unable to compute tau.")
self._compute_theta()
def probability_density(self, X):
r"""Compute probability density function for given copula family.
The probability density(PDF) for the Clayton family of copulas correspond to the formula:
.. math:: 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}}
Args:
X (numpy.ndarray)
Returns:
numpy.ndarray: Probability density for the input values.
"""
self.check_fit()
U, V = split_matrix(X)
a = (self.theta + 1) * np.power(np.multiply(U, V), -(self.theta + 1))
b = np.power(U, -self.theta) + np.power(V, -self.theta) - 1
c = -(2 * self.theta + 1) / self.theta
return a * np.power(b, c)
