329 lines
9.0 KiB
Python
329 lines
9.0 KiB
Python
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""" Pickand's dependence functions as generators for EV-copulas
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Created on Wed Jan 27 14:33:40 2021
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Author: Josef Perktold
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License: BSD-3
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"""
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import numpy as np
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from scipy import stats
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from statsmodels.tools.numdiff import _approx_fprime_cs_scalar, approx_hess
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class PickandDependence:
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def __call__(self, *args, **kwargs):
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return self.evaluate(*args, **kwargs)
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def evaluate(self, t, *args):
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raise NotImplementedError
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def deriv(self, t, *args):
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"""First derivative of the dependence function
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implemented through numerical differentiation
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"""
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t = np.atleast_1d(t)
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return _approx_fprime_cs_scalar(t, self.evaluate)
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def deriv2(self, t, *args):
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"""Second derivative of the dependence function
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implemented through numerical differentiation
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"""
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if np.size(t) == 1:
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d2 = approx_hess([t], self.evaluate, args=args)[0]
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else:
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d2 = np.array([approx_hess([ti], self.evaluate, args=args)[0, 0]
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for ti in t])
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return d2
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class AsymLogistic(PickandDependence):
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'''asymmetric logistic model of Tawn 1988
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special case: a1=a2=1 : Gumbel
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restrictions:
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- theta in (0,1]
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- a1, a2 in [0,1]
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'''
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k_args = 3
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def _check_args(self, a1, a2, theta):
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condth = (theta > 0) and (theta <= 1)
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conda1 = (a1 >= 0) and (a1 <= 1)
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conda2 = (a2 >= 0) and (a2 <= 1)
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return condth and conda1 and conda2
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def evaluate(self, t, a1, a2, theta):
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# if not np.all(_check_args(a1, a2, theta)):
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# raise ValueError('invalid args')
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transf = (1 - a2) * (1-t)
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transf += (1 - a1) * t
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transf += ((a1 * t)**(1./theta) + (a2 * (1-t))**(1./theta))**theta
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return transf
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def deriv(self, t, a1, a2, theta):
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b = theta
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d1 = ((a1 * (a1 * t)**(1/b - 1) - a2 * (a2 * (1 - t))**(1/b - 1)) *
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((a1 * t)**(1/b) + (a2 * (1 - t))**(1/b))**(b - 1) - a1 + a2)
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return d1
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def deriv2(self, t, a1, a2, theta):
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b = theta
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d2 = ((1 - b) * (a1 * t)**(1/b) * (a2 * (1 - t))**(1/b) *
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((a1 * t)**(1/b) + (a2 * (1 - t))**(1/b))**(b - 2)
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)/(b * (1 - t)**2 * t**2)
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return d2
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transform_tawn = AsymLogistic()
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class AsymNegLogistic(PickandDependence):
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'''asymmetric negative logistic model of Joe 1990
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special case: a1=a2=1 : symmetric negative logistic of Galambos 1978
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restrictions:
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- theta in (0,inf)
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- a1, a2 in (0,1]
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'''
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k_args = 3
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def _check_args(self, a1, a2, theta):
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condth = (theta > 0)
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conda1 = (a1 > 0) and (a1 <= 1)
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conda2 = (a2 > 0) and (a2 <= 1)
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return condth and conda1 and conda2
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def evaluate(self, t, a1, a2, theta):
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# if not np.all(self._check_args(a1, a2, theta)):
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# raise ValueError('invalid args')
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a1, a2 = a2, a1
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transf = 1 - ((a1 * (1-t))**(-1./theta) +
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(a2 * t)**(-1./theta))**(-theta)
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return transf
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def deriv(self, t, a1, a2, theta):
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a1, a2 = a2, a1
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m1 = -1 / theta
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m2 = m1 - 1
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# (a1^(-1/θ) (1 - t)^(-1/θ - 1) - a2^(-1/θ) t^(-1/θ - 1))*
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# (a1^(-1/θ) (1 - t)^(-1/θ) + (a2 t)^(-1/θ))^(-θ - 1)
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d1 = (a1**m1 * (1 - t)**m2 - a2**m1 * t**m2) * (
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(a1 * (1 - t))**m1 + (a2 * t)**m1)**(-theta - 1)
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return d1
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def deriv2(self, t, a1, a2, theta):
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b = theta
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a1, a2 = a2, a1
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a1tp = (a1 * (1 - t))**(1/b)
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a2tp = (a2 * t)**(1/b)
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a1tn = (a1 * (1 - t))**(-1/b)
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a2tn = (a2 * t)**(-1/b)
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t1 = (b + 1) * a2tp * a1tp * (a1tn + a2tn)**(-b)
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t2 = b * (1 - t)**2 * t**2 * (a1tp + a2tp)**2
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d2 = t1 / t2
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return d2
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transform_joe = AsymNegLogistic()
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class AsymMixed(PickandDependence):
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'''asymmetric mixed model of Tawn 1988
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special case: k=0, theta in [0,1] : symmetric mixed model of
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Tiago de Oliveira 1980
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restrictions:
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- theta > 0
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- theta + 3*k > 0
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- theta + k <= 1
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- theta + 2*k <= 1
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'''
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k_args = 2
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def _check_args(self, theta, k):
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condth = (theta >= 0)
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cond1 = (theta + 3*k > 0) and (theta + k <= 1) and (theta + 2*k <= 1)
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return condth & cond1
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def evaluate(self, t, theta, k):
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transf = 1 - (theta + k) * t + theta * t*t + k * t**3
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return transf
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def deriv(self, t, theta, k):
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d_dt = - (theta + k) + 2 * theta * t + 3 * k * t**2
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return d_dt
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def deriv2(self, t, theta, k):
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d2_dt2 = 2 * theta + 6 * k * t
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return d2_dt2
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# backwards compatibility for now
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transform_tawn2 = AsymMixed()
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class AsymBiLogistic(PickandDependence):
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'''bilogistic model of Coles and Tawn 1994, Joe, Smith and Weissman 1992
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restrictions:
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- (beta, delta) in (0,1)^2 or
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- (beta, delta) in (-inf,0)^2
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not vectorized because of numerical integration
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'''
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k_args = 2
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def _check_args(self, beta, delta):
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cond1 = (beta > 0) and (beta <= 1) and (delta > 0) and (delta <= 1)
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cond2 = (beta < 0) and (delta < 0)
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return cond1 | cond2
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def evaluate(self, t, beta, delta):
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# if not np.all(_check_args(beta, delta)):
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# raise ValueError('invalid args')
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def _integrant(w):
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term1 = (1 - beta) * np.power(w, -beta) * (1-t)
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term2 = (1 - delta) * np.power(1-w, -delta) * t
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return np.maximum(term1, term2)
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from scipy.integrate import quad
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transf = quad(_integrant, 0, 1)[0]
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return transf
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transform_bilogistic = AsymBiLogistic()
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class HR(PickandDependence):
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'''model of Huesler Reiss 1989
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special case: a1=a2=1 : symmetric negative logistic of Galambos 1978
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restrictions:
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- lambda in (0,inf)
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'''
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k_args = 1
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def _check_args(self, lamda):
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cond = (lamda > 0)
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return cond
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def evaluate(self, t, lamda):
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# if not np.all(self._check_args(lamda)):
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# raise ValueError('invalid args')
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term = np.log((1. - t) / t) * 0.5 / lamda
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from scipy.stats import norm
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# use special if I want to avoid stats import
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transf = ((1 - t) * norm._cdf(lamda + term) +
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t * norm._cdf(lamda - term))
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return transf
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def _derivs(self, t, lamda, order=(1, 2)):
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if not isinstance(order, (int, np.integer)):
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if (1 in order) and (2 in order):
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order = -1
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else:
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raise ValueError("order should be 1, 2, or (1,2)")
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dn = 1 / np.sqrt(2 * np.pi)
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a = lamda
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g = np.log((1. - t) / t) * 0.5 / a
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gd1 = 1 / (2 * a * (t - 1) * t)
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gd2 = (0.5 - t) / (a * ((1 - t) * t)**2)
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# f = stats.norm.cdf(t)
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# fd1 = np.exp(-t**2 / 2) / sqrt(2 * np.pi) # stats.norm.pdf(t)
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# fd2 = fd1 * t
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tp = a + g
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fp = stats.norm.cdf(tp)
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fd1p = np.exp(-tp**2 / 2) * dn # stats.norm.pdf(t)
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fd2p = -fd1p * tp
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tn = a - g
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fn = stats.norm.cdf(tn)
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fd1n = np.exp(-tn**2 / 2) * dn # stats.norm.pdf(t)
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fd2n = -fd1n * tn
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if order in (1, -1):
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# d1 = g'(t) (-t f'(a - g(t)) - (t - 1) f'(a + g(t))) + f(a - g(t))
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# - f(a + g(t))
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d1 = gd1 * (-t * fd1n - (t - 1) * fd1p) + fn - fp
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if order in (2, -1):
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# d2 = g'(t)^2 (t f''(a - g(t)) - (t - 1) f''(a + g(t))) +
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# (-(t - 1) g''(t) - 2 g'(t)) f'(a + g(t)) -
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# (t g''(t) + 2 g'(t)) f'(a - g(t))
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d2 = (gd1**2 * (t * fd2n - (t - 1) * fd2p) +
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(-(t - 1) * gd2 - 2 * gd1) * fd1p -
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(t * gd2 + 2 * gd1) * fd1n
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)
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if order == 1:
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return d1
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elif order == 2:
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return d2
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elif order == -1:
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return (d1, d2)
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def deriv(self, t, lamda):
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return self._derivs(t, lamda, 1)
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def deriv2(self, t, lamda):
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return self._derivs(t, lamda, 2)
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transform_hr = HR()
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# def transform_tev(t, rho, df):
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class TEV(PickandDependence):
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'''t-EV model of Demarta and McNeil 2005
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restrictions:
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- rho in (-1,1)
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- x > 0
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'''
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k_args = 2
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def _check_args(self, rho, df):
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x = df # alias, Genest and Segers use chi, copual package uses df
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cond1 = (x > 0)
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cond2 = (rho > 0) and (rho < 1)
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return cond1 and cond2
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def evaluate(self, t, rho, df):
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x = df # alias, Genest and Segers use chi, copual package uses df
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# if not np.all(self, _check_args(rho, x)):
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# raise ValueError('invalid args')
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from scipy.stats import t as stats_t
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# use special if I want to avoid stats import
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term1 = (np.power(t/(1.-t), 1./x) - rho) # for t
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term2 = (np.power((1.-t)/t, 1./x) - rho) # for 1-t
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term0 = np.sqrt(1. + x) / np.sqrt(1 - rho*rho)
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z1 = term0 * term1
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z2 = term0 * term2
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transf = t * stats_t._cdf(z1, x+1) + (1 - t) * stats_t._cdf(z2, x+1)
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return transf
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transform_tev = TEV()
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