import numpy as np
import sympy
from copul.family.archimedean.biv_archimedean_copula import BivArchimedeanCopula
from copul.family.frechet.biv_independence_copula import BivIndependenceCopula
from copul.wrapper.sympy_wrapper import SymPyFuncWrapper
[docs]
class Nelsen17(BivArchimedeanCopula):
ac = BivArchimedeanCopula
theta_interval = sympy.Interval(-np.inf, np.inf, left_open=True, right_open=True)
special_cases = {-1: BivIndependenceCopula}
invalid_params = {0}
@property
def is_absolutely_continuous(self) -> bool:
return True
@property
def _raw_generator(self):
return -sympy.log(
((1 + self.t) ** (-self.theta) - 1) / (2 ** (-self.theta) - 1)
)
@property
def _raw_inv_generator(self):
theta = self.theta
y = self.y
return (2**theta * sympy.exp(y) / (2**theta * sympy.exp(y) - 2**theta + 1)) ** (
1 / theta
) - 1
@property
def _cdf_expr(self):
v = self.v
u = self.u
theta = self.theta
return (
1
+ ((1 + u) ** (-theta) - 1)
* ((1 + v) ** (-theta) - 1)
/ (2 ** (-theta) - 1)
) ** (-1 / theta) - 1
@property
def pdf(self):
theta = self.theta
u = self.u
v = self.v
pdf = (
2**theta
* theta
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
+ 4**theta
* (theta + 1)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
) / (
(
(
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
/ ((2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta)
)
** (1 / theta)
* (u + 1)
* (v + 1)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 3
)
return SymPyFuncWrapper(pdf)
@property
def first_deriv_of_inv_gen(self):
theta = self.theta
y = self.y
return sympy.simplify(
(2**theta * sympy.exp(y) / (2**theta * sympy.exp(y) - 2**theta + 1))
** (1 / theta)
* (2**theta * (-(2**theta) + 1))
/ (2**theta * theta * (2**theta * sympy.exp(y) - 2**theta + 1))
)
@property
def second_deriv_of_inv_gen(self):
theta = self.theta
y = self.y
return sympy.simplify(
(2**theta * sympy.exp(y) / (2**theta * sympy.exp(y) - 2**theta + 1))
** (1 / theta)
* (
2**theta * theta * (-(2**theta) + 1) * sympy.exp(y)
- 2 ** (theta + 1) * theta * (-(2**theta) + 1) * sympy.exp(y)
+ (2**theta - 1) ** 2
)
/ (theta**2 * (2**theta * sympy.exp(y) - 2**theta + 1) ** 2)
)
@property
def first_deriv_of_ci_char(self):
theta = self.theta
y = self.y
return sympy.simplify(
(2**theta * (-(2**theta) + 1) - 4**theta * theta * sympy.exp(y))
/ (2**theta * theta * (2**theta * sympy.exp(y) - 2**theta + 1))
)
[docs]
def first_deriv_of_tp2_char(self):
theta = self.theta
y = self.y
a = 2**theta * sympy.exp(y) - 2**theta + 1
b = 2**theta * theta * sympy.exp(y) + 2**theta - 1
return (
2**theta * theta**2 * a * sympy.exp(y)
- 2 ** (theta + 1) * theta * b * sympy.exp(y)
+ (1 - 2**theta) * b
) / (theta * a * b)
[docs]
def second_deriv_of_tp2_char(self):
theta = self.theta
y = self.y
a = 2**theta * sympy.exp(y) - 2**theta + 1
b = 2**theta * theta * sympy.exp(y) + 2**theta - 1
numerator = (
(2**theta - 1) * theta**2 * a**2
+ 2**theta * b**2
+ 2 ** (theta + 1) * theta * b**2
- theta**2 * a**2
- 2 * theta * b**2
- b**2
)
# sub_expr = a**2 * theta**2 + (1 + 2 * theta) * b**2
# ey_part = (
# 2 * 2 ** (2 * theta) * theta**3 * sympy.exp(2 * y)
# + 2 * 2 ** (2 * theta) * theta**2 * sympy.exp(2 * y)
# + 2 * 2 ** (2 * theta) * theta**2 * sympy.exp(y)
# + 2 * 2 ** (2 * theta) * theta * sympy.exp(y)
# - 2 * 2**theta * theta**2 * sympy.exp(y)
# - 2 * 2**theta * theta * sympy.exp(y)
# )
# ey_critical = (
# 2**theta * theta**2 * sympy.exp(y)
# + 2**theta * theta * sympy.exp(y)
# + 2**theta * theta
# + 2**theta
# - theta
# - 1
# )
# numerator = (2**theta - 1)(sub_expr)
return 2**theta * numerator * sympy.exp(y) / (theta * a**2 * b**2)
[docs]
def deriv_of_log_density(self):
theta = self.theta
u = self.u
v = self.v
return (
-3
* theta
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
- (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
- (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 2 ** (theta + 1)
* theta**2
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
+ 4**theta
* theta
* (theta + 1)
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
+ theta
* (theta + 1)
* (4 * u + 4) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((v + 1) ** theta - 1)
)
) / (
(u + 1)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
)
[docs]
def density_of_log_density(self):
theta = self.theta
v = self.v
u = self.u
return (
(
-3
* theta
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
* (
2
* 2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
- 2
* theta
* (2**theta - 1)
* (u + 1) ** theta
* (v + 1) ** theta
/ (v + 1)
)
- 4**theta
* theta
* (theta + 1)
* (v + 1) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
/ (v + 1)
- 4**theta
* (theta + 1)
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta
* ((u + 1) * (v + 1)) ** theta
* (1 - 2**theta)
/ (v + 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
- 3
* theta
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
theta * ((u + 1) * (v + 1)) ** theta * (1 - 2**theta) / (v + 1)
+ theta * (2 * u + 2) ** theta * (v + 1) ** theta / (v + 1)
)
- (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
-(2**theta)
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta * ((u + 1) * (v + 1)) ** theta * (1 - 2**theta) / (v + 1)
+ theta * ((u + 1) * (v + 1)) ** theta * (2**theta - 1) / (v + 1)
+ theta * (2 * u + 2) ** theta * (v + 1) ** theta / (v + 1)
)
- (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
* (
2
* 2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
- 2
* theta
* (2**theta - 1)
* (u + 1) ** theta
* (v + 1) ** theta
/ (v + 1)
)
- 2 ** (theta + 1)
* theta**2
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta
* ((u + 1) * (v + 1)) ** theta
* (1 - 2**theta)
/ (v + 1)
)
- 2 ** (theta + 1)
* theta**2
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* (
theta * ((u + 1) * (v + 1)) ** theta * (1 - 2**theta) / (v + 1)
+ theta * (2 * u + 2) ** theta * (v + 1) ** theta / (v + 1)
)
+ 4**theta
* theta**2
* (theta + 1)
* (v + 1) ** theta
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* ((u + 1) ** theta - 1)
/ (v + 1)
+ 4**theta
* theta
* (theta + 1)
* (
theta * ((u + 1) * (v + 1)) ** theta * (1 - 2**theta) / (v + 1)
+ theta * (2 * u + 2) ** theta * (v + 1) ** theta / (v + 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
- 4**theta
* theta
* (theta + 1)
* (v + 1) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
/ (v + 1)
- 4**theta
* (theta + 1)
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta
* ((u + 1) * (v + 1)) ** theta
* (1 - 2**theta)
/ (v + 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
+ theta**2
* (theta + 1)
* (4 * u + 4) ** theta
* (v + 1) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
/ (v + 1)
+ theta
* (theta + 1)
* (4 * u + 4) ** theta
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta
* ((u + 1) * (v + 1)) ** theta
* (1 - 2**theta)
/ (v + 1)
)
* ((v + 1) ** theta - 1)
)
- (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta * ((u + 1) * (v + 1)) ** theta * (1 - 2**theta) / (v + 1)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 2 ** (theta + 1)
* theta**2
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
+ 4**theta
* theta
* (theta + 1)
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
+ theta
* (theta + 1)
* (4 * u + 4) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((v + 1) ** theta - 1)
)
- (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
* (
2
* 2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
- 2
* theta
* (2**theta - 1)
* (u + 1) ** theta
* (v + 1) ** theta
/ (v + 1)
)
- 4**theta
* theta
* (theta + 1)
* (v + 1) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
/ (v + 1)
- 4**theta
* (theta + 1)
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta
* ((u + 1) * (v + 1)) ** theta
* (1 - 2**theta)
/ (v + 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
)
/ (
(u + 1)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
)
+ (
-(2**theta)
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
+ theta * (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta / (v + 1)
)
* (
-3
* theta
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
- (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
- (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 2 ** (theta + 1)
* theta**2
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
+ 4**theta
* theta
* (theta + 1)
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
+ theta
* (theta + 1)
* (4 * u + 4) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((v + 1) ** theta - 1)
)
)
/ (
(u + 1)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
)
+ (
-3
* theta
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
- (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
* (
-(2**theta) * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
- (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 2 ** (theta + 1)
* theta**2
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
+ 4**theta
* theta
* (theta + 1)
* (
((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
+ (2 * u + 2) ** theta * ((v + 1) ** theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
+ theta
* (theta + 1)
* (4 * u + 4) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
+ ((u + 1) * (v + 1)) ** theta * (1 - 2**theta)
)
* ((v + 1) ** theta - 1)
)
)
* (
-(2**theta)
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
* (
2
* 2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
- 2
* theta
* (2**theta - 1)
* (u + 1) ** theta
* (v + 1) ** theta
/ (v + 1)
)
+ 4**theta
* theta
* (theta + 1)
* (v + 1) ** theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
/ (v + 1)
+ 4**theta
* (theta + 1)
* (
2**theta
* theta
* (v + 1) ** theta
* ((u + 1) ** theta - 1)
/ (v + 1)
- theta * ((u + 1) * (v + 1)) ** theta * (2**theta - 1) / (v + 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
/ (
(u + 1)
* (
2**theta
* theta
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
** 2
- 4**theta
* (theta + 1)
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- ((u + 1) * (v + 1)) ** theta * (2**theta - 1)
)
* ((u + 1) ** theta - 1)
* ((v + 1) ** theta - 1)
)
** 2
* (
2**theta * ((u + 1) ** theta - 1) * ((v + 1) ** theta - 1)
- (2**theta - 1) * (u + 1) ** theta * (v + 1) ** theta
)
)
)
[docs]
def lambda_L(self):
return 0
[docs]
def lambda_U(self):
return 0