def setUp(self):
self.X = np.array([[2641.16233666, 180.2425623],
[921.14476418, 192.35609972],
[-651.32239137, 150.24830291],
[1223.63536668, 156.62123653],
[3233.37342355, 173.80311908],
[1373.22400821, 191.0922843],
[1959.28188858, 163.22252158],
[1076.99295365, 190.73280428],
[2029.25100261, 158.52982435],
[1835.52188141, 163.0101334],
[1170.03850556, 205.24904026],
[739.42628394, 175.42916046],
[1866.65810627, 208.31821984],
[3703.49786503, 178.98351969],
[1719.45232017, 160.50981075],
[258.90206528, 163.19294974],
[219.42363944, 173.30395132],
[609.90212377, 215.18996298],
[1618.44207239, 164.71141696],
[2323.2775272, 178.84973821],
[3251.78732274, 182.99902513],
[1430.63989981, 217.5796917],
[-180.57028875, 201.56983421],
[-592.84497457, 174.92272693]])
self.copula = Frank()
def setUp(self):
self.X = np.array([
[0.2, 0.1],
[0.2, 0.3],
[0.4, 0.5],
[0.6, 0.4],
[0.8, 0.6],
[0.8, 0.9],
])
self.copula = Frank()
def test_cdf_zero_if_single_arg_is_zero(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1: -1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_zero_if_arg_zero(instance)
def test_cdf_value_if_all_other_arg_are_one(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1: -1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_single_arg_not_one(instance, tolerance=1E-03)
def test_sample(self, uniform_mock):
"""Sample use the inverse-transform method to generate new samples."""
# Setup
instance = Frank()
instance.tau = 0.5
instance.theta = instance.compute_theta()
uniform_mock.return_value = np.array([0.1, 0.2, 0.4, 0.6, 0.8])
expected_result = np.array([
[0.0312640840463779, 0.1],
[0.1007998170183327, 0.2],
[0.3501836319841291, 0.4],
[0.6498163680158703, 0.6],
[0.8992001829816683, 0.8]
])
expected_uniform_call_args_list = [
((0, 1, 5), {}),
((0, 1, 5), {})
]
# Run
result = instance.sample(5)
# Check
assert isinstance(result, np.ndarray)
assert result.shape == (5, 2)
compare_nested_iterables(result, expected_result)
assert uniform_mock.call_args_list == expected_uniform_call_args_list
class TestFrank(TestCase):
def setUp(self):
self.X = np.array([
[0.2, 0.1],
[0.2, 0.3],
[0.4, 0.5],
[0.6, 0.4],
[0.8, 0.6],
[0.8, 0.9],
])
self.copula = Frank()
def test_fit(self):
"""On fit, theta and tau attributes are set."""
self.copula.fit(self.X)
self.assertAlmostEqual(self.copula.tau, 0.7877, places=3)
self.assertAlmostEqual(self.copula.theta, 17.0227, places=3)
def test_probability_density(self):
"""Probability_density returns the probability density for the given values."""
# Setup
self.copula.fit(self.X)
expected_result = np.array([4.4006, 4.2302])
# Run
result = self.copula.probability_density(np.array([
[0.2, 0.2],
[0.6, 0.61]
]))
# Check
assert isinstance(result, np.ndarray)
assert np.isclose(result, expected_result, rtol=0.05).all()
def test_cumulative_distribution(self):
"""Cumulative_density returns the probability distribution value for a point."""
# Setup
self.copula.fit(self.X)
expected_result = np.array([0.1602, 0.5641])
# Run
result = self.copula.cumulative_distribution(np.array([
[0.2, 0.2],
[0.6, 0.61]
]))
# Check
assert isinstance(result, np.ndarray)
assert np.isclose(result, expected_result, rtol=0.05).all()
@patch('copulas.bivariate.base.np.random.uniform')
def test_sample(self, uniform_mock):
"""Sample use the inverse-transform method to generate new samples."""
# Setup
instance = Frank()
instance.tau = 0.5
instance.theta = instance.compute_theta()
uniform_mock.return_value = np.array([0.1, 0.2, 0.4, 0.6, 0.8])
expected_result = np.array([
[0.0312640840463779, 0.1],
[0.1007998170183327, 0.2],
[0.3501836319841291, 0.4],
[0.6498163680158703, 0.6],
[0.8992001829816683, 0.8]
])
expected_uniform_call_args_list = [
((0, 1, 5), {}),
((0, 1, 5), {})
]
# Run
result = instance.sample(5)
# Check
assert isinstance(result, np.ndarray)
assert result.shape == (5, 2)
compare_nested_iterables(result, expected_result)
assert uniform_mock.call_args_list == expected_uniform_call_args_list
def test_cdf_zero_if_single_arg_is_zero(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1: -1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_zero_if_arg_zero(instance)
def test_cdf_value_if_all_other_arg_are_one(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1: -1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_single_arg_not_one(instance, tolerance=1E-03)
class TestFrank(TestCase):
def setUp(self):
self.X = np.array([[2641.16233666, 180.2425623],
[921.14476418, 192.35609972],
[-651.32239137, 150.24830291],
[1223.63536668, 156.62123653],
[3233.37342355, 173.80311908],
[1373.22400821, 191.0922843],
[1959.28188858, 163.22252158],
[1076.99295365, 190.73280428],
[2029.25100261, 158.52982435],
[1835.52188141, 163.0101334],
[1170.03850556, 205.24904026],
[739.42628394, 175.42916046],
[1866.65810627, 208.31821984],
[3703.49786503, 178.98351969],
[1719.45232017, 160.50981075],
[258.90206528, 163.19294974],
[219.42363944, 173.30395132],
[609.90212377, 215.18996298],
[1618.44207239, 164.71141696],
[2323.2775272, 178.84973821],
[3251.78732274, 182.99902513],
[1430.63989981, 217.5796917],
[-180.57028875, 201.56983421],
[-592.84497457, 174.92272693]])
self.copula = Frank()
def test_fit(self):
"""On fit, theta and tau attributes are set."""
# Setup
expected_theta = 0.1307082
expected_tau = 0.01449275
# Run
self.copula.fit(self.X)
actual_theta = self.copula.theta
actual_tau = self.copula.tau
# Check
self.assertAlmostEqual(actual_theta, expected_theta, places=3)
self.assertAlmostEqual(actual_tau, expected_tau)
def test_probability_density(self):
"""Probability_density returns the probability density for the given values."""
# Setup
self.copula.fit(self.X)
expected_result = 0.999672586804842
# Run
result = self.copula.probability_density(np.array([[0.1, 0.5]]))
# Check
assert np.isclose(result, expected_result).all()
assert isinstance(result, np.ndarray)
def test_cumulative_distribution(self):
"""Cumulative_density returns the probability distribution value for a point."""
# Setup
self.copula.fit(self.X)
expected_result = 0.05147003
# Run
result = self.copula.cumulative_distribution(np.array([[0.1, 0.5]]))
# Check
assert np.isclose(result, expected_result).all()
assert isinstance(result, np.ndarray)
@patch('copulas.bivariate.base.np.random.uniform')
def test_sample(self, uniform_mock):
"""Sample use the inverse-transform method to generate new samples."""
# Setup
instance = Frank()
instance.tau = 0.5
instance.theta = instance.compute_theta()
uniform_mock.return_value = np.array([0.1, 0.2, 0.4, 0.6, 0.8])
expected_result = np.array([[0.0312640840463779, 0.1],
[0.1007998170183327, 0.2],
[0.3501836319841291, 0.4],
[0.6498163680158703, 0.6],
[0.8992001829816683, 0.8]])
expected_uniform_call_args_list = [((0, 1, 5), {}), ((0, 1, 5), {})]
# Run
result = instance.sample(5)
# Check
assert isinstance(result, np.ndarray)
assert result.shape == (5, 2)
compare_nested_iterables(result, expected_result)
assert uniform_mock.call_args_list == expected_uniform_call_args_list
def test_cdf_zero_if_single_arg_is_zero(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1:-1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_zero_if_arg_zero(instance)
def test_cdf_value_if_all_other_arg_are_one(self):
"""Test of the analytical properties of copulas on a range of values of theta."""
# Setup
instance = Frank()
tau_values = np.linspace(-1.0, 1.0, 20)[1:-1]
# Run/Check
for tau in tau_values:
instance.tau = tau
instance.theta = instance.compute_theta()
copula_single_arg_not_one(instance, tolerance=1E-03)
def select_copula(X):
r"""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 :math:`\{(u_j, v_j )\}, j=1,2,...n` , is as follows:
1. Estimate the most likely parameter :math:`\theta` of each copula candidate for the given
dataset.
2. Construct :math:`R(z|\theta)`. Calculate the area under the tail for each of the copula
candidates.
3. Compare the areas: :math:`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 :math:`L`.
5. Finally the sum of empirical upper and lower tail functions is compared against
:math:`R + L`. Scores of the three comparisons are summed and the candidate with the
highest value is selected.
Args:
X(np.ndarray): Matrix of shape (n,2).
Returns:
copula: Best copula that fits for it.
"""
frank = Frank()
frank.fit(X)
if frank.tau <= 0:
return frank
copula_candidates = [frank]
# append copulas into the candidate list
for copula_class in [Clayton, Gumbel]:
try:
copula = copula_class()
copula.tau = frank.tau
copula._compute_theta()
copula_candidates.append(copula)
except ValueError:
pass
left_tail, empirical_left_aut, right_tail, empirical_right_aut = _compute_empirical(
X)
candidate_left_auts, candidate_right_auts = _compute_candidates(
copula_candidates, left_tail, right_tail)
empirical_aut = np.concatenate((empirical_left_aut, empirical_right_aut))
candidate_auts = [
np.concatenate((l, r))
for l, r in zip(candidate_left_auts, candidate_right_auts)
]
# compute L2 distance from empirical distribution
diff_left = [
np.sum((empirical_left_aut - l)**2) for l in candidate_left_auts
]
diff_right = [
np.sum((empirical_right_aut - r)**2) for r in candidate_right_auts
]
diff_both = [
np.sum((empirical_aut - candidate)**2) for candidate in candidate_auts
]
# calcule ranks
score_left = pd.Series(diff_left).rank(ascending=False)
score_right = pd.Series(diff_right).rank(ascending=False)
score_both = pd.Series(diff_both).rank(ascending=False)
score = score_left + score_right + score_both
selected_copula = np.argmax(score.values)
return copula_candidates[selected_copula]