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def test_orthogonal_target(self):
"""
Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa
"""
A = self.str2matrix("""
.830 -.396
.818 -.469
.777 -.470
.798 -.401
.786 .500
.672 .458
.594 .444
.647 .333
""")
H = self.str2matrix("""
.8 -.3
.8 -.4
.7 -.4
.9 -.4
.8 .5
.6 .4
.5 .4
.6 .3
""")
def vgQ(L=None, A=None, T=None):
return vgQ_target(H, L=L, A=A, T=T)
L, phi, T, table = GPA(A, vgQ=vgQ, rotation_method='orthogonal')
T_analytic = target_rotation(A, H)
self.assertTrue(np.allclose(T, T_analytic, atol=1e-05)) | Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa | test_orthogonal_target | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def get_quartimin_example(cls):
A = cls.get_A()
table_required = cls.str2matrix("""
0.00000 0.42806 -0.46393 1.00000
1.00000 0.41311 -0.57313 0.25000
2.00000 0.38238 -0.36652 0.50000
3.00000 0.31850 -0.21011 0.50000
4.00000 0.20937 -0.13838 0.50000
5.00000 0.12379 -0.35583 0.25000
6.00000 0.04289 -0.53244 0.50000
7.00000 0.01098 -0.86649 0.50000
8.00000 0.00566 -1.65798 0.50000
9.00000 0.00558 -2.13212 0.25000
10.00000 0.00557 -2.49020 0.25000
11.00000 0.00557 -2.84585 0.25000
12.00000 0.00557 -3.20320 0.25000
13.00000 0.00557 -3.56143 0.25000
14.00000 0.00557 -3.92005 0.25000
15.00000 0.00557 -4.27885 0.25000
16.00000 0.00557 -4.63772 0.25000
17.00000 0.00557 -4.99663 0.25000
18.00000 0.00557 -5.35555 0.25000
""")
L_required = cls.str2matrix("""
0.891822 0.056015
0.953680 -0.023246
0.929150 -0.046503
0.876683 0.033658
0.013701 0.925000
-0.017265 0.821253
-0.052445 0.764953
0.085890 0.683115
""")
return A, table_required, L_required | )
L_required = cls.str2matrix( | get_quartimin_example | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def get_biquartimin_example(cls):
A = cls.get_A()
table_required = cls.str2matrix("""
0.00000 0.21632 -0.54955 1.00000
1.00000 0.19519 -0.46174 0.50000
2.00000 0.09479 -0.16365 1.00000
3.00000 -0.06302 -0.32096 0.50000
4.00000 -0.21304 -0.46562 1.00000
5.00000 -0.33199 -0.33287 1.00000
6.00000 -0.35108 -0.63990 0.12500
7.00000 -0.35543 -1.20916 0.12500
8.00000 -0.35568 -2.61213 0.12500
9.00000 -0.35568 -2.97910 0.06250
10.00000 -0.35568 -3.32645 0.06250
11.00000 -0.35568 -3.66021 0.06250
12.00000 -0.35568 -3.98564 0.06250
13.00000 -0.35568 -4.30635 0.06250
14.00000 -0.35568 -4.62451 0.06250
15.00000 -0.35568 -4.94133 0.06250
16.00000 -0.35568 -5.25745 0.06250
""")
L_required = cls.str2matrix("""
1.01753 -0.13657
1.11338 -0.24643
1.09200 -0.26890
1.00676 -0.16010
-0.26534 1.11371
-0.26972 0.99553
-0.29341 0.93561
-0.10806 0.80513
""")
return A, table_required, L_required | )
L_required = cls.str2matrix( | get_biquartimin_example | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def get_biquartimin_example_derivative_free(cls):
A = cls.get_A()
table_required = cls.str2matrix("""
0.00000 0.21632 -0.54955 1.00000
1.00000 0.19519 -0.46174 0.50000
2.00000 0.09479 -0.16365 1.00000
3.00000 -0.06302 -0.32096 0.50000
4.00000 -0.21304 -0.46562 1.00000
5.00000 -0.33199 -0.33287 1.00000
6.00000 -0.35108 -0.63990 0.12500
7.00000 -0.35543 -1.20916 0.12500
8.00000 -0.35568 -2.61213 0.12500
9.00000 -0.35568 -2.97910 0.06250
10.00000 -0.35568 -3.32645 0.06250
11.00000 -0.35568 -3.66021 0.06250
12.00000 -0.35568 -3.98564 0.06250
13.00000 -0.35568 -4.30634 0.06250
14.00000 -0.35568 -4.62451 0.06250
15.00000 -0.35568 -4.94133 0.06250
16.00000 -0.35568 -6.32435 0.12500
""")
L_required = cls.str2matrix("""
1.01753 -0.13657
1.11338 -0.24643
1.09200 -0.26890
1.00676 -0.16010
-0.26534 1.11371
-0.26972 0.99553
-0.29342 0.93561
-0.10806 0.80513
""")
return A, table_required, L_required | )
L_required = cls.str2matrix( | get_biquartimin_example_derivative_free | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def get_quartimax_example_derivative_free(cls):
A = cls.get_A()
table_required = cls.str2matrix("""
0.00000 -0.72073 -0.65498 1.00000
1.00000 -0.88561 -0.34614 2.00000
2.00000 -1.01992 -1.07152 1.00000
3.00000 -1.02237 -1.51373 0.50000
4.00000 -1.02269 -1.96205 0.50000
5.00000 -1.02273 -2.41116 0.50000
6.00000 -1.02273 -2.86037 0.50000
7.00000 -1.02273 -3.30959 0.50000
8.00000 -1.02273 -3.75881 0.50000
9.00000 -1.02273 -4.20804 0.50000
10.00000 -1.02273 -4.65726 0.50000
11.00000 -1.02273 -5.10648 0.50000
""")
L_required = cls.str2matrix("""
0.89876 0.19482
0.93394 0.12974
0.90213 0.10386
0.87651 0.17128
0.31558 0.87647
0.25113 0.77349
0.19801 0.71468
0.30786 0.65933
""")
return A, table_required, L_required | )
L_required = cls.str2matrix( | get_quartimax_example_derivative_free | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def test_orthomax(self):
"""
Quartimax example
http://www.stat.ucla.edu/research/gpa
"""
A = self.get_A()
def vgQ(L=None, A=None, T=None):
return orthomax_objective(L=L, A=A, T=T, gamma=0, return_gradient=True)
L, phi, T, table = GPA(A, vgQ=vgQ, rotation_method='orthogonal')
table_required = self.str2matrix("""
0.00000 -0.72073 -0.65498 1.00000
1.00000 -0.88561 -0.34614 2.00000
2.00000 -1.01992 -1.07152 1.00000
3.00000 -1.02237 -1.51373 0.50000
4.00000 -1.02269 -1.96205 0.50000
5.00000 -1.02273 -2.41116 0.50000
6.00000 -1.02273 -2.86037 0.50000
7.00000 -1.02273 -3.30959 0.50000
8.00000 -1.02273 -3.75881 0.50000
9.00000 -1.02273 -4.20804 0.50000
10.00000 -1.02273 -4.65726 0.50000
11.00000 -1.02273 -5.10648 0.50000
""")
L_required = self.str2matrix("""
0.89876 0.19482
0.93394 0.12974
0.90213 0.10386
0.87651 0.17128
0.31558 0.87647
0.25113 0.77349
0.19801 0.71468
0.30786 0.65933
""")
self.assertTrue(np.allclose(table, table_required, atol=1e-05))
self.assertTrue(np.allclose(L, L_required, atol=1e-05))
# oblimin criterion gives same result
def vgQ(L=None, A=None, T=None):
return oblimin_objective(L=L, A=A, T=T, gamma=0, rotation_method='orthogonal', return_gradient=True)
L_oblimin, phi2, T2, table2 = GPA(A, vgQ=vgQ,
rotation_method='orthogonal')
self.assertTrue(np.allclose(L, L_oblimin, atol=1e-05))
# derivative free quartimax
out = self.get_quartimax_example_derivative_free()
A, table_required, L_required = out
def ff(L=None, A=None, T=None):
return orthomax_objective(L=L, A=A, T=T, gamma=0, return_gradient=False)
L, phi, T, table = GPA(A, ff=ff, rotation_method='orthogonal')
self.assertTrue(np.allclose(table, table_required, atol=1e-05))
self.assertTrue(np.allclose(L, L_required, atol=1e-05)) | Quartimax example
http://www.stat.ucla.edu/research/gpa | test_orthomax | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def test_equivalence_orthomax_oblimin(self):
"""
These criteria should be equivalent when restricted to orthogonal
rotation.
See Hartman 1976 page 299.
"""
A = self.get_A()
gamma = 0 # quartimax
def vgQ(L=None, A=None, T=None):
return orthomax_objective(L=L, A=A, T=T, gamma=gamma, return_gradient=True)
L_orthomax, phi, T, table = GPA(
A, vgQ=vgQ, rotation_method='orthogonal')
def vgQ(L=None, A=None, T=None):
return oblimin_objective(L=L, A=A, T=T, gamma=gamma, rotation_method='orthogonal', return_gradient=True)
L_oblimin, phi2, T2, table2 = GPA(A, vgQ=vgQ,
rotation_method='orthogonal')
self.assertTrue(np.allclose(L_orthomax, L_oblimin, atol=1e-05))
gamma = 1 # varimax
def vgQ(L=None, A=None, T=None):
return orthomax_objective(L=L, A=A, T=T, gamma=gamma, return_gradient=True)
L_orthomax, phi, T, table = GPA(
A, vgQ=vgQ, rotation_method='orthogonal')
def vgQ(L=None, A=None, T=None):
return oblimin_objective(L=L, A=A, T=T, gamma=gamma, rotation_method='orthogonal', return_gradient=True)
L_oblimin, phi2, T2, table2 = GPA(
A, vgQ=vgQ, rotation_method='orthogonal')
self.assertTrue(np.allclose(L_orthomax, L_oblimin, atol=1e-05)) | These criteria should be equivalent when restricted to orthogonal
rotation.
See Hartman 1976 page 299. | test_equivalence_orthomax_oblimin | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def test_orthogonal_target(self):
"""
Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa
"""
A = self.get_A()
H = self.str2matrix("""
.8 -.3
.8 -.4
.7 -.4
.9 -.4
.8 .5
.6 .4
.5 .4
.6 .3
""")
def vgQ(L=None, A=None, T=None):
return vgQ_target(H, L=L, A=A, T=T)
L, phi, T, table = GPA(A, vgQ=vgQ, rotation_method='orthogonal')
table_required = self.str2matrix("""
0.00000 0.05925 -0.61244 1.00000
1.00000 0.05444 -1.14701 0.12500
2.00000 0.05403 -1.68194 0.12500
3.00000 0.05399 -2.21689 0.12500
4.00000 0.05399 -2.75185 0.12500
5.00000 0.05399 -3.28681 0.12500
6.00000 0.05399 -3.82176 0.12500
7.00000 0.05399 -4.35672 0.12500
8.00000 0.05399 -4.89168 0.12500
9.00000 0.05399 -5.42664 0.12500
""")
L_required = self.str2matrix("""
0.84168 -0.37053
0.83191 -0.44386
0.79096 -0.44611
0.80985 -0.37650
0.77040 0.52371
0.65774 0.47826
0.58020 0.46189
0.63656 0.35255
""")
self.assertTrue(np.allclose(table, table_required, atol=1e-05))
self.assertTrue(np.allclose(L, L_required, atol=1e-05))
def ff(L=None, A=None, T=None):
return ff_target(H, L=L, A=A, T=T)
L2, phi, T2, table = GPA(A, ff=ff, rotation_method='orthogonal')
self.assertTrue(np.allclose(L, L2, atol=1e-05))
self.assertTrue(np.allclose(T, T2, atol=1e-05))
def vgQ(L=None, A=None, T=None):
return vgQ_target(H, L=L, A=A, T=T, rotation_method='oblique')
L, phi, T, table = GPA(A, vgQ=vgQ, rotation_method='oblique')
def ff(L=None, A=None, T=None):
return ff_target(H, L=L, A=A, T=T, rotation_method='oblique')
L2, phi, T2, table = GPA(A, ff=ff, rotation_method='oblique')
self.assertTrue(np.allclose(L, L2, atol=1e-05))
self.assertTrue(np.allclose(T, T2, atol=1e-05)) | Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa | test_orthogonal_target | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def test_orthogonal_partial_target(self):
"""
Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa
"""
A = self.get_A()
H = self.str2matrix("""
.8 -.3
.8 -.4
.7 -.4
.9 -.4
.8 .5
.6 .4
.5 .4
.6 .3
""")
W = self.str2matrix("""
1 0
0 1
0 0
1 1
1 0
1 0
0 1
1 0
""")
def vgQ(L=None, A=None, T=None):
return vgQ_partial_target(H, W, L=L, A=A, T=T)
L, phi, T, table = GPA(A, vgQ=vgQ, rotation_method='orthogonal')
table_required = self.str2matrix("""
0.00000 0.02559 -0.84194 1.00000
1.00000 0.02203 -1.27116 0.25000
2.00000 0.02154 -1.71198 0.25000
3.00000 0.02148 -2.15713 0.25000
4.00000 0.02147 -2.60385 0.25000
5.00000 0.02147 -3.05114 0.25000
6.00000 0.02147 -3.49863 0.25000
7.00000 0.02147 -3.94619 0.25000
8.00000 0.02147 -4.39377 0.25000
9.00000 0.02147 -4.84137 0.25000
10.00000 0.02147 -5.28897 0.25000
""")
L_required = self.str2matrix("""
0.84526 -0.36228
0.83621 -0.43571
0.79528 -0.43836
0.81349 -0.36857
0.76525 0.53122
0.65303 0.48467
0.57565 0.46754
0.63308 0.35876
""")
self.assertTrue(np.allclose(table, table_required, atol=1e-05))
self.assertTrue(np.allclose(L, L_required, atol=1e-05))
def ff(L=None, A=None, T=None):
return ff_partial_target(H, W, L=L, A=A, T=T)
L2, phi, T2, table = GPA(A, ff=ff, rotation_method='orthogonal')
self.assertTrue(np.allclose(L, L2, atol=1e-05))
self.assertTrue(np.allclose(T, T2, atol=1e-05)) | Rotation towards target matrix example
http://www.stat.ucla.edu/research/gpa | test_orthogonal_partial_target | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def test_methods(self):
"""
Quartimax derivative free example
http://www.stat.ucla.edu/research/gpa
"""
# orthomax, oblimin and CF are tested indirectly
methods = ['quartimin', 'biquartimin',
'quartimax', 'biquartimax', 'varimax', 'equamax',
'parsimax', 'parsimony',
'target', 'partial_target']
for method in methods:
method_args = []
if method == 'target':
method_args = [self.get_H(), 'orthogonal']
self._test_template(method, *method_args)
method_args = [self.get_H(), 'oblique']
self._test_template(method, *method_args)
method_args = [self.get_H(), 'orthogonal']
self._test_template(method, *method_args,
algorithm2='analytic')
elif method == 'partial_target':
method_args = [self.get_H(), self.get_W()]
self._test_template(method, *method_args) | Quartimax derivative free example
http://www.stat.ucla.edu/research/gpa | test_methods | python | statsmodels/statsmodels | statsmodels/multivariate/factor_rotation/tests/test_rotation.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/multivariate/factor_rotation/tests/test_rotation.py | BSD-3-Clause |
def _faa_di_bruno_partitions(n):
"""
Return all non-negative integer solutions of the diophantine equation
n*k_n + ... + 2*k_2 + 1*k_1 = n (1)
Parameters
----------
n : int
the r.h.s. of Eq. (1)
Returns
-------
partitions : list
Each solution is itself a list of the form `[(m, k_m), ...]`
for non-zero `k_m`. Notice that the index `m` is 1-based.
Examples:
---------
>>> _faa_di_bruno_partitions(2)
[[(1, 2)], [(2, 1)]]
>>> for p in _faa_di_bruno_partitions(4):
... assert 4 == sum(m * k for (m, k) in p)
"""
if n < 1:
raise ValueError("Expected a positive integer; got %s instead" % n)
try:
return _faa_di_bruno_cache[n]
except KeyError:
# TODO: higher order terms
# solve Eq. (31) from Blinninkov & Moessner here
raise NotImplementedError('Higher order terms not yet implemented.') | Return all non-negative integer solutions of the diophantine equation
n*k_n + ... + 2*k_2 + 1*k_1 = n (1)
Parameters
----------
n : int
the r.h.s. of Eq. (1)
Returns
-------
partitions : list
Each solution is itself a list of the form `[(m, k_m), ...]`
for non-zero `k_m`. Notice that the index `m` is 1-based.
Examples:
---------
>>> _faa_di_bruno_partitions(2)
[[(1, 2)], [(2, 1)]]
>>> for p in _faa_di_bruno_partitions(4):
... assert 4 == sum(m * k for (m, k) in p) | _faa_di_bruno_partitions | python | statsmodels/statsmodels | statsmodels/distributions/edgeworth.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/edgeworth.py | BSD-3-Clause |
def cumulant_from_moments(momt, n):
"""Compute n-th cumulant given moments.
Parameters
----------
momt : array_like
`momt[j]` contains `(j+1)`-th moment.
These can be raw moments around zero, or central moments
(in which case, `momt[0]` == 0).
n : int
which cumulant to calculate (must be >1)
Returns
-------
kappa : float
n-th cumulant.
"""
if n < 1:
raise ValueError("Expected a positive integer. Got %s instead." % n)
if len(momt) < n:
raise ValueError("%s-th cumulant requires %s moments, "
"only got %s." % (n, n, len(momt)))
kappa = 0.
for p in _faa_di_bruno_partitions(n):
r = sum(k for (m, k) in p)
term = (-1)**(r - 1) * factorial(r - 1)
for (m, k) in p:
term *= np.power(momt[m - 1] / factorial(m), k) / factorial(k)
kappa += term
kappa *= factorial(n)
return kappa | Compute n-th cumulant given moments.
Parameters
----------
momt : array_like
`momt[j]` contains `(j+1)`-th moment.
These can be raw moments around zero, or central moments
(in which case, `momt[0]` == 0).
n : int
which cumulant to calculate (must be >1)
Returns
-------
kappa : float
n-th cumulant. | cumulant_from_moments | python | statsmodels/statsmodels | statsmodels/distributions/edgeworth.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/edgeworth.py | BSD-3-Clause |
def prob2cdf_grid(probs):
"""Cumulative probabilities from cell provabilites on a grid
Parameters
----------
probs : array_like
Rectangular grid of cell probabilities.
Returns
-------
cdf : ndarray
Grid of cumulative probabilities with same shape as probs.
"""
cdf = np.asarray(probs).copy()
k = cdf.ndim
for i in range(k):
cdf = cdf.cumsum(axis=i)
return cdf | Cumulative probabilities from cell provabilites on a grid
Parameters
----------
probs : array_like
Rectangular grid of cell probabilities.
Returns
-------
cdf : ndarray
Grid of cumulative probabilities with same shape as probs. | prob2cdf_grid | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def cdf2prob_grid(cdf, prepend=0):
"""Cell probabilities from cumulative probabilities on a grid.
Parameters
----------
cdf : array_like
Grid of cumulative probabilities with same shape as probs.
Returns
-------
probs : ndarray
Rectangular grid of cell probabilities.
"""
if prepend is None:
prepend = np._NoValue
prob = np.asarray(cdf).copy()
k = prob.ndim
for i in range(k):
prob = np.diff(prob, prepend=prepend, axis=i)
return prob | Cell probabilities from cumulative probabilities on a grid.
Parameters
----------
cdf : array_like
Grid of cumulative probabilities with same shape as probs.
Returns
-------
probs : ndarray
Rectangular grid of cell probabilities. | cdf2prob_grid | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def average_grid(values, coords=None, _method="slicing"):
"""Compute average for each cell in grid using endpoints
Parameters
----------
values : array_like
Values on a grid that will average over corner points of each cell.
coords : None or list of array_like
Grid coordinates for each axis use to compute volumne of cell.
If None, then averaged values are not rescaled.
_method : {"slicing", "convolve"}
Grid averaging is implemented using numpy "slicing" or using
scipy.signal "convolve".
Returns
-------
Grid with averaged cell values.
"""
k_dim = values.ndim
if _method == "slicing":
p = values.copy()
for d in range(k_dim):
# average (p[:-1] + p[1:]) / 2 over each axis
sl1 = [slice(None, None, None)] * k_dim
sl2 = [slice(None, None, None)] * k_dim
sl1[d] = slice(None, -1, None)
sl2[d] = slice(1, None, None)
sl1 = tuple(sl1)
sl2 = tuple(sl2)
p = (p[sl1] + p[sl2]) / 2
elif _method == "convolve":
from scipy import signal
p = signal.convolve(values, 0.5**k_dim * np.ones([2] * k_dim),
mode="valid")
if coords is not None:
dx = np.array(1)
for d in range(k_dim):
dx = dx[..., None] * np.diff(coords[d])
p = p * dx
return p | Compute average for each cell in grid using endpoints
Parameters
----------
values : array_like
Values on a grid that will average over corner points of each cell.
coords : None or list of array_like
Grid coordinates for each axis use to compute volumne of cell.
If None, then averaged values are not rescaled.
_method : {"slicing", "convolve"}
Grid averaging is implemented using numpy "slicing" or using
scipy.signal "convolve".
Returns
-------
Grid with averaged cell values. | average_grid | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def nearest_matrix_margins(mat, maxiter=100, tol=1e-8):
"""nearest matrix with uniform margins
Parameters
----------
mat : array_like, 2-D
Matrix that will be converted to have uniform margins.
Currently, `mat` has to be two dimensional.
maxiter : in
Maximum number of iterations.
tol : float
Tolerance for convergence, defined for difference between largest and
smallest margin in each dimension.
Returns
-------
ndarray, nearest matrix with uniform margins.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change.
changed in 0.14 to support k_dim > 2.
"""
pc = np.asarray(mat)
converged = False
for _ in range(maxiter):
pc0 = pc.copy()
for ax in range(pc.ndim):
axs = tuple([i for i in range(pc.ndim) if not i == ax])
pc0 /= pc.sum(axis=axs, keepdims=True)
pc = pc0
pc /= pc.sum()
# check convergence
mptps = []
for ax in range(pc.ndim):
axs = tuple([i for i in range(pc.ndim) if not i == ax])
marg = pc.sum(axis=axs, keepdims=False)
mptps.append(np.ptp(marg))
if max(mptps) < tol:
converged = True
break
if not converged:
from statsmodels.tools.sm_exceptions import ConvergenceWarning
warnings.warn("Iterations did not converge, maxiter reached",
ConvergenceWarning)
return pc | nearest matrix with uniform margins
Parameters
----------
mat : array_like, 2-D
Matrix that will be converted to have uniform margins.
Currently, `mat` has to be two dimensional.
maxiter : in
Maximum number of iterations.
tol : float
Tolerance for convergence, defined for difference between largest and
smallest margin in each dimension.
Returns
-------
ndarray, nearest matrix with uniform margins.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change.
changed in 0.14 to support k_dim > 2. | nearest_matrix_margins | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def _rankdata_no_ties(x):
"""rankdata without ties for 2-d array
This is a simplified version for ranking data if there are no ties.
Works vectorized across columns.
See Also
--------
scipy.stats.rankdata
"""
nobs, k_vars = x.shape
ranks = np.ones((nobs, k_vars))
sidx = np.argsort(x, axis=0)
ranks[sidx, np.arange(k_vars)] = np.arange(1, nobs + 1)[:, None]
return ranks | rankdata without ties for 2-d array
This is a simplified version for ranking data if there are no ties.
Works vectorized across columns.
See Also
--------
scipy.stats.rankdata | _rankdata_no_ties | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def frequencies_fromdata(data, k_bins, use_ranks=True):
"""count of observations in bins (histogram)
currently only for bivariate data
Parameters
----------
data : array_like
Bivariate data with observations in rows and two columns. Binning is
in unit rectangle [0, 1]^2. If use_rank is False, then data should be
in unit interval.
k_bins : int
Number of bins along each dimension in the histogram
use_ranks : bool
If use_rank is True, then data will be converted to ranks without
tie handling.
Returns
-------
bin counts : ndarray
Frequencies are the number of observations in a given bin.
Bin counts are a 2-dim array with k_bins rows and k_bins columns.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change.
"""
data = np.asarray(data)
k_dim = data.shape[-1]
k = k_bins + 1
g2 = _Grid([k] * k_dim, eps=0)
if use_ranks:
data = _rankdata_no_ties(data) / (data.shape[0] + 1)
# alternatives: scipy handles ties, but uses np.apply_along_axis
# rvs = stats.rankdata(rvs, axis=0) / (rvs.shape[0] + 1)
# rvs = (np.argsort(np.argsort(rvs, axis=0), axis=0) + 1
# ) / (rvs.shape[0] + 1)
freqr, _ = np.histogramdd(data, bins=g2.x_marginal)
return freqr | count of observations in bins (histogram)
currently only for bivariate data
Parameters
----------
data : array_like
Bivariate data with observations in rows and two columns. Binning is
in unit rectangle [0, 1]^2. If use_rank is False, then data should be
in unit interval.
k_bins : int
Number of bins along each dimension in the histogram
use_ranks : bool
If use_rank is True, then data will be converted to ranks without
tie handling.
Returns
-------
bin counts : ndarray
Frequencies are the number of observations in a given bin.
Bin counts are a 2-dim array with k_bins rows and k_bins columns.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change. | frequencies_fromdata | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def approx_copula_pdf(copula, k_bins=10, force_uniform=True, use_pdf=False):
"""Histogram probabilities as approximation to a copula density.
Parameters
----------
copula : instance
Instance of a copula class. Only the ``pdf`` method is used.
k_bins : int
Number of bins along each dimension in the approximating histogram.
force_uniform : bool
If true, then the pdf grid will be adjusted to have uniform margins
using `nearest_matrix_margin`.
If false, then no adjustment is done and the margins may not be exactly
uniform.
use_pdf : bool
If false, then the grid cell probabilities will be computed from the
copula cdf.
If true, then the density, ``pdf``, is used and cell probabilities
are approximated by averaging the pdf of the cell corners. This is
only useful if the cdf is not available.
Returns
-------
bin probabilites : ndarray
Probability that random variable falls in given bin. This corresponds
to a discrete distribution, and is not scaled to bin size to form a
piecewise uniform, histogram density.
Bin probabilities are a k-dim array with k_bins segments in each
dimensionrows.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change.
"""
k_dim = copula.k_dim
k = k_bins + 1
ks = tuple([k] * k_dim)
if use_pdf:
g = _Grid([k] * k_dim, eps=0.1 / k_bins)
pdfg = copula.pdf(g.x_flat).reshape(*ks)
# correct for bin size
pdfg *= 1 / k**k_dim
ag = average_grid(pdfg)
if force_uniform:
pdf_grid = nearest_matrix_margins(ag, maxiter=100, tol=1e-8)
else:
pdf_grid = ag / ag.sum()
else:
g = _Grid([k] * k_dim, eps=1e-6)
cdfg = copula.cdf(g.x_flat).reshape(*ks)
# correct for bin size
pdf_grid = cdf2prob_grid(cdfg, prepend=None)
# TODO: check boundary approximation, eg. undefined at zero
# for now just normalize
pdf_grid /= pdf_grid.sum()
return pdf_grid | Histogram probabilities as approximation to a copula density.
Parameters
----------
copula : instance
Instance of a copula class. Only the ``pdf`` method is used.
k_bins : int
Number of bins along each dimension in the approximating histogram.
force_uniform : bool
If true, then the pdf grid will be adjusted to have uniform margins
using `nearest_matrix_margin`.
If false, then no adjustment is done and the margins may not be exactly
uniform.
use_pdf : bool
If false, then the grid cell probabilities will be computed from the
copula cdf.
If true, then the density, ``pdf``, is used and cell probabilities
are approximated by averaging the pdf of the cell corners. This is
only useful if the cdf is not available.
Returns
-------
bin probabilites : ndarray
Probability that random variable falls in given bin. This corresponds
to a discrete distribution, and is not scaled to bin size to form a
piecewise uniform, histogram density.
Bin probabilities are a k-dim array with k_bins segments in each
dimensionrows.
Notes
-----
This function is intended for internal use and will be generalized in
future. API will change. | approx_copula_pdf | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def _eval_bernstein_1d(x, fvals, method="binom"):
"""Evaluate 1-dimensional bernstein polynomial given grid of values.
experimental, comparing methods
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
method: "binom", "beta" or "bpoly"
Method to construct Bernstein polynomial basis, used for comparison
of parameterizations.
- "binom" uses pmf of Binomial distribution
- "beta" uses pdf of Beta distribution
- "bpoly" uses one interval in scipy.interpolate.BPoly
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis.
"""
k_terms = fvals.shape[-1]
xx = np.asarray(x)
k = np.arange(k_terms).astype(float)
n = k_terms - 1.
if method.lower() == "binom":
# Divide by 0 RuntimeWarning here
with warnings.catch_warnings():
warnings.simplefilter("ignore", RuntimeWarning)
poly_base = stats.binom.pmf(k, n, xx[..., None])
bp_values = (fvals * poly_base).sum(-1)
elif method.lower() == "bpoly":
bpb = interpolate.BPoly(fvals[:, None], [0., 1])
bp_values = bpb(x)
elif method.lower() == "beta":
# Divide by 0 RuntimeWarning here
with warnings.catch_warnings():
warnings.simplefilter("ignore", RuntimeWarning)
poly_base = stats.beta.pdf(xx[..., None], k + 1, n - k + 1) / (n + 1)
bp_values = (fvals * poly_base).sum(-1)
else:
raise ValueError("method not recogized")
return bp_values | Evaluate 1-dimensional bernstein polynomial given grid of values.
experimental, comparing methods
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
method: "binom", "beta" or "bpoly"
Method to construct Bernstein polynomial basis, used for comparison
of parameterizations.
- "binom" uses pmf of Binomial distribution
- "beta" uses pdf of Beta distribution
- "bpoly" uses one interval in scipy.interpolate.BPoly
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis. | _eval_bernstein_1d | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def _eval_bernstein_2d(x, fvals):
"""Evaluate 2-dimensional bernstein polynomial given grid of values
experimental
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis.
"""
k_terms = fvals.shape
k_dim = fvals.ndim
if k_dim != 2:
raise ValueError("`fval` needs to be 2-dimensional")
xx = np.atleast_2d(x)
if xx.shape[1] != 2:
raise ValueError("x needs to be bivariate and have 2 columns")
x1, x2 = xx.T
n1, n2 = k_terms[0] - 1, k_terms[1] - 1
k1 = np.arange(k_terms[0]).astype(float)
k2 = np.arange(k_terms[1]).astype(float)
# we are building a nobs x n1 x n2 array
poly_base = (stats.binom.pmf(k1[None, :, None], n1, x1[:, None, None]) *
stats.binom.pmf(k2[None, None, :], n2, x2[:, None, None]))
bp_values = (fvals * poly_base).sum(-1).sum(-1)
return bp_values | Evaluate 2-dimensional bernstein polynomial given grid of values
experimental
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis. | _eval_bernstein_2d | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def _eval_bernstein_dd(x, fvals):
"""Evaluate d-dimensional bernstein polynomial given grid of valuesv
experimental
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis.
"""
k_terms = fvals.shape
k_dim = fvals.ndim
xx = np.atleast_2d(x)
# The following loop is a tricky
# we add terms for each x and expand dimension of poly base in each
# iteration using broadcasting
poly_base = np.zeros(x.shape[0])
for i in range(k_dim):
ki = np.arange(k_terms[i]).astype(float)
for _ in range(i+1):
ki = ki[..., None]
ni = k_terms[i] - 1
xi = xx[:, i]
poly_base = poly_base[None, ...] + stats.binom._logpmf(ki, ni, xi)
poly_base = np.exp(poly_base)
bp_values = fvals.T[..., None] * poly_base
for i in range(k_dim):
bp_values = bp_values.sum(0)
return bp_values | Evaluate d-dimensional bernstein polynomial given grid of valuesv
experimental
Parameters
----------
x : array_like
Values at which to evaluate the Bernstein polynomial.
fvals : ndarray
Grid values of coefficients for Bernstein polynomial basis in the
weighted sum.
Returns
-------
Bernstein polynomial at evaluation points, weighted sum of Bernstein
polynomial basis. | _eval_bernstein_dd | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def _ecdf_mv(data, method="seq", use_ranks=True):
"""
Multivariate empiricial distribution function, empirical copula
Notes
-----
Method "seq" is faster than method "brute", but supports mainly bivariate
case. Speed advantage of "seq" is increasing in number of observations
and decreasing in number of variables.
(see Segers ...)
Warning: This does not handle ties. The ecdf is based on univariate ranks
without ties. The assignment of ranks to ties depends on the sorting
algorithm and the initial ordering of the data.
When the original data is used instead of ranks, then method "brute"
computes the correct ecdf counts even in the case of ties.
"""
x = np.asarray(data)
n = x.shape[0]
if use_ranks:
x = _rankdata_no_ties(x) / n
if method == "brute":
count = [((x <= x[i]).all(1)).sum() for i in range(n)]
count = np.asarray(count)
elif method.startswith("seq"):
sort_idx0 = np.argsort(x[:, 0])
x_s0 = x[sort_idx0]
x1 = x_s0[:, 1:]
count_smaller = [(x1[:i] <= x1[i]).all(1).sum() + 1 for i in range(n)]
count = np.empty(x.shape[0])
count[sort_idx0] = count_smaller
else:
raise ValueError("method not available")
return count, x | Multivariate empiricial distribution function, empirical copula
Notes
-----
Method "seq" is faster than method "brute", but supports mainly bivariate
case. Speed advantage of "seq" is increasing in number of observations
and decreasing in number of variables.
(see Segers ...)
Warning: This does not handle ties. The ecdf is based on univariate ranks
without ties. The assignment of ranks to ties depends on the sorting
algorithm and the initial ordering of the data.
When the original data is used instead of ranks, then method "brute"
computes the correct ecdf counts even in the case of ties. | _ecdf_mv | python | statsmodels/statsmodels | statsmodels/distributions/tools.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/tools.py | BSD-3-Clause |
def get_distr(self, params):
"""frozen distribution instance of the discrete distribution.
"""
args = params
distr = self.distr(*args)
return distr | frozen distribution instance of the discrete distribution. | get_distr | python | statsmodels/statsmodels | statsmodels/distributions/discrete.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/discrete.py | BSD-3-Clause |
def from_data(cls, data, k_bins):
"""Create distribution instance from data using histogram binning.
Classmethod to construct a distribution instance.
Parameters
----------
data : array_like
Data with observation in rows and random variables in columns.
Data can be 1-dimensional in the univariate case.
k_bins : int or list
Number or edges of bins to be used in numpy histogramdd.
If k_bins is a scalar int, then the number of bins of each
component will be equal to it.
Returns
-------
Instance of a Bernstein distribution
"""
data = np.asarray(data)
if np.any(data < 0) or np.any(data > 1):
raise ValueError("data needs to be in [0, 1]")
if data.ndim == 1:
data = data[:, None]
k_dim = data.shape[1]
if np.size(k_bins) == 1:
k_bins = [k_bins] * k_dim
bins = [np.linspace(-1 / ni, 1, ni + 2) for ni in k_bins]
c, e = np.histogramdd(data, bins=bins, density=False)
# TODO: check when we have zero observations, which bin?
# check bins start at 0 exept leading bin
assert all([ei[1] == 0 for ei in e])
c /= len(data)
cdf_grid = prob2cdf_grid(c)
return cls(cdf_grid) | Create distribution instance from data using histogram binning.
Classmethod to construct a distribution instance.
Parameters
----------
data : array_like
Data with observation in rows and random variables in columns.
Data can be 1-dimensional in the univariate case.
k_bins : int or list
Number or edges of bins to be used in numpy histogramdd.
If k_bins is a scalar int, then the number of bins of each
component will be equal to it.
Returns
-------
Instance of a Bernstein distribution | from_data | python | statsmodels/statsmodels | statsmodels/distributions/bernstein.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/bernstein.py | BSD-3-Clause |
def cdf(self, x):
"""cdf values evaluated at x.
Parameters
----------
x : array_like
Points of multivariate random variable at which cdf is evaluated.
This can be a single point with length equal to the dimension of
the random variable, or two dimensional with points (observations)
in rows and random variables in columns.
In the univariate case, a 1-dimensional x will be interpreted as
different points for evaluation.
Returns
-------
pdf values
Notes
-----
Warning: 2-dim x with many points can be memory intensive because
currently the bernstein polynomials will be evaluated in a fully
vectorized computation.
"""
x = np.asarray(x)
if x.ndim == 1 and self.k_dim == 1:
x = x[:, None]
cdf_ = _eval_bernstein_dd(x, self.cdf_grid)
return cdf_ | cdf values evaluated at x.
Parameters
----------
x : array_like
Points of multivariate random variable at which cdf is evaluated.
This can be a single point with length equal to the dimension of
the random variable, or two dimensional with points (observations)
in rows and random variables in columns.
In the univariate case, a 1-dimensional x will be interpreted as
different points for evaluation.
Returns
-------
pdf values
Notes
-----
Warning: 2-dim x with many points can be memory intensive because
currently the bernstein polynomials will be evaluated in a fully
vectorized computation. | cdf | python | statsmodels/statsmodels | statsmodels/distributions/bernstein.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/bernstein.py | BSD-3-Clause |
def pdf(self, x):
"""pdf values evaluated at x.
Parameters
----------
x : array_like
Points of multivariate random variable at which pdf is evaluated.
This can be a single point with length equal to the dimension of
the random variable, or two dimensional with points (observations)
in rows and random variables in columns.
In the univariate case, a 1-dimensional x will be interpreted as
different points for evaluation.
Returns
-------
cdf values
Notes
-----
Warning: 2-dim x with many points can be memory intensive because
currently the bernstein polynomials will be evaluated in a fully
vectorized computation.
"""
x = np.asarray(x)
if x.ndim == 1 and self.k_dim == 1:
x = x[:, None]
# TODO: check usage of k_grid_product. Should this go into eval?
pdf_ = self.k_grid_product * _eval_bernstein_dd(x, self.prob_grid)
return pdf_ | pdf values evaluated at x.
Parameters
----------
x : array_like
Points of multivariate random variable at which pdf is evaluated.
This can be a single point with length equal to the dimension of
the random variable, or two dimensional with points (observations)
in rows and random variables in columns.
In the univariate case, a 1-dimensional x will be interpreted as
different points for evaluation.
Returns
-------
cdf values
Notes
-----
Warning: 2-dim x with many points can be memory intensive because
currently the bernstein polynomials will be evaluated in a fully
vectorized computation. | pdf | python | statsmodels/statsmodels | statsmodels/distributions/bernstein.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/bernstein.py | BSD-3-Clause |
def get_marginal(self, idx):
"""Get marginal BernsteinDistribution.
Parameters
----------
idx : int or list of int
Index or indices of the component for which the marginal
distribution is returned.
Returns
-------
BernsteinDistribution instance for the marginal distribution.
"""
# univariate
if self.k_dim == 1:
return self
sl = [-1] * self.k_dim
if np.shape(idx) == ():
idx = [idx]
for ii in idx:
sl[ii] = slice(None, None, None)
cdf_m = self.cdf_grid[tuple(sl)]
bpd_marginal = BernsteinDistribution(cdf_m)
return bpd_marginal | Get marginal BernsteinDistribution.
Parameters
----------
idx : int or list of int
Index or indices of the component for which the marginal
distribution is returned.
Returns
-------
BernsteinDistribution instance for the marginal distribution. | get_marginal | python | statsmodels/statsmodels | statsmodels/distributions/bernstein.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/bernstein.py | BSD-3-Clause |
def rvs(self, nobs):
"""Generate random numbers from distribution.
Parameters
----------
nobs : int
Number of random observations to generate.
"""
rvs_mnl = np.random.multinomial(nobs, self.prob_grid.flatten())
k_comp = self.k_dim
rvs_m = []
for i in range(len(rvs_mnl)):
if rvs_mnl[i] != 0:
idx = np.unravel_index(i, self.prob_grid.shape)
rvsi = []
for j in range(k_comp):
n = self.k_grid[j]
xgi = self._grid.x_marginal[j][idx[j]]
# Note: x_marginal starts at 0
# x_marginal ends with 1 but that is not used by idx
rvsi.append(stats.beta.rvs(n * xgi + 1, n * (1-xgi) + 0,
size=rvs_mnl[i]))
rvs_m.append(np.column_stack(rvsi))
rvsm = np.concatenate(rvs_m)
return rvsm | Generate random numbers from distribution.
Parameters
----------
nobs : int
Number of random observations to generate. | rvs | python | statsmodels/statsmodels | statsmodels/distributions/bernstein.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/bernstein.py | BSD-3-Clause |
def monotone_fn_inverter(fn, x, vectorized=True, **keywords):
"""
Given a monotone function fn (no checking is done to verify monotonicity)
and a set of x values, return an linearly interpolated approximation
to its inverse from its values on x.
"""
x = np.asarray(x)
if vectorized:
y = fn(x, **keywords)
else:
y = []
for _x in x:
y.append(fn(_x, **keywords))
y = np.array(y)
a = np.argsort(y)
return interp1d(y[a], x[a]) | Given a monotone function fn (no checking is done to verify monotonicity)
and a set of x values, return an linearly interpolated approximation
to its inverse from its values on x. | monotone_fn_inverter | python | statsmodels/statsmodels | statsmodels/distributions/empirical_distribution.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/empirical_distribution.py | BSD-3-Clause |
def _make_index(prob,size):
"""
Returns a boolean index for given probabilities.
Notes
-----
prob = [.75,.25] means that there is a 75% chance of the first column
being True and a 25% chance of the second column being True. The
columns are mutually exclusive.
"""
rv = np.random.uniform(size=(size,1))
cumprob = np.cumsum(prob)
return np.logical_and(np.r_[0,cumprob[:-1]] <= rv, rv < cumprob) | Returns a boolean index for given probabilities.
Notes
-----
prob = [.75,.25] means that there is a 75% chance of the first column
being True and a 25% chance of the second column being True. The
columns are mutually exclusive. | _make_index | python | statsmodels/statsmodels | statsmodels/distributions/mixture_rvs.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/mixture_rvs.py | BSD-3-Clause |
def mixture_rvs(prob, size, dist, kwargs=None):
"""
Sample from a mixture of distributions.
Parameters
----------
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> from scipy import stats
>>> prob = [.75,.25]
>>> Y = mixture_rvs(prob, 5000, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5)))
"""
if len(prob) != len(dist):
raise ValueError("You must provide as many probabilities as distributions")
if not np.allclose(np.sum(prob), 1):
raise ValueError("prob does not sum to 1")
if kwargs is None:
kwargs = ({},)*len(prob)
idx = _make_index(prob,size)
sample = np.empty(size)
for i in range(len(prob)):
sample_idx = idx[...,i]
sample_size = sample_idx.sum()
loc = kwargs[i].get('loc',0)
scale = kwargs[i].get('scale',1)
args = kwargs[i].get('args',())
sample[sample_idx] = dist[i].rvs(*args, **dict(loc=loc,scale=scale,
size=sample_size))
return sample | Sample from a mixture of distributions.
Parameters
----------
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> from scipy import stats
>>> prob = [.75,.25]
>>> Y = mixture_rvs(prob, 5000, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5))) | mixture_rvs | python | statsmodels/statsmodels | statsmodels/distributions/mixture_rvs.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/mixture_rvs.py | BSD-3-Clause |
def pdf(self, x, prob, dist, kwargs=None):
"""
pdf a mixture of distributions.
Parameters
----------
x : array_like
Array containing locations where the PDF should be evaluated
prob : array_like
Probability of sampling from each distribution in dist
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> import numpy as np
>>> from scipy import stats
>>> from statsmodels.distributions.mixture_rvs import MixtureDistribution
>>> x = np.arange(-4.0, 4.0, 0.01)
>>> prob = [.75,.25]
>>> mixture = MixtureDistribution()
>>> Y = mixture.pdf(x, prob, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5)))
"""
if len(prob) != len(dist):
raise ValueError("You must provide as many probabilities as distributions")
if not np.allclose(np.sum(prob), 1):
raise ValueError("prob does not sum to 1")
if kwargs is None:
kwargs = ({},)*len(prob)
for i in range(len(prob)):
loc = kwargs[i].get('loc',0)
scale = kwargs[i].get('scale',1)
args = kwargs[i].get('args',())
if i == 0: #assume all broadcast the same as the first dist
pdf_ = prob[i] * dist[i].pdf(x, *args, loc=loc, scale=scale)
else:
pdf_ += prob[i] * dist[i].pdf(x, *args, loc=loc, scale=scale)
return pdf_ | pdf a mixture of distributions.
Parameters
----------
x : array_like
Array containing locations where the PDF should be evaluated
prob : array_like
Probability of sampling from each distribution in dist
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> import numpy as np
>>> from scipy import stats
>>> from statsmodels.distributions.mixture_rvs import MixtureDistribution
>>> x = np.arange(-4.0, 4.0, 0.01)
>>> prob = [.75,.25]
>>> mixture = MixtureDistribution()
>>> Y = mixture.pdf(x, prob, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5))) | pdf | python | statsmodels/statsmodels | statsmodels/distributions/mixture_rvs.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/mixture_rvs.py | BSD-3-Clause |
def cdf(self, x, prob, dist, kwargs=None):
"""
cdf of a mixture of distributions.
Parameters
----------
x : array_like
Array containing locations where the CDF should be evaluated
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> import numpy as np
>>> from scipy import stats
>>> from statsmodels.distributions.mixture_rvs import MixtureDistribution
>>> x = np.arange(-4.0, 4.0, 0.01)
>>> prob = [.75,.25]
>>> mixture = MixtureDistribution()
>>> Y = mixture.pdf(x, prob, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5)))
"""
if len(prob) != len(dist):
raise ValueError("You must provide as many probabilities as distributions")
if not np.allclose(np.sum(prob), 1):
raise ValueError("prob does not sum to 1")
if kwargs is None:
kwargs = ({},)*len(prob)
for i in range(len(prob)):
loc = kwargs[i].get('loc',0)
scale = kwargs[i].get('scale',1)
args = kwargs[i].get('args',())
if i == 0: #assume all broadcast the same as the first dist
cdf_ = prob[i] * dist[i].cdf(x, *args, loc=loc, scale=scale)
else:
cdf_ += prob[i] * dist[i].cdf(x, *args, loc=loc, scale=scale)
return cdf_ | cdf of a mixture of distributions.
Parameters
----------
x : array_like
Array containing locations where the CDF should be evaluated
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions objects from scipy.stats.
kwargs : tuple of dicts, optional
A tuple of dicts. Each dict in kwargs can have keys loc, scale, and
args to be passed to the respective distribution in dist. If not
provided, the distribution defaults are used.
Examples
--------
Say we want 5000 random variables from mixture of normals with two
distributions norm(-1,.5) and norm(1,.5) and we want to sample from the
first with probability .75 and the second with probability .25.
>>> import numpy as np
>>> from scipy import stats
>>> from statsmodels.distributions.mixture_rvs import MixtureDistribution
>>> x = np.arange(-4.0, 4.0, 0.01)
>>> prob = [.75,.25]
>>> mixture = MixtureDistribution()
>>> Y = mixture.pdf(x, prob, dist=[stats.norm, stats.norm],
... kwargs = (dict(loc=-1,scale=.5),dict(loc=1,scale=.5))) | cdf | python | statsmodels/statsmodels | statsmodels/distributions/mixture_rvs.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/mixture_rvs.py | BSD-3-Clause |
def mv_mixture_rvs(prob, size, dist, nvars, **kwargs):
"""
Sample from a mixture of multivariate distributions.
Parameters
----------
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions instances with callable method rvs.
nvargs : int
dimension of the multivariate distribution, could be inferred instead
kwargs : tuple of dicts, optional
ignored
Examples
--------
Say we want 2000 random variables from mixture of normals with two
multivariate normal distributions, and we want to sample from the
first with probability .4 and the second with probability .6.
import statsmodels.sandbox.distributions.mv_normal as mvd
cov3 = np.array([[ 1. , 0.5 , 0.75],
[ 0.5 , 1.5 , 0.6 ],
[ 0.75, 0.6 , 2. ]])
mu = np.array([-1, 0.0, 2.0])
mu2 = np.array([4, 2.0, 2.0])
mvn3 = mvd.MVNormal(mu, cov3)
mvn32 = mvd.MVNormal(mu2, cov3/2., 4)
rvs = mix.mv_mixture_rvs([0.4, 0.6], 2000, [mvn3, mvn32], 3)
"""
if len(prob) != len(dist):
raise ValueError("You must provide as many probabilities as distributions")
if not np.allclose(np.sum(prob), 1):
raise ValueError("prob does not sum to 1")
if kwargs is None:
kwargs = ({},)*len(prob)
idx = _make_index(prob,size)
sample = np.empty((size, nvars))
for i in range(len(prob)):
sample_idx = idx[...,i]
sample_size = sample_idx.sum()
#loc = kwargs[i].get('loc',0)
#scale = kwargs[i].get('scale',1)
#args = kwargs[i].get('args',())
# use int to avoid numpy bug with np.random.multivariate_normal
sample[sample_idx] = dist[i].rvs(size=int(sample_size))
return sample | Sample from a mixture of multivariate distributions.
Parameters
----------
prob : array_like
Probability of sampling from each distribution in dist
size : int
The length of the returned sample.
dist : array_like
An iterable of distributions instances with callable method rvs.
nvargs : int
dimension of the multivariate distribution, could be inferred instead
kwargs : tuple of dicts, optional
ignored
Examples
--------
Say we want 2000 random variables from mixture of normals with two
multivariate normal distributions, and we want to sample from the
first with probability .4 and the second with probability .6.
import statsmodels.sandbox.distributions.mv_normal as mvd
cov3 = np.array([[ 1. , 0.5 , 0.75],
[ 0.5 , 1.5 , 0.6 ],
[ 0.75, 0.6 , 2. ]])
mu = np.array([-1, 0.0, 2.0])
mu2 = np.array([4, 2.0, 2.0])
mvn3 = mvd.MVNormal(mu, cov3)
mvn32 = mvd.MVNormal(mu2, cov3/2., 4)
rvs = mix.mv_mixture_rvs([0.4, 0.6], 2000, [mvn3, mvn32], 3) | mv_mixture_rvs | python | statsmodels/statsmodels | statsmodels/distributions/mixture_rvs.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/mixture_rvs.py | BSD-3-Clause |
def deriv(self, t, *args):
"""First derivative of the dependence function
implemented through numerical differentiation
"""
t = np.atleast_1d(t)
return _approx_fprime_cs_scalar(t, self.evaluate) | First derivative of the dependence function
implemented through numerical differentiation | deriv | python | statsmodels/statsmodels | statsmodels/distributions/copula/depfunc_ev.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/depfunc_ev.py | BSD-3-Clause |
def deriv2(self, t, *args):
"""Second derivative of the dependence function
implemented through numerical differentiation
"""
if np.size(t) == 1:
d2 = approx_hess([t], self.evaluate, args=args)[0]
else:
d2 = np.array([approx_hess([ti], self.evaluate, args=args)[0, 0]
for ti in t])
return d2 | Second derivative of the dependence function
implemented through numerical differentiation | deriv2 | python | statsmodels/statsmodels | statsmodels/distributions/copula/depfunc_ev.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/depfunc_ev.py | BSD-3-Clause |
def rvs(self, nobs=1, cop_args=None, marg_args=None, random_state=None):
"""Draw `n` in the half-open interval ``[0, 1)``.
Sample the joint distribution.
Parameters
----------
nobs : int, optional
Number of samples to generate in the parameter space.
Default is 1.
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
Returns
-------
sample : array_like (n, d)
Sample from the joint distribution.
Notes
-----
The random samples are generated by creating a sample with uniform
margins from the copula, and using ``ppf`` to convert uniform margins
to the one specified by the marginal distribution.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state
"""
if cop_args is None:
cop_args = self.cop_args
if marg_args is None:
marg_args = [()] * self.k_vars
sample = self.copula.rvs(nobs=nobs, args=cop_args,
random_state=random_state)
for i, dist in enumerate(self.marginals):
sample[:, i] = dist.ppf(0.5 + (1 - 1e-10) * (sample[:, i] - 0.5),
*marg_args[i])
return sample | Draw `n` in the half-open interval ``[0, 1)``.
Sample the joint distribution.
Parameters
----------
nobs : int, optional
Number of samples to generate in the parameter space.
Default is 1.
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
Returns
-------
sample : array_like (n, d)
Sample from the joint distribution.
Notes
-----
The random samples are generated by creating a sample with uniform
margins from the copula, and using ``ppf`` to convert uniform margins
to the one specified by the marginal distribution.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state | rvs | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def cdf(self, y, cop_args=None, marg_args=None):
"""CDF of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
cdf values
"""
y = np.asarray(y)
if cop_args is None:
cop_args = self.cop_args
if marg_args is None:
marg_args = [()] * y.shape[-1]
cdf_marg = []
for i in range(self.k_vars):
cdf_marg.append(self.marginals[i].cdf(y[..., i], *marg_args[i]))
u = np.column_stack(cdf_marg)
if y.ndim == 1:
u = u.squeeze()
return self.copula.cdf(u, cop_args) | CDF of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
cdf values | cdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def pdf(self, y, cop_args=None, marg_args=None):
"""PDF of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
pdf values
"""
return np.exp(self.logpdf(y, cop_args=cop_args, marg_args=marg_args)) | PDF of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute created when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
pdf values | pdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def logpdf(self, y, cop_args=None, marg_args=None):
"""Log-pdf of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute creating when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
log-pdf values
"""
y = np.asarray(y)
if cop_args is None:
cop_args = self.cop_args
if marg_args is None:
marg_args = tuple([()] * y.shape[-1])
lpdf = 0.0
cdf_marg = []
for i in range(self.k_vars):
lpdf += self.marginals[i].logpdf(y[..., i], *marg_args[i])
cdf_marg.append(self.marginals[i].cdf(y[..., i], *marg_args[i]))
u = np.column_stack(cdf_marg)
if y.ndim == 1:
u = u.squeeze()
lpdf += self.copula.logpdf(u, cop_args)
return lpdf | Log-pdf of copula distribution.
Parameters
----------
y : array_like
Values of random variable at which to evaluate cdf.
If 2-dimensional, then components of multivariate random variable
need to be in columns
cop_args : tuple
Copula parameters. If None, then the copula parameters will be
taken from the ``cop_args`` attribute creating when initiializing
the instance.
marg_args : list of tuples
Parameters for the marginal distributions. It can be None if none
of the marginal distributions have parameters, otherwise it needs
to be a list of tuples with the same length has the number of
marginal distributions. The list can contain empty tuples for
marginal distributions that do not take parameter arguments.
Returns
-------
log-pdf values | logpdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def rvs(self, nobs=1, args=(), random_state=None):
"""Draw `n` in the half-open interval ``[0, 1)``.
Marginals are uniformly distributed.
Parameters
----------
nobs : int, optional
Number of samples to generate from the copula. Default is 1.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
Returns
-------
sample : array_like (nobs, d)
Sample from the copula.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state
"""
raise NotImplementedError | Draw `n` in the half-open interval ``[0, 1)``.
Marginals are uniformly distributed.
Parameters
----------
nobs : int, optional
Number of samples to generate from the copula. Default is 1.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
Returns
-------
sample : array_like (nobs, d)
Sample from the copula.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state | rvs | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def pdf(self, u, args=()):
"""Probability density function of copula.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
pdf : ndarray, (nobs, k_dim)
Copula pdf evaluated at points ``u``.
""" | Probability density function of copula.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
pdf : ndarray, (nobs, k_dim)
Copula pdf evaluated at points ``u``. | pdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def logpdf(self, u, args=()):
"""Log of copula pdf, loglikelihood.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
cdf : ndarray, (nobs, k_dim)
Copula log-pdf evaluated at points ``u``.
"""
return np.log(self.pdf(u, *args)) | Log of copula pdf, loglikelihood.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
cdf : ndarray, (nobs, k_dim)
Copula log-pdf evaluated at points ``u``. | logpdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def cdf(self, u, args=()):
"""Cumulative distribution function evaluated at points u.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
cdf : ndarray, (nobs, k_dim)
Copula cdf evaluated at points ``u``.
""" | Cumulative distribution function evaluated at points u.
Parameters
----------
u : array_like, 2-D
Points of random variables in unit hypercube at which method is
evaluated.
The second (or last) dimension should be the same as the dimension
of the random variable, e.g. 2 for bivariate copula.
args : tuple
Arguments for copula parameters. The number of arguments depends
on the copula.
Returns
-------
cdf : ndarray, (nobs, k_dim)
Copula cdf evaluated at points ``u``. | cdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def plot_scatter(self, sample=None, nobs=500, random_state=None, ax=None):
"""Sample the copula and plot.
Parameters
----------
sample : array-like, optional
The sample to plot. If not provided (the default), a sample
is generated.
nobs : int, optional
Number of samples to generate from the copula.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
ax : AxesSubplot, optional
If given, this subplot is used to plot in instead of a new figure
being created.
Returns
-------
fig : Figure
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
sample : array_like (n, d)
Sample from the copula.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state
"""
if self.k_dim != 2:
raise ValueError("Can only plot 2-dimensional Copula.")
if sample is None:
sample = self.rvs(nobs=nobs, random_state=random_state)
fig, ax = utils.create_mpl_ax(ax)
ax.scatter(sample[:, 0], sample[:, 1])
ax.set_xlabel('u')
ax.set_ylabel('v')
return fig, sample | Sample the copula and plot.
Parameters
----------
sample : array-like, optional
The sample to plot. If not provided (the default), a sample
is generated.
nobs : int, optional
Number of samples to generate from the copula.
random_state : {None, int, numpy.random.Generator}, optional
If `seed` is None then the legacy singleton NumPy generator.
This will change after 0.13 to use a fresh NumPy ``Generator``,
so you should explicitly pass a seeded ``Generator`` if you
need reproducible results.
If `seed` is an int, a new ``Generator`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` instance then that instance is
used.
ax : AxesSubplot, optional
If given, this subplot is used to plot in instead of a new figure
being created.
Returns
-------
fig : Figure
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
sample : array_like (n, d)
Sample from the copula.
See Also
--------
statsmodels.tools.rng_qrng.check_random_state | plot_scatter | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def plot_pdf(self, ticks_nbr=10, ax=None):
"""Plot the PDF.
Parameters
----------
ticks_nbr : int, optional
Number of color isolines for the PDF. Default is 10.
ax : AxesSubplot, optional
If given, this subplot is used to plot in instead of a new figure
being created.
Returns
-------
fig : Figure
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
"""
from matplotlib import pyplot as plt
if self.k_dim != 2:
import warnings
warnings.warn("Plotting 2-dimensional Copula.")
n_samples = 100
eps = 1e-4
uu, vv = np.meshgrid(np.linspace(eps, 1 - eps, n_samples),
np.linspace(eps, 1 - eps, n_samples))
points = np.vstack([uu.ravel(), vv.ravel()]).T
data = self.pdf(points).T.reshape(uu.shape)
min_ = np.nanpercentile(data, 5)
max_ = np.nanpercentile(data, 95)
fig, ax = utils.create_mpl_ax(ax)
vticks = np.linspace(min_, max_, num=ticks_nbr)
range_cbar = [min_, max_]
cs = ax.contourf(uu, vv, data, vticks,
antialiased=True, vmin=range_cbar[0],
vmax=range_cbar[1])
ax.set_xlabel("u")
ax.set_ylabel("v")
ax.set_xlim(0, 1)
ax.set_ylim(0, 1)
ax.set_aspect('equal')
cbar = plt.colorbar(cs, ticks=vticks)
cbar.set_label('p')
fig.tight_layout()
return fig | Plot the PDF.
Parameters
----------
ticks_nbr : int, optional
Number of color isolines for the PDF. Default is 10.
ax : AxesSubplot, optional
If given, this subplot is used to plot in instead of a new figure
being created.
Returns
-------
fig : Figure
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected. | plot_pdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def tau_simulated(self, nobs=1024, random_state=None):
"""Kendall's tau based on simulated samples.
Returns
-------
tau : float
Kendall's tau.
"""
x = self.rvs(nobs, random_state=random_state)
return stats.kendalltau(x[:, 0], x[:, 1])[0] | Kendall's tau based on simulated samples.
Returns
-------
tau : float
Kendall's tau. | tau_simulated | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def fit_corr_param(self, data):
"""Copula correlation parameter using Kendall's tau of sample data.
Parameters
----------
data : array_like
Sample data used to fit `theta` using Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical.
If k_dim > 2, then average tau is used.
"""
x = np.asarray(data)
if x.shape[1] == 2:
tau = stats.kendalltau(x[:, 0], x[:, 1])[0]
else:
k = self.k_dim
taus = [stats.kendalltau(x[..., i], x[..., j])[0]
for i in range(k) for j in range(i+1, k)]
tau = np.mean(taus)
return self._arg_from_tau(tau) | Copula correlation parameter using Kendall's tau of sample data.
Parameters
----------
data : array_like
Sample data used to fit `theta` using Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical.
If k_dim > 2, then average tau is used. | fit_corr_param | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def _arg_from_tau(self, tau):
"""Compute correlation parameter from tau.
Parameters
----------
tau : float
Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical.
"""
raise NotImplementedError | Compute correlation parameter from tau.
Parameters
----------
tau : float
Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical. | _arg_from_tau | python | statsmodels/statsmodels | statsmodels/distributions/copula/copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/copulas.py | BSD-3-Clause |
def tau(self, corr=None):
"""Bivariate kendall's tau based on correlation coefficient.
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Kendall's tau that corresponds to pearson correlation in the
elliptical copula.
"""
if corr is None:
corr = self.corr
if corr.shape == (2, 2):
corr = corr[0, 1]
rho = 2 * np.arcsin(corr) / np.pi
return rho | Bivariate kendall's tau based on correlation coefficient.
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Kendall's tau that corresponds to pearson correlation in the
elliptical copula. | tau | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def corr_from_tau(self, tau):
"""Pearson correlation from kendall's tau.
Parameters
----------
tau : array_like
Kendall's tau correlation coefficient.
Returns
-------
Pearson correlation coefficient for given tau in elliptical
copula. This can be used as parameter for an elliptical copula.
"""
corr = np.sin(tau * np.pi / 2)
return corr | Pearson correlation from kendall's tau.
Parameters
----------
tau : array_like
Kendall's tau correlation coefficient.
Returns
-------
Pearson correlation coefficient for given tau in elliptical
copula. This can be used as parameter for an elliptical copula. | corr_from_tau | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def fit_corr_param(self, data):
"""Copula correlation parameter using Kendall's tau of sample data.
Parameters
----------
data : array_like
Sample data used to fit `theta` using Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical.
If k_dim > 2, then average tau is used.
"""
x = np.asarray(data)
if x.shape[1] == 2:
tau = stats.kendalltau(x[:, 0], x[:, 1])[0]
else:
k = self.k_dim
tau = np.eye(k)
for i in range(k):
for j in range(i+1, k):
tau_ij = stats.kendalltau(x[..., i], x[..., j])[0]
tau[i, j] = tau[j, i] = tau_ij
return self._arg_from_tau(tau) | Copula correlation parameter using Kendall's tau of sample data.
Parameters
----------
data : array_like
Sample data used to fit `theta` using Kendall's tau.
Returns
-------
corr_param : float
Correlation parameter of the copula, ``theta`` in Archimedean and
pearson correlation in elliptical.
If k_dim > 2, then average tau is used. | fit_corr_param | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def dependence_tail(self, corr=None):
"""
Bivariate tail dependence parameter.
Joe (2014) p. 182
Parameters
----------
corr : any
Tail dependence for Gaussian copulas is always zero.
Argument will be ignored
Returns
-------
Lower and upper tail dependence coefficients of the copula with given
Pearson correlation coefficient.
"""
return 0, 0 | Bivariate tail dependence parameter.
Joe (2014) p. 182
Parameters
----------
corr : any
Tail dependence for Gaussian copulas is always zero.
Argument will be ignored
Returns
-------
Lower and upper tail dependence coefficients of the copula with given
Pearson correlation coefficient. | dependence_tail | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def spearmans_rho(self, corr=None):
"""
Bivariate Spearman's rho based on correlation coefficient.
Joe (2014) p. 182
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Spearman's rho that corresponds to pearson correlation in the
elliptical copula.
"""
if corr is None:
corr = self.corr
if corr.shape == (2, 2):
corr = corr[0, 1]
tau = 6 * np.arcsin(corr / 2) / np.pi
return tau | Bivariate Spearman's rho based on correlation coefficient.
Joe (2014) p. 182
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Spearman's rho that corresponds to pearson correlation in the
elliptical copula. | spearmans_rho | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def dependence_tail(self, corr=None):
"""
Bivariate tail dependence parameter.
Joe (2014) p. 182
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Lower and upper tail dependence coefficients of the copula with given
Pearson correlation coefficient.
"""
if corr is None:
corr = self.corr
if corr.shape == (2, 2):
corr = corr[0, 1]
df = self.df
t = - np.sqrt((df + 1) * (1 - corr) / 1 + corr)
# Note self.distr_uv is frozen, df cannot change, use stats.t instead
lam = 2 * stats.t.cdf(t, df + 1)
return lam, lam | Bivariate tail dependence parameter.
Joe (2014) p. 182
Parameters
----------
corr : None or float
Pearson correlation. If corr is None, then the correlation will be
taken from the copula attribute.
Returns
-------
Lower and upper tail dependence coefficients of the copula with given
Pearson correlation coefficient. | dependence_tail | python | statsmodels/statsmodels | statsmodels/distributions/copula/elliptical.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/elliptical.py | BSD-3-Clause |
def _debyem1_expansion(x):
"""Debye function minus 1, Taylor series approximation around zero
function is not used
"""
x = np.asarray(x)
# Expansion derived using Wolfram alpha
dm1 = (-x/4 + x**2/36 - x**4/3600 + x**6/211680 - x**8/10886400 +
x**10/526901760 - x**12 * 691/16999766784000)
return dm1 | Debye function minus 1, Taylor series approximation around zero
function is not used | _debyem1_expansion | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def tau_frank(theta):
"""Kendall's tau for Frank Copula
This uses Taylor series expansion for theta <= 1.
Parameters
----------
theta : float
Parameter of the Frank copula. (not vectorized)
Returns
-------
tau : float, tau for given theta
"""
if theta <= 1:
tau = _tau_frank_expansion(theta)
else:
debye_value = _debye(theta)
tau = 1 + 4 * (debye_value - 1) / theta
return tau | Kendall's tau for Frank Copula
This uses Taylor series expansion for theta <= 1.
Parameters
----------
theta : float
Parameter of the Frank copula. (not vectorized)
Returns
-------
tau : float, tau for given theta | tau_frank | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def cdf(self, u, args=()):
"""Evaluate cdf of Archimedean copula."""
args = self._handle_args(args)
u = self._handle_u(u)
axis = -1
phi = self.transform.evaluate
phi_inv = self.transform.inverse
cdfv = phi_inv(phi(u, *args).sum(axis), *args)
# clip numerical noise
out = cdfv if isinstance(cdfv, np.ndarray) else None
cdfv = np.clip(cdfv, 0., 1., out=out) # inplace if possible
return cdfv | Evaluate cdf of Archimedean copula. | cdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def pdf(self, u, args=()):
"""Evaluate pdf of Archimedean copula."""
u = self._handle_u(u)
args = self._handle_args(args)
axis = -1
phi_d1 = self.transform.deriv
if u.shape[-1] == 2:
psi_d = self.transform.deriv2_inverse
elif u.shape[-1] == 3:
psi_d = self.transform.deriv3_inverse
elif u.shape[-1] == 4:
psi_d = self.transform.deriv4_inverse
else:
# will raise NotImplementedError if not available
k = u.shape[-1]
def psi_d(*args):
return self.transform.derivk_inverse(k, *args)
psi = self.transform.evaluate(u, *args).sum(axis)
pdfv = np.prod(phi_d1(u, *args), axis)
pdfv *= (psi_d(psi, *args))
# use abs, I'm not sure yet about where to add signs
return np.abs(pdfv) | Evaluate pdf of Archimedean copula. | pdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def logpdf(self, u, args=()):
"""Evaluate log pdf of multivariate Archimedean copula."""
u = self._handle_u(u)
args = self._handle_args(args)
axis = -1
phi_d1 = self.transform.deriv
if u.shape[-1] == 2:
psi_d = self.transform.deriv2_inverse
elif u.shape[-1] == 3:
psi_d = self.transform.deriv3_inverse
elif u.shape[-1] == 4:
psi_d = self.transform.deriv4_inverse
else:
# will raise NotImplementedError if not available
k = u.shape[-1]
def psi_d(*args):
return self.transform.derivk_inverse(k, *args)
psi = self.transform.evaluate(u, *args).sum(axis)
# I need np.abs because derivatives are negative,
# is this correct for mv?
logpdfv = np.sum(np.log(np.abs(phi_d1(u, *args))), axis)
logpdfv += np.log(np.abs(psi_d(psi, *args)))
return logpdfv | Evaluate log pdf of multivariate Archimedean copula. | logpdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def cdfcond_2g1(self, u, args=()):
"""Conditional cdf of second component given the value of first.
"""
u = self._handle_u(u)
th, = self._handle_args(args)
if u.shape[-1] == 2:
# bivariate case
u1, u2 = u[..., 0], u[..., 1]
cdfc = np.exp(- th * u1)
cdfc /= np.expm1(-th) / np.expm1(- th * u2) + np.expm1(- th * u1)
return cdfc
else:
raise NotImplementedError("u needs to be bivariate (2 columns)") | Conditional cdf of second component given the value of first. | cdfcond_2g1 | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def ppfcond_2g1(self, q, u1, args=()):
"""Conditional pdf of second component given the value of first.
"""
u1 = np.asarray(u1)
th, = self._handle_args(args)
if u1.shape[-1] == 1:
# bivariate case, conditional on value of first variable
ppfc = - np.log(1 + np.expm1(- th) /
((1 / q - 1) * np.exp(-th * u1) + 1)) / th
return ppfc
else:
raise NotImplementedError("u needs to be bivariate (2 columns)") | Conditional pdf of second component given the value of first. | ppfcond_2g1 | python | statsmodels/statsmodels | statsmodels/distributions/copula/archimedean.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/archimedean.py | BSD-3-Clause |
def copula_bv_ev(u, transform, args=()):
'''generic bivariate extreme value copula
'''
u, v = u
return np.exp(np.log(u * v) * (transform(np.log(u)/np.log(u*v), *args))) | generic bivariate extreme value copula | copula_bv_ev | python | statsmodels/statsmodels | statsmodels/distributions/copula/extreme_value.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/extreme_value.py | BSD-3-Clause |
def cdf(self, u, args=()):
"""Evaluate cdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
CDF values at evaluation points.
"""
# currently only Bivariate
u, v = np.asarray(u).T
args = self._handle_args(args)
cdfv = np.exp(np.log(u * v) *
self.transform(np.log(u)/np.log(u*v), *args))
return cdfv | Evaluate cdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
CDF values at evaluation points. | cdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/extreme_value.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/extreme_value.py | BSD-3-Clause |
def pdf(self, u, args=()):
"""Evaluate pdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
PDF values at evaluation points.
"""
tr = self.transform
u1, u2 = np.asarray(u).T
args = self._handle_args(args)
log_u12 = np.log(u1 * u2)
t = np.log(u1) / log_u12
cdf = self.cdf(u, args)
dep = tr(t, *args)
d1 = tr.deriv(t, *args)
d2 = tr.deriv2(t, *args)
pdf_ = cdf / (u1 * u2) * ((dep + (1 - t) * d1) * (dep - t * d1) -
d2 * (1 - t) * t / log_u12)
return pdf_ | Evaluate pdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
PDF values at evaluation points. | pdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/extreme_value.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/extreme_value.py | BSD-3-Clause |
def logpdf(self, u, args=()):
"""Evaluate log-pdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
Log-pdf values at evaluation points.
"""
return np.log(self.pdf(u, args=args)) | Evaluate log-pdf of bivariate extreme value copula.
Parameters
----------
u : array_like
Values of random bivariate random variable, each defined on [0, 1],
for which cdf is computed.
Can be two dimensional with multivariate components in columns and
observation in rows.
args : tuple
Required parameters for the copula. The meaning and number of
parameters in the tuple depends on the specific copula.
Returns
-------
Log-pdf values at evaluation points. | logpdf | python | statsmodels/statsmodels | statsmodels/distributions/copula/extreme_value.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/extreme_value.py | BSD-3-Clause |
def conditional_2g1(self, u, args=()):
"""conditional distribution
not yet implemented
C2|1(u2|u1) := ∂C(u1, u2) / ∂u1 = C(u1, u2) / u1 * (A(t) − t A'(t))
where t = np.log(v)/np.log(u*v)
"""
raise NotImplementedError | conditional distribution
not yet implemented
C2|1(u2|u1) := ∂C(u1, u2) / ∂u1 = C(u1, u2) / u1 * (A(t) − t A'(t))
where t = np.log(v)/np.log(u*v) | conditional_2g1 | python | statsmodels/statsmodels | statsmodels/distributions/copula/extreme_value.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/extreme_value.py | BSD-3-Clause |
def rvs_kernel(sample, size, bw=1, k_func=None, return_extras=False):
"""Random sampling from empirical copula using Beta distribution
Parameters
----------
sample : ndarray
Sample of multivariate observations in (o, 1) interval.
size : int
Number of observations to simulate.
bw : float
Bandwidth for Beta sampling. The beta copula corresponds to a kernel
estimate of the distribution. bw=1 corresponds to the empirical beta
copula. A small bandwidth like bw=0.001 corresponds to small noise
added to the empirical distribution. Larger bw, e.g. bw=10 corresponds
to kernel estimate with more smoothing.
k_func : None or callable
The default kernel function is currently a beta function with 1 added
to the first beta parameter.
return_extras : bool
If this is False, then only the random sample will be returned.
If true, then extra information is returned that is mainly of interest
for verification.
Returns
-------
rvs : ndarray
Multivariate sample with ``size`` observations drawn from the Beta
Copula.
Notes
-----
Status: experimental, API will change.
"""
# vectorized for observations
n = sample.shape[0]
if k_func is None:
kfunc = _kernel_rvs_beta1
idx = np.random.randint(0, n, size=size)
xi = sample[idx]
krvs = np.column_stack([kfunc(xii, bw) for xii in xi.T])
if return_extras:
return krvs, idx, xi
else:
return krvs | Random sampling from empirical copula using Beta distribution
Parameters
----------
sample : ndarray
Sample of multivariate observations in (o, 1) interval.
size : int
Number of observations to simulate.
bw : float
Bandwidth for Beta sampling. The beta copula corresponds to a kernel
estimate of the distribution. bw=1 corresponds to the empirical beta
copula. A small bandwidth like bw=0.001 corresponds to small noise
added to the empirical distribution. Larger bw, e.g. bw=10 corresponds
to kernel estimate with more smoothing.
k_func : None or callable
The default kernel function is currently a beta function with 1 added
to the first beta parameter.
return_extras : bool
If this is False, then only the random sample will be returned.
If true, then extra information is returned that is mainly of interest
for verification.
Returns
-------
rvs : ndarray
Multivariate sample with ``size`` observations drawn from the Beta
Copula.
Notes
-----
Status: experimental, API will change. | rvs_kernel | python | statsmodels/statsmodels | statsmodels/distributions/copula/other_copulas.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/other_copulas.py | BSD-3-Clause |
def clear_cache(self):
"""clear cache of Sterling numbers
"""
self._cache = {} | clear cache of Sterling numbers | clear_cache | python | statsmodels/statsmodels | statsmodels/distributions/copula/_special.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/_special.py | BSD-3-Clause |
def clear_cache(self):
"""clear cache of Sterling numbers
"""
self._cache = {} | clear cache of Sterling numbers | clear_cache | python | statsmodels/statsmodels | statsmodels/distributions/copula/_special.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/_special.py | BSD-3-Clause |
def li3(z):
"""Polylogarithm for negative integer order -3
Li(-3, z)
"""
return z * (1 + 4 * z + z**2) / (1 - z)**4 | Polylogarithm for negative integer order -3
Li(-3, z) | li3 | python | statsmodels/statsmodels | statsmodels/distributions/copula/_special.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/_special.py | BSD-3-Clause |
def li4(z):
"""Polylogarithm for negative integer order -4
Li(-4, z)
"""
return z * (1 + z) * (1 + 10 * z + z**2) / (1 - z)**5 | Polylogarithm for negative integer order -4
Li(-4, z) | li4 | python | statsmodels/statsmodels | statsmodels/distributions/copula/_special.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/_special.py | BSD-3-Clause |
def lin(n, z):
"""Polylogarithm for negative integer order -n
Li(-n, z)
https://en.wikipedia.org/wiki/Polylogarithm#Particular_values
"""
if np.size(z) > 1:
z = np.array(z)[..., None]
k = np.arange(n+1)
st2 = np.array([sterling2(n + 1, ki + 1) for ki in k])
res = (-1)**(n+1) * np.sum(factorial(k) * st2 * (-1 / (1 - z))**(k+1),
axis=-1)
return res | Polylogarithm for negative integer order -n
Li(-n, z)
https://en.wikipedia.org/wiki/Polylogarithm#Particular_values | lin | python | statsmodels/statsmodels | statsmodels/distributions/copula/_special.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/distributions/copula/_special.py | BSD-3-Clause |
def _next_regular(target):
"""
Find the next regular number greater than or equal to target.
Regular numbers are composites of the prime factors 2, 3, and 5.
Also known as 5-smooth numbers or Hamming numbers, these are the optimal
size for inputs to FFTPACK.
Target must be a positive integer.
"""
if target <= 6:
return target
# Quickly check if it's already a power of 2
if not (target & (target - 1)):
return target
match = float("inf") # Anything found will be smaller
p5 = 1
while p5 < target:
p35 = p5
while p35 < target:
# Ceiling integer division, avoiding conversion to float
# (quotient = ceil(target / p35))
quotient = -(-target // p35)
# Quickly find next power of 2 >= quotient
p2 = 2 ** ((quotient - 1).bit_length())
N = p2 * p35
if N == target:
return N
elif N < match:
match = N
p35 *= 3
if p35 == target:
return p35
if p35 < match:
match = p35
p5 *= 5
if p5 == target:
return p5
if p5 < match:
match = p5
return match | Find the next regular number greater than or equal to target.
Regular numbers are composites of the prime factors 2, 3, and 5.
Also known as 5-smooth numbers or Hamming numbers, these are the optimal
size for inputs to FFTPACK.
Target must be a positive integer. | _next_regular | python | statsmodels/statsmodels | statsmodels/compat/scipy.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/scipy.py | BSD-3-Clause |
def _valarray(shape, value=np.nan, typecode=None):
"""Return an array of all value."""
out = np.ones(shape, dtype=bool) * value
if typecode is not None:
out = out.astype(typecode)
if not isinstance(out, np.ndarray):
out = np.asarray(out)
return out | Return an array of all value. | _valarray | python | statsmodels/statsmodels | statsmodels/compat/scipy.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/scipy.py | BSD-3-Clause |
def pytest_warns(
warning: type[Warning] | tuple[type[Warning], ...] | None
) -> WarningsChecker | NoWarningsChecker:
"""
Parameters
----------
warning : {None, Warning, Tuple[Warning]}
None if no warning is produced, or a single or multiple Warnings
Returns
-------
cm
"""
if warning is None:
return NoWarningsChecker()
else:
assert warning is not None
return warns(warning) | Parameters
----------
warning : {None, Warning, Tuple[Warning]}
None if no warning is produced, or a single or multiple Warnings
Returns
-------
cm | pytest_warns | python | statsmodels/statsmodels | statsmodels/compat/pytest.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/pytest.py | BSD-3-Clause |
def _squeeze_output(out):
"""
Remove single-dimensional entries from array and convert to scalar,
if necessary.
"""
out = out.squeeze()
if out.ndim == 0:
out = out[()]
return out | Remove single-dimensional entries from array and convert to scalar,
if necessary. | _squeeze_output | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _eigvalsh_to_eps(spectrum, cond=None, rcond=None):
"""
Determine which eigenvalues are "small" given the spectrum.
This is for compatibility across various linear algebra functions
that should agree about whether or not a Hermitian matrix is numerically
singular and what is its numerical matrix rank.
This is designed to be compatible with scipy.linalg.pinvh.
Parameters
----------
spectrum : 1d ndarray
Array of eigenvalues of a Hermitian matrix.
cond, rcond : float, optional
Cutoff for small eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are
considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
eps : float
Magnitude cutoff for numerical negligibility.
"""
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = spectrum.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
eps = cond * np.max(abs(spectrum))
return eps | Determine which eigenvalues are "small" given the spectrum.
This is for compatibility across various linear algebra functions
that should agree about whether or not a Hermitian matrix is numerically
singular and what is its numerical matrix rank.
This is designed to be compatible with scipy.linalg.pinvh.
Parameters
----------
spectrum : 1d ndarray
Array of eigenvalues of a Hermitian matrix.
cond, rcond : float, optional
Cutoff for small eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are
considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
eps : float
Magnitude cutoff for numerical negligibility. | _eigvalsh_to_eps | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _pinv_1d(v, eps=1e-5):
"""
A helper function for computing the pseudoinverse.
Parameters
----------
v : iterable of numbers
This may be thought of as a vector of eigenvalues or singular values.
eps : float
Values with magnitude no greater than eps are considered negligible.
Returns
-------
v_pinv : 1d float ndarray
A vector of pseudo-inverted numbers.
"""
return np.array([0 if abs(x) <= eps else 1/x for x in v], dtype=float) | A helper function for computing the pseudoinverse.
Parameters
----------
v : iterable of numbers
This may be thought of as a vector of eigenvalues or singular values.
eps : float
Values with magnitude no greater than eps are considered negligible.
Returns
-------
v_pinv : 1d float ndarray
A vector of pseudo-inverted numbers. | _pinv_1d | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def random_state(self):
""" Get or set the RandomState object for generating random variates.
This can be either None, int, a RandomState instance, or a
np.random.Generator instance.
If None (or np.random), use the RandomState singleton used by
np.random.
If already a RandomState or Generator instance, use it.
If an int, use a new RandomState instance seeded with seed.
"""
return self._random_state | Get or set the RandomState object for generating random variates.
This can be either None, int, a RandomState instance, or a
np.random.Generator instance.
If None (or np.random), use the RandomState singleton used by
np.random.
If already a RandomState or Generator instance, use it.
If an int, use a new RandomState instance seeded with seed. | random_state | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def __call__(self, mean=None, cov=1, allow_singular=False, seed=None):
"""
Create a frozen multivariate normal distribution.
See `multivariate_normal_frozen` for more information.
"""
return multivariate_normal_frozen(mean, cov,
allow_singular=allow_singular,
seed=seed) | Create a frozen multivariate normal distribution.
See `multivariate_normal_frozen` for more information. | __call__ | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _process_parameters(self, dim, mean, cov):
"""
Infer dimensionality from mean or covariance matrix, ensure that
mean and covariance are full vector resp. matrix.
"""
# Try to infer dimensionality
if dim is None:
if mean is None:
if cov is None:
dim = 1
else:
cov = np.asarray(cov, dtype=float)
if cov.ndim < 2:
dim = 1
else:
dim = cov.shape[0]
else:
mean = np.asarray(mean, dtype=float)
dim = mean.size
else:
if not np.isscalar(dim):
raise ValueError("Dimension of random variable must be "
"a scalar.")
# Check input sizes and return full arrays for mean and cov if
# necessary
if mean is None:
mean = np.zeros(dim)
mean = np.asarray(mean, dtype=float)
if cov is None:
cov = 1.0
cov = np.asarray(cov, dtype=float)
if dim == 1:
mean.shape = (1,)
cov.shape = (1, 1)
if mean.ndim != 1 or mean.shape[0] != dim:
raise ValueError("Array 'mean' must be a vector of length %d." %
dim)
if cov.ndim == 0:
cov = cov * np.eye(dim)
elif cov.ndim == 1:
cov = np.diag(cov)
elif cov.ndim == 2 and cov.shape != (dim, dim):
rows, cols = cov.shape
if rows != cols:
msg = ("Array 'cov' must be square if it is two dimensional,"
" but cov.shape = %s." % str(cov.shape))
else:
msg = ("Dimension mismatch: array 'cov' is of shape %s,"
" but 'mean' is a vector of length %d.")
msg = msg % (str(cov.shape), len(mean))
raise ValueError(msg)
elif cov.ndim > 2:
raise ValueError("Array 'cov' must be at most two-dimensional,"
" but cov.ndim = %d" % cov.ndim)
return dim, mean, cov | Infer dimensionality from mean or covariance matrix, ensure that
mean and covariance are full vector resp. matrix. | _process_parameters | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _process_quantiles(self, x, dim):
"""
Adjust quantiles array so that last axis labels the components of
each data point.
"""
x = np.asarray(x, dtype=float)
if x.ndim == 0:
x = x[np.newaxis]
elif x.ndim == 1:
if dim == 1:
x = x[:, np.newaxis]
else:
x = x[np.newaxis, :]
return x | Adjust quantiles array so that last axis labels the components of
each data point. | _process_quantiles | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _logpdf(self, x, mean, prec_U, log_det_cov, rank):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function
mean : ndarray
Mean of the distribution
prec_U : ndarray
A decomposition such that np.dot(prec_U, prec_U.T)
is the precision matrix, i.e. inverse of the covariance matrix.
log_det_cov : float
Logarithm of the determinant of the covariance matrix
rank : int
Rank of the covariance matrix.
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead.
"""
dev = x - mean
maha = np.sum(np.square(np.dot(dev, prec_U)), axis=-1)
return -0.5 * (rank * _LOG_2PI + log_det_cov + maha) | Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability
density function
mean : ndarray
Mean of the distribution
prec_U : ndarray
A decomposition such that np.dot(prec_U, prec_U.T)
is the precision matrix, i.e. inverse of the covariance matrix.
log_det_cov : float
Logarithm of the determinant of the covariance matrix
rank : int
Rank of the covariance matrix.
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'logpdf' instead. | _logpdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def logpdf(self, x, mean=None, cov=1, allow_singular=False):
"""
Log of the multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Log of the probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
psd = _PSD(cov, allow_singular=allow_singular)
out = self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank)
return _squeeze_output(out) | Log of the multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Log of the probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s | logpdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def pdf(self, x, mean=None, cov=1, allow_singular=False):
"""
Multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
psd = _PSD(cov, allow_singular=allow_singular)
out = np.exp(self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank))
return _squeeze_output(out) | Multivariate normal probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Probability density function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s | pdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _cdf(self, x, mean, cov, maxpts, abseps, releps):
"""
Parameters
----------
x : ndarray
Points at which to evaluate the cumulative distribution function.
mean : ndarray
Mean of the distribution
cov : array_like
Covariance matrix of the distribution
maxpts: integer
The maximum number of points to use for integration
abseps: float
Absolute error tolerance
releps: float
Relative error tolerance
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'cdf' instead.
.. versionadded:: 1.0.0
"""
lower = np.full(mean.shape, -np.inf)
# mvnun expects 1-d arguments, so process points sequentially
def func1d(x_slice):
return mvn.mvnun(lower, x_slice, mean, cov, maxpts, abseps, releps)[0]
out = np.apply_along_axis(func1d, -1, x)
return _squeeze_output(out) | Parameters
----------
x : ndarray
Points at which to evaluate the cumulative distribution function.
mean : ndarray
Mean of the distribution
cov : array_like
Covariance matrix of the distribution
maxpts: integer
The maximum number of points to use for integration
abseps: float
Absolute error tolerance
releps: float
Relative error tolerance
Notes
-----
As this function does no argument checking, it should not be
called directly; use 'cdf' instead.
.. versionadded:: 1.0.0 | _cdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def logcdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
abseps=1e-5, releps=1e-5):
"""
Log of the multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Log of the cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
# Use _PSD to check covariance matrix
_PSD(cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * dim
out = np.log(self._cdf(x, mean, cov, maxpts, abseps, releps))
return out | Log of the multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Log of the cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0 | logcdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def cdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
abseps=1e-5, releps=1e-5):
"""
Multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
# Use _PSD to check covariance matrix
_PSD(cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * dim
out = self._cdf(x, mean, cov, maxpts, abseps, releps)
return out | Multivariate normal cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvn_doc_default_callparams)s
maxpts: integer, optional
The maximum number of points to use for integration
(default `1000000*dim`)
abseps: float, optional
Absolute error tolerance (default 1e-5)
releps: float, optional
Relative error tolerance (default 1e-5)
Returns
-------
cdf : ndarray or scalar
Cumulative distribution function evaluated at `x`
Notes
-----
%(_mvn_doc_callparams_note)s
.. versionadded:: 1.0.0 | cdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def rvs(self, mean=None, cov=1, size=1, random_state=None):
"""
Draw random samples from a multivariate normal distribution.
Parameters
----------
%(_mvn_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `N`), where `N` is the
dimension of the random variable.
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
random_state = self._get_random_state(random_state)
out = random_state.multivariate_normal(mean, cov, size)
return _squeeze_output(out) | Draw random samples from a multivariate normal distribution.
Parameters
----------
%(_mvn_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `N`), where `N` is the
dimension of the random variable.
Notes
-----
%(_mvn_doc_callparams_note)s | rvs | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def entropy(self, mean=None, cov=1):
"""
Compute the differential entropy of the multivariate normal.
Parameters
----------
%(_mvn_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the multivariate normal distribution
Notes
-----
%(_mvn_doc_callparams_note)s
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
_, logdet = np.linalg.slogdet(2 * np.pi * np.e * cov)
return 0.5 * logdet | Compute the differential entropy of the multivariate normal.
Parameters
----------
%(_mvn_doc_default_callparams)s
Returns
-------
h : scalar
Entropy of the multivariate normal distribution
Notes
-----
%(_mvn_doc_callparams_note)s | entropy | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def __init__(self, mean=None, cov=1, allow_singular=False, seed=None,
maxpts=None, abseps=1e-5, releps=1e-5):
"""
Create a frozen multivariate normal distribution.
Parameters
----------
mean : array_like, optional
Mean of the distribution (default zero)
cov : array_like, optional
Covariance matrix of the distribution (default one)
allow_singular : bool, optional
If this flag is True then tolerate a singular
covariance matrix (default False).
seed : {None, int, `~np.random.RandomState`, `~np.random.Generator`}, optional
This parameter defines the object to use for drawing random
variates.
If `seed` is `None` the `~np.random.RandomState` singleton is used.
If `seed` is an int, a new ``RandomState`` instance is used, seeded
with seed.
If `seed` is already a ``RandomState`` or ``Generator`` instance,
then that object is used.
Default is None.
maxpts: integer, optional
The maximum number of points to use for integration of the
cumulative distribution function (default `1000000*dim`)
abseps: float, optional
Absolute error tolerance for the cumulative distribution function
(default 1e-5)
releps: float, optional
Relative error tolerance for the cumulative distribution function
(default 1e-5)
Examples
--------
When called with the default parameters, this will create a 1D random
variable with mean 0 and covariance 1:
>>> from scipy.stats import multivariate_normal
>>> r = multivariate_normal()
>>> r.mean
array([ 0.])
>>> r.cov
array([[1.]])
"""
self._dist = multivariate_normal_gen(seed)
self.dim, self.mean, self.cov = self._dist._process_parameters(
None, mean, cov)
self.cov_info = _PSD(self.cov, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * self.dim
self.maxpts = maxpts
self.abseps = abseps
self.releps = releps | Create a frozen multivariate normal distribution.
Parameters
----------
mean : array_like, optional
Mean of the distribution (default zero)
cov : array_like, optional
Covariance matrix of the distribution (default one)
allow_singular : bool, optional
If this flag is True then tolerate a singular
covariance matrix (default False).
seed : {None, int, `~np.random.RandomState`, `~np.random.Generator`}, optional
This parameter defines the object to use for drawing random
variates.
If `seed` is `None` the `~np.random.RandomState` singleton is used.
If `seed` is an int, a new ``RandomState`` instance is used, seeded
with seed.
If `seed` is already a ``RandomState`` or ``Generator`` instance,
then that object is used.
Default is None.
maxpts: integer, optional
The maximum number of points to use for integration of the
cumulative distribution function (default `1000000*dim`)
abseps: float, optional
Absolute error tolerance for the cumulative distribution function
(default 1e-5)
releps: float, optional
Relative error tolerance for the cumulative distribution function
(default 1e-5)
Examples
--------
When called with the default parameters, this will create a 1D random
variable with mean 0 and covariance 1:
>>> from scipy.stats import multivariate_normal
>>> r = multivariate_normal()
>>> r.mean
array([ 0.])
>>> r.cov
array([[1.]]) | __init__ | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def entropy(self):
"""
Computes the differential entropy of the multivariate normal.
Returns
-------
h : scalar
Entropy of the multivariate normal distribution
"""
log_pdet = self.cov_info.log_pdet
rank = self.cov_info.rank
return 0.5 * (rank * (_LOG_2PI + 1) + log_pdet) | Computes the differential entropy of the multivariate normal.
Returns
-------
h : scalar
Entropy of the multivariate normal distribution | entropy | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def __init__(self, seed=None):
"""
Initialize a multivariate t-distributed random variable.
Parameters
----------
seed : Random state.
"""
super().__init__(seed)
self.__doc__ = doccer.docformat(self.__doc__, mvt_docdict_params)
self._random_state = check_random_state(seed) | Initialize a multivariate t-distributed random variable.
Parameters
----------
seed : Random state. | __init__ | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def __call__(self, loc=None, shape=1, df=1, allow_singular=False,
seed=None):
"""
Create a frozen multivariate t-distribution. See
`multivariate_t_frozen` for parameters.
"""
if df == np.inf:
return multivariate_normal_frozen(mean=loc, cov=shape,
allow_singular=allow_singular,
seed=seed)
return multivariate_t_frozen(loc=loc, shape=shape, df=df,
allow_singular=allow_singular, seed=seed) | Create a frozen multivariate t-distribution. See
`multivariate_t_frozen` for parameters. | __call__ | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def pdf(self, x, loc=None, shape=1, df=1, allow_singular=False):
"""
Multivariate t-distribution probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the probability density function.
%(_mvt_doc_default_callparams)s
Returns
-------
pdf : Probability density function evaluated at `x`.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.pdf(x, loc, shape, df)
array([0.00075713])
"""
dim, loc, shape, df = self._process_parameters(loc, shape, df)
x = self._process_quantiles(x, dim)
shape_info = _PSD(shape, allow_singular=allow_singular)
logpdf = self._logpdf(x, loc, shape_info.U, shape_info.log_pdet, df,
dim, shape_info.rank)
return np.exp(logpdf) | Multivariate t-distribution probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the probability density function.
%(_mvt_doc_default_callparams)s
Returns
-------
pdf : Probability density function evaluated at `x`.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.pdf(x, loc, shape, df)
array([0.00075713]) | pdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def logpdf(self, x, loc=None, shape=1, df=1):
"""
Log of the multivariate t-distribution probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
%(_mvt_doc_default_callparams)s
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.logpdf(x, loc, shape, df)
array([-7.1859802])
See Also
--------
pdf : Probability density function.
"""
dim, loc, shape, df = self._process_parameters(loc, shape, df)
x = self._process_quantiles(x, dim)
shape_info = _PSD(shape)
return self._logpdf(x, loc, shape_info.U, shape_info.log_pdet, df, dim,
shape_info.rank) | Log of the multivariate t-distribution probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
%(_mvt_doc_default_callparams)s
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.logpdf(x, loc, shape, df)
array([-7.1859802])
See Also
--------
pdf : Probability density function. | logpdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def _logpdf(self, x, loc, prec_U, log_pdet, df, dim, rank):
"""Utility method `pdf`, `logpdf` for parameters.
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability density
function.
loc : ndarray
Location of the distribution.
prec_U : ndarray
A decomposition such that `np.dot(prec_U, prec_U.T)` is the inverse
of the shape matrix.
log_pdet : float
Logarithm of the determinant of the shape matrix.
df : float
Degrees of freedom of the distribution.
dim : int
Dimension of the quantiles x.
rank : int
Rank of the shape matrix.
Notes
-----
As this function does no argument checking, it should not be called
directly; use 'logpdf' instead.
"""
if df == np.inf:
return multivariate_normal._logpdf(x, loc, prec_U, log_pdet, rank)
dev = x - loc
maha = np.square(np.dot(dev, prec_U)).sum(axis=-1)
t = 0.5 * (df + dim)
A = gammaln(t)
B = gammaln(0.5 * df)
C = dim/2. * np.log(df * np.pi)
D = 0.5 * log_pdet
E = -t * np.log(1 + (1./df) * maha)
return _squeeze_output(A - B - C - D + E) | Utility method `pdf`, `logpdf` for parameters.
Parameters
----------
x : ndarray
Points at which to evaluate the log of the probability density
function.
loc : ndarray
Location of the distribution.
prec_U : ndarray
A decomposition such that `np.dot(prec_U, prec_U.T)` is the inverse
of the shape matrix.
log_pdet : float
Logarithm of the determinant of the shape matrix.
df : float
Degrees of freedom of the distribution.
dim : int
Dimension of the quantiles x.
rank : int
Rank of the shape matrix.
Notes
-----
As this function does no argument checking, it should not be called
directly; use 'logpdf' instead. | _logpdf | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
def rvs(self, loc=None, shape=1, df=1, size=1, random_state=None):
"""
Draw random samples from a multivariate t-distribution.
Parameters
----------
%(_mvt_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `P`), where `P` is the
dimension of the random variable.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.rvs(loc, shape, df)
array([[0.93477495, 3.00408716]])
"""
# For implementation details, see equation (3):
#
# Hofert, "On Sampling from the Multivariatet Distribution", 2013
# http://rjournal.github.io/archive/2013-2/hofert.pdf
#
dim, loc, shape, df = self._process_parameters(loc, shape, df)
if random_state is not None:
rng = check_random_state(random_state)
else:
rng = self._random_state
if np.isinf(df):
x = np.ones(size)
else:
x = rng.chisquare(df, size=size) / df
z = rng.multivariate_normal(np.zeros(dim), shape, size=size)
samples = loc + z / np.sqrt(x)[:, None]
return _squeeze_output(samples) | Draw random samples from a multivariate t-distribution.
Parameters
----------
%(_mvt_doc_default_callparams)s
size : integer, optional
Number of samples to draw (default 1).
%(_doc_random_state)s
Returns
-------
rvs : ndarray or scalar
Random variates of size (`size`, `P`), where `P` is the
dimension of the random variable.
Examples
--------
>>> from scipy.stats import multivariate_t
>>> x = [0.4, 5]
>>> loc = [0, 1]
>>> shape = [[1, 0.1], [0.1, 1]]
>>> df = 7
>>> multivariate_t.rvs(loc, shape, df)
array([[0.93477495, 3.00408716]]) | rvs | python | statsmodels/statsmodels | statsmodels/compat/_scipy_multivariate_t.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/compat/_scipy_multivariate_t.py | BSD-3-Clause |
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