def _t(u, rho, nu):
""" Generates values of the T copula
Inputs:
u -- u is an N-by-P matrix of values in [0,1], representing N
points in the P-dimensional unit hypercube.
rho -- a P-by-P correlation matrix.
nu -- degrees of freedom for T Copula
Outputs:
y -- the value of the T Copula
"""
n = u.shape[0]
p = u.shape[1]
loIntegrationVal = -40
lo = np.full((1,p), loIntegrationVal) # more accuracy, but slower :/
hi = t.ppf(u, nu)
mu = np.zeros(p)
y = np.zeros(n)
for ii in np.arange(n):
x = hi[ii,:]
x[x<-40] = -40
p = mvt.mvstdtprob(lo[0], x, rho, nu)
y[ii] = p
return y
def _t(u, rho, nu):
""" Generates values of the T copula
Inputs:
u -- u is an N-by-P matrix of values in [0,1], representing N
points in the P-dimensional unit hypercube.
rho -- a P-by-P correlation matrix.
nu -- degrees of freedom for T Copula
Outputs:
y -- the value of the T Copula
"""
n = u.shape[0]
p = u.shape[1]
loIntegrationVal = -40
lo = np.full((1, p), loIntegrationVal) # more accuracy, but slower :/
hi = t.ppf(u, nu)
mu = np.zeros(p)
y = np.zeros(n)
for ii in np.arange(n):
x = hi[ii, :]
x[x < -40] = -40
p = mvt.mvstdtprob(lo[0], x, rho, nu)
y[ii] = p
return y
def test_mvn_mvt_2(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
probmvn_R = 0.6472497 #reported error approx. 7.7e-08
probmvt_R = 0.5881863 #highest reported error up to approx. 1.99e-06
assert_almost_equal(probmvt_R, mvstdtprob(a, b, corr2, df), 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a, b, corr2, abseps=1e-5), 4)
def test_mvn_mvt_2(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
probmvn_R = 0.6472497 #reported error approx. 7.7e-08
probmvt_R = 0.5881863 #highest reported error up to approx. 1.99e-06
assert_almost_equal(probmvt_R, mvstdtprob(a, b, corr2, df), 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a, b, corr2, abseps=1e-5), 4)
def test_mvn_mvt_5(self):
a, bl = self.a, self.b
df = self.df
corr2 = self.corr2
#unequal integration bounds
#print "ue"
a3 = np.array([0.5, -0.5, 0.5])
probmvn_R = 0.06910487 #using higher precision in R, error approx. 3.5e-08
probmvt_R = 0.05797867 #using higher precision in R, error approx. 5.8e-08
assert_almost_equal(mvstdtprob(a3, a3+1, corr2, df), probmvt_R, 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a3, a3+1, corr2,
maxpts=100000, abseps=1e-5), 4)
def test_mvn_mvt_5(self):
a, bl = self.a, self.b
df = self.df
corr2 = self.corr2
#unequal integration bounds
#print "ue"
a3 = np.array([0.5, -0.5, 0.5])
probmvn_R = 0.06910487 #using higher precision in R, error approx. 3.5e-08
probmvt_R = 0.05797867 #using higher precision in R, error approx. 5.8e-08
assert_almost_equal(mvstdtprob(a3, a3+1, corr2, df), probmvt_R, 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a3, a3+1, corr2,
maxpts=100000, abseps=1e-5), 4)
def test_mvn_mvt_3(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
#from -inf
#print 'from -inf'
a2 = a.copy()
a2[:] = -np.inf
probmvn_R = 0.9961141 #using higher precision in R, error approx. 6.866163e-07
probmvt_R = 0.9522146 #using higher precision in R, error approx. 1.6e-07
assert_almost_equal(probmvt_R, mvstdtprob(a2, b, corr2, df), 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a2, b, corr2, maxpts=100000,
abseps=1e-5), 4)
def test_mvn_mvt_3(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
#from -inf
#print 'from -inf'
a2 = a.copy()
a2[:] = -np.inf
probmvn_R = 0.9961141 #using higher precision in R, error approx. 6.866163e-07
probmvt_R = 0.9522146 #using higher precision in R, error approx. 1.6e-07
assert_almost_equal(probmvt_R, mvstdtprob(a2, b, corr2, df), 4)
assert_almost_equal(probmvn_R, mvstdnormcdf(a2, b, corr2, maxpts=100000,
abseps=1e-5), 4)
def test_mvn_mvt_1(self):
a, b = self.a, self.b
df = self.df
corr_equal = self.corr_equal
#result from R, mvtnorm with option
#algorithm = GenzBretz(maxpts = 100000, abseps = 0.000001, releps = 0)
# or higher
probmvt_R = 0.60414 #report, ed error approx. 7.5e-06
probmvn_R = 0.673970 #reported error approx. 6.4e-07
assert_almost_equal(probmvt_R, mvstdtprob(a, b, corr_equal, df), 4)
assert_almost_equal(probmvn_R,
mvstdnormcdf(a, b, corr_equal, abseps=1e-5), 4)
mvn_high = mvstdnormcdf(a, b, corr_equal, abseps=1e-8, maxpts=10000000)
assert_almost_equal(probmvn_R, mvn_high, 5)
def test_mvn_mvt_4(self):
a, bl = self.a, self.b
df = self.df
corr2 = self.corr2
#from 0 to inf
#print '0 inf'
a2 = a.copy()
a2[:] = -np.inf
probmvn_R = 0.1666667 #error approx. 6.1e-08
probmvt_R = 0.1666667 #error approx. 8.2e-08
assert_almost_equal(probmvt_R, mvstdtprob(np.zeros(3), -a2, corr2, df), 4)
assert_almost_equal(probmvn_R,
mvstdnormcdf(np.zeros(3), -a2, corr2,
maxpts=100000, abseps=1e-5), 4)
def test_mvn_mvt_1(self):
a, b = self.a, self.b
df = self.df
corr_equal = self.corr_equal
#result from R, mvtnorm with option
#algorithm = GenzBretz(maxpts = 100000, abseps = 0.000001, releps = 0)
# or higher
probmvt_R = 0.60414 #report, ed error approx. 7.5e-06
probmvn_R = 0.673970 #reported error approx. 6.4e-07
assert_almost_equal(probmvt_R, mvstdtprob(a, b, corr_equal, df), 4)
assert_almost_equal(probmvn_R,
mvstdnormcdf(a, b, corr_equal, abseps=1e-5), 4)
mvn_high = mvstdnormcdf(a, b, corr_equal, abseps=1e-8, maxpts=10000000)
assert_almost_equal(probmvn_R, mvn_high, 5)
def test_mvn_mvt_4(self):
a, bl = self.a, self.b
df = self.df
corr2 = self.corr2
#from 0 to inf
#print '0 inf'
a2 = a.copy()
a2[:] = -np.inf
probmvn_R = 0.1666667 #error approx. 6.1e-08
probmvt_R = 0.1666667 #error approx. 8.2e-08
assert_almost_equal(probmvt_R, mvstdtprob(np.zeros(3), -a2, corr2, df), 4)
assert_almost_equal(probmvn_R,
mvstdnormcdf(np.zeros(3), -a2, corr2,
maxpts=100000, abseps=1e-5), 4)
def test_mvn_mvt_3(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
a2 = a.copy()
a2[:] = -np.inf
# using higher precision in R, error approx. 6.866163e-07
probmvn_R = 0.9961141
# using higher precision in R, error approx. 1.6e-07
probmvt_R = 0.9522146
quadkwds = {'epsabs': 1e-08}
probmvt = mvstdtprob(a2, b, corr2, df, quadkwds=quadkwds)
assert_allclose(probmvt_R, probmvt, atol=5e-4)
probmvn = mvstdnormcdf(a2, b, corr2, maxpts=100000, abseps=1e-5)
assert_allclose(probmvn_R, probmvn, atol=1e-4)
def test_mvn_mvt_3(self):
a, b = self.a, self.b
df = self.df
corr2 = self.corr2
a2 = a.copy()
a2[:] = -np.inf
# using higher precision in R, error approx. 6.866163e-07
probmvn_R = 0.9961141
# using higher precision in R, error approx. 1.6e-07
probmvt_R = 0.9522146
quadkwds = {'epsabs': 1e-08}
probmvt = mvstdtprob(a2, b, corr2, df, quadkwds=quadkwds)
assert_allclose(probmvt_R, probmvt, atol=5e-4)
probmvn = mvstdnormcdf(a2, b, corr2, maxpts=100000, abseps=1e-5)
assert_allclose(probmvn_R, probmvn, atol=1e-4)
def cdf(self, x, **kwds):
'''cumulative distribution function
Parameters
----------
x : array_like
can be 1d or 2d, if 2d, then each row is taken as independent
multivariate random vector
kwds : dict
contains options for the numerical calculation of the cdf
Returns
-------
cdf : float or array
probability density value of each random vector
'''
lower = -np.inf * np.ones_like(x)
#std_sigma = np.sqrt(np.diag(self.sigma))
upper = (x - self.mean) / self.std_sigma
return mvstdtprob(lower, upper, self.corr, self.df, **kwds)
def cdf(self, x, **kwds):
'''cumulative distribution function
Parameters
----------
x : array_like
can be 1d or 2d, if 2d, then each row is taken as independent
multivariate random vector
kwds : dict
contains options for the numerical calculation of the cdf
Returns
-------
cdf : float or array
probability density value of each random vector
'''
lower = -np.inf * np.ones_like(x)
#std_sigma = np.sqrt(np.diag(self.sigma))
upper = (x - self.mean)/self.std_sigma
return mvstdtprob(lower, upper, self.corr, self.df, **kwds)