def test_special_cases():
assert_equal(sc.owens_t(5, 0), 0)
assert_allclose(sc.owens_t(0, 5), 0.5*np.arctan(5)/np.pi,
rtol=5e-14)
# Target value is 0.5*Phi(5)*(1 - Phi(5)) for Phi the CDF of the
# standard normal distribution
assert_allclose(sc.owens_t(5, 1), 1.4332574485503512543e-07,
rtol=5e-14)
def test_special_cases():
assert_equal(sc.owens_t(5, 0), 0)
assert_allclose(sc.owens_t(0, 5), 0.5*np.arctan(5)/np.pi,
rtol=5e-14)
# Target value is 0.5*Phi(5)*(1 - Phi(5)) for Phi the CDF of the
# standard normal distribution
assert_allclose(sc.owens_t(5, 1), 1.4332574485503512543e-07,
rtol=5e-14)
def Cov2S2(tau, g, strategy=0):
vf = tau2vf(tau, strategy=strategy)
x = np.sqrt((1 - g) / (1 + g))
if strategy:
# |u|<tau
x1 = np.where(np.abs(x) < 1.e-6, 1.e-6, x)
S2 = 2 * vf - 4 * owens_t(tau, x) - 4 * owens_t(tau, 1 / x1)
else:
# u<tau
S2 = vf - 2 * owens_t(tau, x)
return S2
def change_of_var_Phi_withGradients(m, s, dmdx, dsdx):
""" from arrays of mean and std of NORMAL variables X, produce the mean and std of
Phi(X) where Phi is the normal cumulative distribution
E[Phi(X)] = Phi(m/sqrt(1 + s^2))
Var[Phi(X)] = E - E^2 - T (m/sqrt(1 + s^2), m/sqrt(1 + 2*s^2))
RETURNS VARIANCE (and not STD)
"""
assert np.ndim(
dmdx) == np.ndim(m) + 1, 'dim of m: {}, dim of dm: {}'.formate(
np.ndim(m), np.ndim(dmdx))
assert np.ndim(
dsdx) == np.ndim(s) + 1, 'dim of s: {}, dim of ds: {}'.formate(
np.ndim(s), np.ndim(dsdx))
v = np.square(s)
a = m / np.sqrt(1 + v)
h = 1 / np.sqrt(1 + 2 * v)
mcdf = stats.norm.cdf(a)
vcdf = mcdf - np.square(mcdf) - 2 * special.owens_t(a, h)
# Carefull
dadx = dmdx / np.sqrt(
1 + v[:, :, None]) - a[:, :, None] * s[:, :, None] * dsdx / (
1 + v[:, :, None])
#dhdx = - 2 * s * dsdx * h / (1+2*v)
A = np.exp(-np.square(a[:, :, None]) / 2)
dmcdfdx = 1 / np.sqrt(2 * np.pi) * A * dadx
dTda = -A / (2 * np.sqrt(2 * np.pi)) * erf(a * h / np.sqrt(2))[:, :, None]
dTdhdhdx = -(np.exp(-np.square(m * h))[:, :, None] * h[:, :, None] *
s[:, :, None] * dsdx) / (2 * np.pi * (1 + v[:, :, None]))
dTdx = dTda * dadx + dTdhdhdx
dvcdfdx = dmcdfdx - 2 * mcdf[:, :, None] * dmcdfdx - 2 * dTdx
return mcdf, vcdf, dmcdfdx, dvcdfdx
def test_infs():
h = 1
res = 0.5*sc.erfc(h/np.sqrt(2))
assert_allclose(sc.owens_t(h, np.inf), res, rtol=5e-14)
assert_allclose(sc.owens_t(h, -np.inf), -res, rtol=5e-14)
assert_equal(sc.owens_t(np.inf, 1), 0)
assert_equal(sc.owens_t(-np.inf, 1), 0)
assert_equal(sc.owens_t(np.inf, np.inf), 0)
assert_equal(sc.owens_t(-np.inf, np.inf), 0)
assert_equal(sc.owens_t(np.inf, -np.inf), -0.0)
assert_equal(sc.owens_t(-np.inf, -np.inf), -0.0)
def test_infs():
h = 1
res = 0.5 * sc.erfc(h / np.sqrt(2))
assert_allclose(sc.owens_t(h, np.inf), res, rtol=5e-14)
assert_allclose(sc.owens_t(h, -np.inf), -res, rtol=5e-14)
assert_equal(sc.owens_t(np.inf, 1), 0)
assert_equal(sc.owens_t(-np.inf, 1), 0)
assert_equal(sc.owens_t(np.inf, np.inf), 0)
assert_equal(sc.owens_t(-np.inf, np.inf), 0)
assert_equal(sc.owens_t(np.inf, -np.inf), -0.0)
assert_equal(sc.owens_t(-np.inf, -np.inf), -0.0)
def change_of_var_Phi(m, s):
""" from arrays of mean and std of NORMAL variables X, produce the mean and std of
Phi(X) where Phi is the normal cumulative distribution
E[Phi(X)] = Phi(m/sqrt(1 + s^2))
Var[Phi(X)] = E - E^2 - T (m/sqrt(1 + s^2), m/sqrt(1 + 2*s^2))
"""
a = m / np.sqrt(1 + np.square(s))
h = 1 / np.sqrt(1 + 2 * np.square(s))
m_cdf = stats.norm.cdf(a)
s_cdf = np.sqrt(m_cdf - np.square(m_cdf) - 2 * special.owens_t(a, h))
return m_cdf, s_cdf
def margin(self, eps=0.0001):
_t = list(np.linspace(-3, 5, 101))
_phi = np.asarray([norm.cdf(t) for t in _t])
_owens_t = np.asarray([owens_t(t, self.alpha) for t in _t])
_cdf = _phi - 2 * _owens_t
idx_l = next(x for x in range(len(_cdf)) if _cdf[x] > eps)
idx_h = next(x for x in range(len(_cdf)) if _cdf[x] > 1 - eps)
low = _t[max(idx_l - 1, 0)]
high = _t[min(idx_h + 1, 100)]
low = (low * self.omega) + self.zeta
high = (high * self.omega) + self.zeta
return low, high
def testOwensTLarge(self, dtype):
seed_stream = test_util.test_seed_stream()
a = tf.random.uniform(shape=[int(1e4)],
minval=100.,
maxval=1000.,
dtype=dtype,
seed=seed_stream())
h = tf.random.uniform(shape=[int(1e4)],
minval=100.,
maxval=1000.,
dtype=dtype,
seed=seed_stream())
a_, h_, owens_t_ = self.evaluate([a, h, tfp.math.owens_t(h, a)])
self.assertAllClose(scipy_special.owens_t(h_, a_), owens_t_)
def test_infs():
h, a = 0, np.inf
# T(0, a) = 1/2π * arctan(a)
res = 1 / (2 * np.pi) * np.arctan(a)
assert_allclose(sc.owens_t(h, a), res, rtol=5e-14)
assert_allclose(sc.owens_t(h, -a), -res, rtol=5e-14)
h = 1
# Refer Owens T function definition in Wikipedia
# https://en.wikipedia.org/wiki/Owen%27s_T_function
# Value approximated through Numerical Integration
# using scipy.integrate.quad
# quad(lambda x: 1/(2*pi)*(exp(-0.5*(1*1)*(1+x*x))/(1+x*x)), 0, inf)
res = 0.07932762696572854
assert_allclose(sc.owens_t(h, np.inf), res, rtol=5e-14)
assert_allclose(sc.owens_t(h, -np.inf), -res, rtol=5e-14)
assert_equal(sc.owens_t(np.inf, 1), 0)
assert_equal(sc.owens_t(-np.inf, 1), 0)
assert_equal(sc.owens_t(np.inf, np.inf), 0)
assert_equal(sc.owens_t(-np.inf, np.inf), 0)
assert_equal(sc.owens_t(np.inf, -np.inf), -0.0)
assert_equal(sc.owens_t(-np.inf, -np.inf), -0.0)
def predict(
self,
x: Union[torch.Tensor, np.ndarray],
probability_space: bool = False
) -> Tuple[torch.Tensor, torch.Tensor]:
"""Query the model for posterior mean and variance.
Args:
x (torch.Tensor): Points at which to predict from the model.
probability_space (bool, optional): Return outputs in units of
response probability instead of latent function value. Defaults to False.
Returns:
Tuple[np.ndarray, np.ndarray]: Posterior mean and variance at queries points.
"""
with torch.no_grad():
post = self.posterior(x)
fmean = post.mean.squeeze()
fvar = post.variance.squeeze()
if probability_space:
if isinstance(self.likelihood, BernoulliLikelihood):
# Probability-space mean and variance for Bernoulli-probit models is
# available in closed form, Proposition 1 in Letham et al. 2022 (AISTATS).
a_star = fmean / torch.sqrt(1 + fvar)
pmean = Normal(0, 1).cdf(a_star)
t_term = torch.tensor(
owens_t(a_star.numpy(), 1 / np.sqrt(1 + 2 * fvar.numpy())),
dtype=a_star.dtype,
)
pvar = pmean - 2 * t_term - pmean.square()
return promote_0d(pmean), promote_0d(pvar)
else:
fsamps = post.sample(torch.Size([10000]))
if hasattr(self.likelihood, "objective"):
psamps = self.likelihood.objective(fsamps)
else:
psamps = norm.cdf(fsamps)
pmean, pvar = psamps.mean(0), psamps.var(0)
return promote_0d(pmean), promote_0d(pvar)
else:
return promote_0d(fmean), promote_0d(fvar)
def variance(u, sigma_0, kappa):
global J0, J1, sigma_1
return J0*J0 * mean_rate(u, sigma_0, kappa) - 2 * special.owens_t(u/np.sqrt(1.0 + sigma_0 + sigma_1 * kappa * kappa), 1/np.sqrt(1.0 + 2*sigma_0 + 2*sigma_1 * kappa * kappa))
def variance(u, sigma):
var = np.maximum(0, gv.J0**2 * ( mean_rate(u, sigma) - 2 * special.owens_t(u/np.sqrt(1.0 + sigma), 1/np.sqrt(1.0 + 2*sigma) ) ) )
return var
def cdf1D(self, x):
if self._check1D():
x = np.squeeze( (x-self.loc) / self.scale )
s = np.squeeze(self.alpha)
return stats.norm.cdf(x) - 2 * special.owens_t(x, s)
def test_symmetries():
np.random.seed(1234)
a, h = np.random.rand(100), np.random.rand(100)
assert_equal(sc.owens_t(h, a), sc.owens_t(-h, a))
assert_equal(sc.owens_t(h, a), -sc.owens_t(h, -a))
def test_nans():
assert_equal(sc.owens_t(20, np.nan), np.nan)
assert_equal(sc.owens_t(np.nan, 20), np.nan)
assert_equal(sc.owens_t(np.nan, np.nan), np.nan)
def test_nans():
assert_equal(sc.owens_t(20, np.nan), np.nan)
assert_equal(sc.owens_t(np.nan, 20), np.nan)
assert_equal(sc.owens_t(np.nan, np.nan), np.nan)
def test_symmetries():
np.random.seed(1234)
a, h = np.random.rand(100), np.random.rand(100)
assert_equal(sc.owens_t(h, a), sc.owens_t(-h, a))
assert_equal(sc.owens_t(h, a), -sc.owens_t(h, -a))
