def test_collocate_1d(self):
x, y = sp.symbols('x y')
length_scale = 0.05
k = sp.exp(-((x - y)**2) / (2 * length_scale**2))
A = lambda f: sp.diff(f, x, x)
Abar = lambda f: sp.diff(f, y, y)
B = lambda f: f
Bbar = lambda f: f
# interior observations: sin(x)*sin(y)
interior = np.linspace(0, 1)[1:-1, None]
exterior = np.array([0., 1.])[:, None]
interior_obs = np.sin(2 * np.pi * interior)
exterior_obs = np.array([0., 0.])[:, None]
print interior.shape, interior_obs.shape
print exterior.shape, exterior_obs.shape
posterior = bayesian_pdes.collocate([A, B], [Abar, Bbar], k, [x, y],
[(interior, interior_obs),
(exterior, exterior_obs)])
mean, cov = posterior(interior)
actual = -1 * np.sin(2 * np.pi * interior) / (4 * np.pi**2)
ci = cov.dot(
0.25 *
np.ones_like(mean)) # represents a 0.25SD CI for each observation
err = np.abs(mean - actual)
np.testing.assert_array_less(err, ci)
def test_collocate_1d(self):
x, y = sp.symbols('x y')
length_scale = 0.05
k = sp.exp(-((x-y)**2) / (2*length_scale**2))
A = lambda f: sp.diff(f, x, x)
Abar = lambda f: sp.diff(f, y, y)
B = lambda f: f
Bbar = lambda f: f
# interior observations: sin(x)*sin(y)
interior = np.linspace(0,1)[1:-1, None]
exterior = np.array([0.,1.])[:, None]
interior_obs = np.sin(2*np.pi*interior)
exterior_obs = np.array([0.,0.])[:, None]
print interior.shape, interior_obs.shape
print exterior.shape, exterior_obs.shape
posterior = bayesian_pdes.collocate(
[A, B],
[Abar, Bbar],
k,
[x,y],
[(interior, interior_obs), (exterior, exterior_obs)]
)
mean, cov = posterior(interior)
actual = -1 * np.sin(2*np.pi*interior) / (4*np.pi**2)
ci = cov.dot(0.25*np.ones_like(mean)) # represents a 0.25SD CI for each observation
err = np.abs(mean - actual)
np.testing.assert_array_less(err, ci)
def test_collocate_2d(self):
x_1, x_2, y_1, y_2 = sp.symbols('x_1 x_2 y_1 y_2')
length_scale = 0.05
k = sp.exp(-((x_1 - y_1)**2 + (x_2 - y_2)**2) / (2 * length_scale**2))
A = lambda f: sp.diff(f, x_1, x_1) + sp.diff(f, x_2, x_2)
Abar = lambda f: sp.diff(f, y_1, y_1) + sp.diff(f, y_2, y_2)
B = lambda f: f
Bbar = lambda f: f
x_interior, y_interior = np.mgrid[0:1:20j, 0:1:20j]
x_interior = x_interior.ravel()
y_interior = y_interior.ravel()
on_bdy = (x_interior == 0) | (x_interior == 1) | (y_interior == 0) | (
y_interior == 1)
interior = np.c_[x_interior[~on_bdy], y_interior[~on_bdy]]
exterior = np.c_[x_interior[on_bdy], y_interior[on_bdy]]
interior_obs = np.sin(2 * np.pi * interior[:, 0]) + np.sin(
2 * np.pi * interior[:, 1])
exterior_obs = np.zeros(exterior.shape[0])
print interior.shape, interior_obs.shape
print exterior.shape, exterior_obs.shape
posterior = bayesian_pdes.collocate([A, B], [Abar, Bbar], k,
[[x_1, x_2], [y_1, y_2]],
[(interior, interior_obs),
(exterior, exterior_obs)])
import time
start = time.time()
# test_x, test_y = np.mgrid[0:1:21j, 0:1:21j]
# test_points = np.c_[test_x.ravel(), test_y.ravel()]
test_points = interior
mean, cov = posterior(test_points)
end = time.time()
print end - start
actual = -(np.sin(2 * np.pi * test_points[:, 0]) +
np.sin(2 * np.pi * test_points[:, 1])) / (4 * np.pi**2)
ci = cov.dot(
0.25 *
np.ones_like(mean)) # represents a 0.25SD CI for each observation
err = np.abs(mean - actual)
np.testing.assert_array_less(err, ci)
def test_collocate_2d(self):
x_1,x_2,y_1,y_2 = sp.symbols('x_1 x_2 y_1 y_2')
length_scale = 0.05
k = sp.exp(-((x_1-y_1)**2+(x_2-y_2)**2) / (2*length_scale**2))
A = lambda f: sp.diff(f, x_1, x_1) + sp.diff(f, x_2, x_2)
Abar = lambda f: sp.diff(f, y_1, y_1) + sp.diff(f, y_2, y_2)
B = lambda f: f
Bbar = lambda f: f
x_interior, y_interior = np.mgrid[0:1:20j, 0:1:20j]
x_interior = x_interior.ravel(); y_interior = y_interior.ravel()
on_bdy = (x_interior == 0) | (x_interior == 1) | (y_interior == 0) | (y_interior == 1)
interior = np.c_[x_interior[~on_bdy], y_interior[~on_bdy]]
exterior = np.c_[x_interior[on_bdy], y_interior[on_bdy]]
interior_obs = np.sin(2*np.pi*interior[:,0]) + np.sin(2*np.pi*interior[:,1])
exterior_obs = np.zeros(exterior.shape[0])
print interior.shape, interior_obs.shape
print exterior.shape, exterior_obs.shape
posterior = bayesian_pdes.collocate(
[A, B],
[Abar, Bbar],
k,
[[x_1, x_2], [y_1, y_2]],
[(interior, interior_obs), (exterior, exterior_obs)]
)
import time
start = time.time()
# test_x, test_y = np.mgrid[0:1:21j, 0:1:21j]
# test_points = np.c_[test_x.ravel(), test_y.ravel()]
test_points = interior
mean, cov = posterior(test_points)
end = time.time()
print end - start
actual = - (np.sin(2*np.pi*test_points[:,0]) + np.sin(2*np.pi*test_points[:,1])) / (4*np.pi**2)
ci = cov.dot(0.25*np.ones_like(mean)) # represents a 0.25SD CI for each observation
err = np.abs(mean - actual)
np.testing.assert_array_less(err, ci)
def construct_posterior(grid, op_system, theta, collocate_args, proposal_dot_mat, debug=False):
design_int = grid.interior_plus_boundary
a_int, a_bdy, a_x, a_y = theta_to_a(theta,
design_int.shape[0],
grid.sensors.shape[0],
proposal_dot_mat
)
augmented_int = np.column_stack([design_int, a_int, a_x, a_y])
augmented_bdy = np.column_stack([grid.sensors, a_bdy, np.nan * np.zeros((a_bdy.shape[0], 2))])
obs = [
(augmented_int, None),
(augmented_bdy, None)
]
posterior = bpdes.collocate(
op_system.operators,
op_system.operators_bar,
obs,
op_system,
collocate_args,
inverter='np'
)
return posterior
def log_likelihood(self,
theta,
observations,
truth,
length_scale,
likelihood_sigma,
use_cache=True):
ls = np.array([length_scale])
interior_points = observations[0][0]
oc = self.get_operator_system(theta, interior_points, use_cache)
posterior = bpdes.collocate(oc.operators,
oc.operators_bar,
observations,
oc,
fun_args=ls)
true_x, true_u = truth
mu, cov = posterior(true_x)
mu = mu.reshape((len(mu), 1))
cov_with_error = cov + likelihood_sigma**2 * np.eye(cov.shape[0])
return stats.multivariate_normal.logpdf(mu.ravel(), true_u.ravel(),
cov_with_error)
