return np.zeros(x.shape[0])
def covariance(xx):
xx = np.atleast_2d(xx)
dd = xx[:, :1] - xx[:, 1:]
ll = np.atleast_2d(integral_range)
return np.exp(- np.sum((dd / ll) ** 2., axis=1))
# Discretization of the random field using Karhunen-Loeve expansion
# from its given theoretical moments
random_field = KarhunenLoeveExpansion(
mean,
covariance,
truncation_order,
[lower_bound, upper_bound],
domain_expand_factor=domain_expand_factor,
verbose=verbose,
galerkin_scheme=galerkin_scheme,
legendre_galerkin_order=legendre_galerkin_order,
legendre_quadrature_order=legendre_quadrature_order)
truncation_order = random_field._truncation_order
# Plot the relative covariance discretization error
res = 50
x1, x2 = np.meshgrid(np.linspace(lower_bound, upper_bound, res),
np.linspace(lower_bound, upper_bound, res))
xx = np.vstack([x1.ravel(), x2.ravel()]).T
approximated_covariance = \
random_field.compute_approximated_covariance(xx)
true_covariance = covariance(xx)
covariance_error = true_covariance - approximated_covariance
xx = np.atleast_2d(xx)
y1 = np.vstack([sample_paths[i](xx[:, :2])
for i in xrange(n_sample_paths)])
y2 = np.vstack([sample_paths[i](xx[:, 2:])
for i in xrange(n_sample_paths)])
cov = np.sum((y1 - y1.mean(axis=0)) * (y2 - y2.mean(axis=0)),
axis=0) / (n_sample_paths - 1.)
return cov
# Discretization of the random field using Karhunen-Loeve expansion
# from its estimated theoretical moments
estimated_random_field = KarhunenLoeveExpansion(
estimated_mean,
estimated_covariance,
truncation_order,
[lower_bound, upper_bound],
domain_expand_factor=1.,
verbose=verbose,
galerkin_scheme=galerkin_scheme,
legendre_galerkin_order=legendre_galerkin_order,
legendre_quadrature_order=legendre_quadrature_order)
truncation_order = estimated_random_field._truncation_order
# Plot eigenvalues and eigenfunctions
for i in xrange(truncation_order):
fig = pl.figure()
ax = Axes3D(fig)
pl.title('Eigensolution \#%d ($\lambda_{%d} = %.2f$)'
% (i, i, estimated_random_field._eigenvalues[i]))
ax.plot_surface(x1, x2,
np.reshape(estimated_random_field._eigenfunctions[i](xx),
(res, res)),
y1 = np.vstack(
[sample_paths[i](xx[:, :2]) for i in xrange(n_sample_paths)])
y2 = np.vstack(
[sample_paths[i](xx[:, 2:]) for i in xrange(n_sample_paths)])
cov = np.sum((y1 - y1.mean(axis=0)) *
(y2 - y2.mean(axis=0)), axis=0) / (n_sample_paths - 1.)
return cov
# Discretization of the random field using Karhunen-Loeve expansion
# from its estimated theoretical moments
estimated_random_field = KarhunenLoeveExpansion(
estimated_mean,
estimated_covariance,
truncation_order, [lower_bound, upper_bound],
domain_expand_factor=1.,
verbose=verbose,
galerkin_scheme=galerkin_scheme,
legendre_galerkin_order=legendre_galerkin_order,
legendre_quadrature_order=legendre_quadrature_order)
truncation_order = estimated_random_field._truncation_order
# Plot eigenvalues and eigenfunctions
for i in xrange(truncation_order):
fig = pl.figure()
ax = Axes3D(fig)
pl.title('Eigensolution \#%d ($\lambda_{%d} = %.2f$)' %
(i, i, estimated_random_field._eigenvalues[i]))
ax.plot_surface(x1,
x2,
np.reshape(estimated_random_field._eigenfunctions[i](xx),
return np.zeros(x.shape[0])
def covariance(xx):
xx = np.atleast_2d(xx)
dd = np.abs(xx[:, :2] - xx[:, 2:])
ll = np.atleast_2d(integral_range)
return np.exp(-np.sum((dd / ll)**2., axis=1))
# Discretization of the random field using Karhunen-Loeve expansion
random_field = KarhunenLoeveExpansion(
mean,
covariance,
truncation_order, [lower_bound, upper_bound],
domain_expand_factor=domain_expand_factor,
verbose=verbose,
galerkin_scheme=galerkin_scheme,
legendre_galerkin_order=legendre_galerkin_order,
legendre_quadrature_order=legendre_quadrature_order)
truncation_order = random_field._truncation_order
# Plot the relative variance discretization error
res = 50
x1, x2 = np.meshgrid(np.linspace(lower_bound[0], upper_bound[0], res),
np.linspace(lower_bound[1], upper_bound[1], res))
xx = np.vstack([x1.ravel(), x2.ravel()]).T
xxxx = np.hstack([xx, xx])
approximated_variance = random_field.compute_approximated_covariance(xxxx)
variance = covariance(xxxx)
variance_error = variance - approximated_variance
