def test_chol_sample(self):
"""
Sample field using Cholesky factorization of the covariance and of
the precision.
"""
#
# Initialize Gaussian Random Field
#
n = 201 # size
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n - 1)
Q = linalg.solve(K, I)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K, mode='covariance')
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Define generating white noise
z = u_cov.iid_gauss(n_samples=10)
u_chol_prec = u_prc.sample(z=z, mode='precision', decomposition='chol')
u_chol_cov = u_cov.sample(z=z, mode='covariance', decomposition='eig')
u_chol_can = u_prc.sample(z=z, mode='canonical', decomposition='chol')
fig, ax = plt.subplots(1, 3, figsize=(7, 3))
ax[0].plot(u_chol_prec, linewidth=0.5)
ax[0].set_title('Precision')
ax[0].axis('tight')
ax[1].plot(u_chol_cov, linewidth=0.5)
ax[1].set_title('Covariance')
ax[1].axis('tight')
ax[2].plot(u_chol_can, linewidth=0.5)
ax[2].set_title('Canonical')
ax[2].axis('tight')
fig.suptitle('Samples')
fig.tight_layout()
fig.subplots_adjust(top=0.8)
fig.savefig('gaussian_field_chol_samples.eps')
def test_eig_condition(self):
"""
Conditioning using Eigen-decomposition
"""
oort = 1 / np.sqrt(2)
V = np.array([[0.5, oort, 0, 0.5], [0.5, 0, -oort, -0.5],
[0.5, -oort, 0, 0.5], [0.5, 0, oort, -0.5]])
# Eigenvalues
d = np.array([4, 3, 2, 0], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
KK = SPDMatrix(K)
# Mean
mu = np.linspace(0, 3, 4)
# Transformation
A = np.array([[1, 2, 3, 4], [2, 4, 6, 8]], dtype=float)
N = V[:, np.abs(d) < 1e-13]
for i in range(A.shape[0]):
ai = A[i].T
vi = ai - N.dot(N.T.dot(ai))
if linalg.norm(vi) > 1e-7:
vi = vi / linalg.norm(vi)
N = np.append(N, vi[:, None], axis=1)
#print(vi)
#print(N)
A = np.array([[1, 0, 0, -1]])
# Check compatibility
P_A = A.T.dot(linalg.solve(A.dot(A.T), A))
#print((1-P_A).dot(0))
#print(A.T - N.dot(N.T.dot(A.T)))
X = GaussianField(4, mean=mu, K=K)
z = X.iid_gauss(n_samples=100)
plt.close('all')
fig = plt.figure()
ax = fig.add_subplot(111)
def test_degenerate_sample(self):
"""
Test support and reduced covariance
"""
oort = 1 / np.sqrt(2)
V = np.array([[0.5, oort, 0, 0.5], [0.5, 0, -oort, -0.5],
[0.5, -oort, 0, 0.5], [0.5, 0, oort, -0.5]])
# Eigenvalues
d = np.array([4, 3, 2, 0], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
# Zero mean Gaussian field
u_ex = GaussianField(4, K=K, mode='covariance', support=V[:, 0:3])
u_im = GaussianField(4, K=K, mode='covariance')
u_im.update_support()
# Check reduced covariances
self.assertTrue(
np.allclose(u_ex.covariance().get_matrix(),
u_im.covariance().get_matrix().toarray()))
# Check supports
V_ex = u_ex.support()
V_im = u_im.support()
# Ensure they have the same sign
for i in range(V_ex.shape[1]):
if V_ex[0, i] < 0:
V_ex[:, i] = -V_ex[:, i]
if V_im[0, i] < 0:
V_im[:, i] = -V_im[:, i]
self.assertTrue(np.allclose(V_ex, V_im))
u_ex.set_support(V_ex)
u_im.set_support(V_im)
# Compare samples
z = u_ex.iid_gauss(n_samples=1)
u_ex_smp = u_ex.sample(z=z, decomposition='chol')
u_im_smp = u_im.sample(z=z, decomposition='chol')
self.assertTrue(np.allclose(u_ex_smp, u_im_smp))
def test_condition_pointswise(self):
"""
Generate samples and random field by conditioning on pointwise data
"""
#
# Initialize Gaussian Random Field
#
# Resolution
max_res = 10
n = 2**max_res + 1 # size
# Hurst parameter
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n - 1)
Q = linalg.solve(K, I)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K, mode='covariance')
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Define generating white noise
z = u_cov.iid_gauss(n_samples=10)
u_obs = u_cov.sample(z=z)
# Index of measured observations
A = np.arange(0, n - 1, 2)
# observed quantities
e = u_obs[A, 0][:, None]
#print('e shape', e.shape)
# change A into matrix
k = len(A)
rows = np.arange(k)
cols = A
vals = np.ones(k)
AA = sp.coo_matrix((vals, (rows, cols)), shape=(k, n - 1)).toarray()
AKAt = AA.dot(K.dot(AA.T))
KAt = K.dot(AA.T)
U, S, Vt = linalg.svd(AA)
#print(U)
#print(S)
#print(Vt)
#print(AA.dot(u_obs)-e)
k = e.shape[0]
Ko = 0.01 * np.identity(k)
# Debug
K = u_cov.covariance()
#U_spp = u_cov.support()
#A_cmp = A.dot(U_spp)
u_cond = u_cov.condition(A, e, Ko=Ko, n_samples=100)
"""