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_gaussian_random_field(self):
"""
Reproduce statistics of Gaussian random field
"""
#
# Define Gaussian Field with degenerate support
#
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, 1], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
mu = np.array([1, 2, 3, 4])[:, None]
# Zero mean Gaussian field
dim = 4
eta = GaussianField(dim, mean=mu, K=K, mode='covariance')
n_vars = eta.covariance().size()
level = 1
# Define Gauss-Hermite physicist's rule exp(-x**2)
grid = Tasmanian.makeGlobalGrid(n_vars, 4, level, "level",
"gauss-hermite-odd")
# Evaluate the Gaussian random field at the Gauss points
z = grid.getPoints()
y = np.sqrt(2) * z
const_norm = np.sqrt(np.pi)**n_vars
# Evaluate the random field at the Gauss points
w = grid.getQuadratureWeights()
etay = eta.sample(z=y.T)
n = grid.getNumPoints()
I = np.zeros(4)
II = np.zeros((4, 4))
for i in range(n):
II += w[i] * np.outer(etay[:, i] - mu.ravel(),
etay[:, i] - mu.ravel())
I += w[i] * etay[:, i]
I /= const_norm
II /= const_norm
self.assertTrue(np.allclose(II, K))
self.assertTrue(np.allclose(I, mu.ravel()))
def test_constructor(self):
#
# Initialize Gaussian Random Field
#
n = 21 # 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 SPD matrices
KK = SPDMatrix(K)
QQ = SPDMatrix(Q)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K)
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Check vitals
self.assertTrue(np.allclose(u_cov.covariance().get_matrix(),\
KK.get_matrix()))
self.assertEqual(u_cov.size(), n - 1)
self.assertTrue(np.allclose(u_prc.precision().get_matrix(), Q))
self.assertTrue(np.allclose(u_cov.mean(), mean))
self.assertEqual(u_prc.mean(n_copies=2).shape[1], 2)
self.assertTrue(np.allclose(u_prc.b(), QQ.dot(u_prc.mean())))
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)
"""