def run():
# Create some data
N = 500
D = 2
# Initial state
x0 = np.array([0.5, -0.5])
# Dynamics (time varying)
A0 = np.array([[.9, -.4], [.4, .9]])
A1 = np.array([[.98, -.1], [.1, .98]])
l = np.linspace(0, 1, N).reshape((-1, 1, 1))
A = (1 - l) * A0 + l * A1
# Innovation covariance matrix
V = np.array([[1, 0], [0, 1]])
# Observation noise covariance matrix
C0 = 10 * np.array([[1, 0], [0, 1]])
C1 = 0.01 * np.array([[1, 0], [0, 1]])
C = (1 - l)**2 * C0 + l**2 * C1
X = np.empty((N, D))
Y = np.empty((N, D))
# Simulate data
x = x0
for n in range(N):
x = np.dot(A[n, :, :], x) + np.random.multivariate_normal(
np.zeros(D), V)
X[n, :] = x
Y[n, :] = x + np.random.multivariate_normal(np.zeros(D), C[n, :, :])
U = np.empty((N, D, D))
UY = np.empty((N, D))
for n in range(N):
U[n, :, :] = np.linalg.inv(C[n, :, :])
UY[n, :] = np.linalg.solve(C[n, :, :], Y[n, :])
# Create iterators for the static matrices
#U = N*(U,)
#A = N*(A,)
V = N * (V, )
(Xh, CovXh) = utils.kalman_filter(UY, U, A, V, np.zeros(2),
10 * np.identity(2))
(Xh, CovXh) = utils.rts_smoother(Xh, CovXh, A, V)
plt.clf()
for d in range(D):
plt.subplot(D, 1, d)
#plt.plot(Xh[:,d], 'r--')
bpplt.errorplot(Xh[:, d], error=2 * np.sqrt(CovXh[:, d, d]))
plt.plot(X[:, d], 'r-')
plt.plot(Y[:, d], '.')
plt.show()
def run():
# Create some data
N = 500
D = 2
# Initial state
x0 = np.array([0.5, -0.5])
# Dynamics (time varying)
A0 = np.array([[.9, -.4], [.4, .9]])
A1 = np.array([[.98, -.1], [.1, .98]])
l = np.linspace(0, 1, N).reshape((-1,1,1))
A = (1-l)*A0 + l*A1
# Innovation covariance matrix
V = np.array([[1, 0], [0, 1]])
# Observation noise covariance matrix
C0 = 10*np.array([[1, 0], [0, 1]])
C1 = 0.01*np.array([[1, 0], [0, 1]])
C = (1-l)**2*C0 + l**2*C1
X = np.empty((N,D))
Y = np.empty((N,D))
# Simulate data
x = x0
for n in range(N):
x = np.dot(A[n,:,:],x) + np.random.multivariate_normal(np.zeros(D), V)
X[n,:] = x
Y[n,:] = x + np.random.multivariate_normal(np.zeros(D), C[n,:,:])
U = np.empty((N,D,D))
UY = np.empty((N,D))
for n in range(N):
U[n,:,:] = np.linalg.inv(C[n,:,:])
UY[n,:] = np.linalg.solve(C[n,:,:], Y[n,:])
# Create iterators for the static matrices
#U = N*(U,)
#A = N*(A,)
V = N*(V,)
(Xh, CovXh) = utils.kalman_filter(UY, U, A, V, np.zeros(2), 10*np.identity(2))
(Xh, CovXh) = utils.rts_smoother(Xh, CovXh, A, V)
plt.clf()
for d in range(D):
plt.subplot(D,1,d)
#plt.plot(Xh[:,d], 'r--')
bpplt.errorplot(Xh[:,d], error=2*np.sqrt(CovXh[:,d,d]))
plt.plot(X[:,d], 'r-')
plt.plot(Y[:,d], '.')
plt.show()
def test_smoothing(self):
"""
Test the posterior estimation of GaussianMarkovChain.
Create time-variant dynamics and compare the results of BayesPy VB
inference and standard Kalman filtering & smoothing.
This is not that useful anymore, because the moments are checked much
better in another test method.
"""
#
# Set up an artificial system
#
# Dimensions
N = 500
D = 2
# Dynamics (time varying)
A0 = np.array([[0.9, -0.4], [0.4, 0.9]])
A1 = np.array([[0.98, -0.1], [0.1, 0.98]])
l = np.linspace(0, 1, N - 1).reshape((-1, 1, 1))
A = (1 - l) * A0 + l * A1
# Innovation covariance matrix (time varying)
v = np.random.rand(D)
V = np.diag(v)
# Observation noise covariance matrix
C = np.identity(D)
#
# Simulate data
#
X = np.empty((N, D))
Y = np.empty((N, D))
x = np.array([0.5, -0.5])
X[0, :] = x
Y[0, :] = x + np.random.multivariate_normal(np.zeros(D), C)
for n in range(N - 1):
x = np.dot(A[n, :, :], x) + np.random.multivariate_normal(np.zeros(D), V)
X[n + 1, :] = x
Y[n + 1, :] = x + np.random.multivariate_normal(np.zeros(D), C)
#
# BayesPy inference
#
# Construct VB model
Xh = GaussianMarkovChain(np.zeros(D), np.identity(D), A, 1 / v, n=N)
Yh = Gaussian(Xh, np.identity(D), plates=(N,))
# Put data
Yh.observe(Y)
# Run inference
Xh.update()
# Store results
Xh_vb = Xh.u[0]
CovXh_vb = Xh.u[1] - Xh_vb[..., np.newaxis, :] * Xh_vb[..., :, np.newaxis]
#
# "The ground truth" using standard Kalman filter and RTS smoother
#
V = N * (V,)
UY = Y
U = N * (C,)
(Xh, CovXh) = utils.kalman_filter(UY, U, A, V, np.zeros(D), np.identity(D))
(Xh, CovXh) = utils.rts_smoother(Xh, CovXh, A, V)
#
# Check results
#
self.assertTrue(np.allclose(Xh_vb, Xh))
self.assertTrue(np.allclose(CovXh_vb, CovXh))
def test_smoothing(self):
"""
Test the posterior estimation of GaussianMarkovChain.
Create time-variant dynamics and compare the results of BayesPy VB
inference and standard Kalman filtering & smoothing.
This is not that useful anymore, because the moments are checked much
better in another test method.
"""
#
# Set up an artificial system
#
# Dimensions
N = 500
D = 2
# Dynamics (time varying)
A0 = np.array([[.9, -.4], [.4, .9]])
A1 = np.array([[.98, -.1], [.1, .98]])
l = np.linspace(0, 1, N - 1).reshape((-1, 1, 1))
A = (1 - l) * A0 + l * A1
# Innovation covariance matrix (time varying)
v = np.random.rand(D)
V = np.diag(v)
# Observation noise covariance matrix
C = np.identity(D)
#
# Simulate data
#
X = np.empty((N, D))
Y = np.empty((N, D))
x = np.array([0.5, -0.5])
X[0, :] = x
Y[0, :] = x + np.random.multivariate_normal(np.zeros(D), C)
for n in range(N - 1):
x = np.dot(A[n, :, :], x) + np.random.multivariate_normal(
np.zeros(D), V)
X[n + 1, :] = x
Y[n + 1, :] = x + np.random.multivariate_normal(np.zeros(D), C)
#
# BayesPy inference
#
# Construct VB model
Xh = GaussianMarkovChain(np.zeros(D), np.identity(D), A, 1 / v, n=N)
Yh = Gaussian(Xh, np.identity(D), plates=(N, ))
# Put data
Yh.observe(Y)
# Run inference
Xh.update()
# Store results
Xh_vb = Xh.u[0]
CovXh_vb = Xh.u[1] - Xh_vb[..., np.newaxis, :] * Xh_vb[..., :,
np.newaxis]
#
# "The ground truth" using standard Kalman filter and RTS smoother
#
V = N * (V, )
UY = Y
U = N * (C, )
(Xh, CovXh) = utils.kalman_filter(UY, U, A, V, np.zeros(D),
np.identity(D))
(Xh, CovXh) = utils.rts_smoother(Xh, CovXh, A, V)
#
# Check results
#
self.assertTrue(np.allclose(Xh_vb, Xh))
self.assertTrue(np.allclose(CovXh_vb, CovXh))