def run(M=10,
N=100,
D_y=3,
D=5,
seed=42,
rotate=False,
maxiter=100,
debug=False,
plot=True):
if seed is not None:
np.random.seed(seed)
# Generate data
w = np.random.normal(0, 1, size=(M, 1, D_y))
x = np.random.normal(0, 1, size=(1, N, D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.2, size=(M, N))
# Construct model
(Y, F, W, X, tau, alpha) = model(M, N, D)
# Data with missing values
mask = random.mask(M, N, p=0.5) # randomly missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha)
# Initialize some nodes randomly
X.initialize_from_random()
W.initialize_from_random()
# Run inference algorithm
if rotate:
# Use rotations to speed up learning
rotW = transformations.RotateGaussianARD(W, alpha)
rotX = transformations.RotateGaussianARD(X)
R = transformations.RotationOptimizer(rotW, rotX, D)
for ind in range(maxiter):
Q.update()
if debug:
R.rotate(check_bound=True, check_gradient=True)
else:
R.rotate()
else:
# Use standard VB-EM alone
Q.update(repeat=maxiter)
# Plot results
if plot:
plt.figure()
bpplt.timeseries_normal(F, scale=2)
bpplt.timeseries(f, color='g', linestyle='-')
bpplt.timeseries(y, color='r', linestyle='None', marker='+')
def run(M=10, N=100, D_y=3, D=5, seed=42, rotate=False, maxiter=100, debug=False):
if seed is not None:
np.random.seed(seed)
# Generate data
w = np.random.normal(0, 1, size=(M,1,D_y))
x = np.random.normal(0, 1, size=(1,N,D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.2, size=(M,N))
# Construct model
(Y, F, W, X, tau, alpha) = model(M, N, D)
# Data with missing values
mask = random.mask(M, N, p=0.5) # randomly missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha)
# Initialize some nodes randomly
X.initialize_from_random()
W.initialize_from_random()
# Run inference algorithm
if rotate:
# Use rotations to speed up learning
rotW = transformations.RotateGaussianARD(W, alpha)
rotX = transformations.RotateGaussianARD(X)
R = transformations.RotationOptimizer(rotW, rotX, D)
for ind in range(maxiter):
Q.update()
if debug:
R.rotate(check_bound=True,
check_gradient=True)
else:
R.rotate()
else:
# Use standard VB-EM alone
Q.update(repeat=maxiter)
# Plot results
plt.figure()
bpplt.timeseries_normal(F, scale=2)
bpplt.timeseries(f, color='g', linestyle='-')
bpplt.timeseries(y, color='r', linestyle='None', marker='+')
plt.show()
def run(M=10, N=100, D_y=3, D=5):
seed = 45
print('seed =', seed)
np.random.seed(seed)
# Generate data
w = np.random.normal(0, 1, size=(M,1,D_y))
x = np.random.normal(0, 1, size=(1,N,D_y))
f = utils.utils.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.5, size=(M,N))
# Construct model
(Y, WX, W, X, tau, alpha) = pca_model(M, N, D)
# Data with missing values
mask = utils.random.mask(M, N, p=0.9) # randomly missing
mask[:,20:40] = False # gap missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha)
# Initialize some nodes randomly
X.initialize_from_value(X.random())
W.initialize_from_value(W.random())
# Inference loop.
Q.update(repeat=100)
# Plot results
plt.clf()
WX_params = WX.get_parameters()
fh = WX_params[0] * np.ones(y.shape)
err_fh = 2*np.sqrt(WX_params[1] + 1/tau.get_moments()[0]) * np.ones(y.shape)
for m in range(M):
plt.subplot(M,1,m+1)
myplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
plt.plot(np.arange(N), f[m], 'g')
plt.plot(np.arange(N), y[m], 'r+')
plt.show()
def pca():
np.random.seed(41)
M = 10
N = 3000
D = 5
# Construct the PCA model
alpha = Gamma(1e-3, 1e-3, plates=(D, ), name='alpha')
W = GaussianARD(0, alpha, plates=(M, 1), shape=(D, ), name='W')
X = GaussianARD(0, 1, plates=(1, N), shape=(D, ), name='X')
tau = Gamma(1e-3, 1e-3, name='tau')
W.initialize_from_random()
F = SumMultiply('d,d->', W, X)
Y = GaussianARD(F, tau, name='Y')
# Observe data
data = np.sum(np.random.randn(M, 1, D - 1) * np.random.randn(1, N, D - 1),
axis=-1) + 1e-1 * np.random.randn(M, N)
Y.observe(data)
# Initialize VB engine
Q = VB(Y, X, W, alpha, tau)
# Take one update step (so phi is ok)
Q.update(repeat=1)
Q.save()
# Run VB-EM
Q.update(repeat=200)
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
# Restore the state
Q.load()
# Run Riemannian conjugate gradient
#Q.optimize(X, alpha, maxiter=100, collapsed=[W, tau])
Q.optimize(W, tau, maxiter=100, collapsed=[X, alpha])
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.show()
def run(M=30, D=5):
# Generate data
y = np.random.randint(D, size=(M, ))
# Construct model
p = nodes.Dirichlet(1 * np.ones(D), name='p')
z = nodes.Categorical(p, plates=(M, ), name='z')
# Observe the data with randomly missing values
mask = random.mask(M, p=0.5)
z.observe(y, mask=mask)
# Run VB-EM
Q = VB(p, z)
Q.update()
# Show results
z.show()
p.show()
def pca():
np.random.seed(41)
M = 10
N = 3000
D = 5
# Construct the PCA model
alpha = Gamma(1e-3, 1e-3, plates=(D,), name='alpha')
W = GaussianARD(0, alpha, plates=(M,1), shape=(D,), name='W')
X = GaussianARD(0, 1, plates=(1,N), shape=(D,), name='X')
tau = Gamma(1e-3, 1e-3, name='tau')
W.initialize_from_random()
F = SumMultiply('d,d->', W, X)
Y = GaussianARD(F, tau, name='Y')
# Observe data
data = np.sum(np.random.randn(M,1,D-1) * np.random.randn(1,N,D-1), axis=-1) + 1e-1 * np.random.randn(M,N)
Y.observe(data)
# Initialize VB engine
Q = VB(Y, X, W, alpha, tau)
# Take one update step (so phi is ok)
Q.update(repeat=1)
Q.save()
# Run VB-EM
Q.update(repeat=200)
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
# Restore the state
Q.load()
# Run Riemannian conjugate gradient
#Q.optimize(X, alpha, maxiter=100, collapsed=[W, tau])
Q.optimize(W, tau, maxiter=100, collapsed=[X, alpha])
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.show()
def run(M=30, D=5):
# Generate data
y = np.random.randint(D, size=(M,))
# Construct model
p = nodes.Dirichlet(1*np.ones(D),
name='p')
z = nodes.Categorical(p,
plates=(M,),
name='z')
# Observe the data with randomly missing values
mask = random.mask(M, p=0.5)
z.observe(y, mask=mask)
# Run VB-EM
Q = VB(p, z)
Q.update()
# Show results
z.show()
p.show()
def run(M=40, N=100, D_y=6, D=8, seed=42, rotate=False, maxiter=1000, debug=False, plot=True):
"""
Run pattern search demo for PCA.
"""
if seed is not None:
np.random.seed(seed)
# Generate data
w = np.random.normal(0, 1, size=(M,1,D_y))
x = np.random.normal(0, 1, size=(1,N,D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.2, size=(M,N))
# Construct model
Q = VB(*(pca.model(M, N, D)))
# Data with missing values
mask = random.mask(M, N, p=0.5) # randomly missing
y[~mask] = np.nan
Q['Y'].observe(y, mask=mask)
# Initialize some nodes randomly
Q['X'].initialize_from_random()
Q['W'].initialize_from_random()
# Use a few VB-EM updates at the beginning
Q.update(repeat=10)
Q.save()
# Standard VB-EM as a baseline
Q.update(repeat=maxiter)
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
# Restore initial state
Q.load()
# Pattern search method for comparison
for n in range(maxiter):
Q.pattern_search('W', 'tau', maxiter=3, collapsed=['X', 'alpha'])
Q.update(repeat=20)
if Q.has_converged():
break
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.xlabel('CPU time (in seconds)')
bpplt.pyplot.ylabel('VB lower bound')
bpplt.pyplot.legend(['VB-EM', 'Pattern search'], loc='lower right')
def run(M=50, N=200, D_y=10, D=20, maxiter=100):
seed = 45
print("seed =", seed)
np.random.seed(seed)
# Generate data (covariance eigenvalues: 1,1,...,1,2^2,3^2,...,(D_y+1)^2
(q, r) = scipy.linalg.qr(np.random.randn(M, M))
C = np.diag(np.arange(2, 2 + D_y))
C = np.ones(M)
C[:D_y] += np.arange(1, 1 + D_y)
y = C[:, np.newaxis] * np.random.randn(M, N)
y = np.dot(q, y)
# Construct model
(Y, WX, W, X, tau, alpha) = pca_model(M, N, D)
# Data with missing values
mask = utils.random.mask(M, N, p=0.9) # randomly missing
mask[:, 20:40] = False # gap missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha, autosave_filename=utils.utils.tempfile())
# Initialize nodes (from prior and randomly)
X.initialize_from_value(X.random())
W.initialize_from_value(W.random())
Q.update(repeat=1)
Q.save()
#
# Run inference with rotations.
#
rotX = transformations.RotateGaussian(X)
rotW = transformations.RotateGaussianARD(W, alpha)
R = transformations.RotationOptimizer(rotX, rotW, D)
for ind in range(maxiter):
Q.update()
R.rotate(
check_gradient=False,
maxiter=10,
verbose=False,
check_bound=Q.compute_lowerbound,
check_bound_terms=Q.compute_lowerbound_terms,
)
L_rot = Q.L
#
# Re-run inference without rotations.
#
Q.load()
Q.update(repeat=maxiter)
L_norot = Q.L
#
# Plot comparison
#
plt.plot(L_rot)
plt.plot(L_norot)
plt.legend(["With rotations", "Without rotations"], loc="lower right")
plt.show()
def run(maxiter=100):
seed = 496#np.random.randint(1000)
print("seed = ", seed)
np.random.seed(seed)
# Simulate some data
D = 3
M = 6
N = 200
c = np.random.randn(M,D)
w = 0.3
a = np.array([[np.cos(w), -np.sin(w), 0],
[np.sin(w), np.cos(w), 0],
[0, 0, 1]])
x = np.empty((N,D))
f = np.empty((M,N))
y = np.empty((M,N))
x[0] = 10*np.random.randn(D)
f[:,0] = np.dot(c,x[0])
y[:,0] = f[:,0] + 3*np.random.randn(M)
for n in range(N-1):
x[n+1] = np.dot(a,x[n]) + np.random.randn(D)
f[:,n+1] = np.dot(c,x[n+1])
y[:,n+1] = f[:,n+1] + 3*np.random.randn(M)
# Create the model
(Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D, N, M)
# Add missing values randomly
mask = random.mask(M, N, p=0.3)
# Add missing values to a period of time
mask[:,30:80] = False
y[~mask] = np.nan # BayesPy doesn't require this. Just for plotting.
# Observe the data
Y.observe(y, mask=mask)
# Initialize nodes (must use some randomness for C)
C.initialize_from_random()
# Run inference
Q = VB(Y, X, C, gamma, A, alpha, tau)
#
# Run inference with rotations.
#
rotA = transformations.RotateGaussianARD(A, alpha)
rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
rotC = transformations.RotateGaussianARD(C, gamma)
R = transformations.RotationOptimizer(rotX, rotC, D)
#maxiter = 84
for ind in range(maxiter):
Q.update()
#print('C term', C.lower_bound_contribution())
R.rotate(maxiter=10,
check_gradient=True,
verbose=False,
check_bound=Q.compute_lowerbound,
#check_bound=None,
check_bound_terms=Q.compute_lowerbound_terms)
#check_bound_terms=None)
X_vb = X.u[0]
varX_vb = utils.diagonal(X.u[1] - X_vb[...,np.newaxis,:] * X_vb[...,:,np.newaxis])
u_CX = CX.get_moments()
CX_vb = u_CX[0]
varCX_vb = u_CX[1] - CX_vb**2
# Show results
plt.figure(3)
plt.clf()
for m in range(M):
plt.subplot(M,1,m+1)
plt.plot(y[m,:], 'r.')
plt.plot(f[m,:], 'b-')
bpplt.errorplot(y=CX_vb[m,:],
error=2*np.sqrt(varCX_vb[m,:]))
plt.figure()
Q.plot_iteration_by_nodes()
def run(N=100000, N_batch=50, seed=42, maxiter=100, plot=True):
"""
Run deterministic annealing demo for 1-D Gaussian mixture.
"""
if seed is not None:
np.random.seed(seed)
# Number of clusters in the model
K = 20
# Dimensionality of the data
D = 5
# Generate data
K_true = 10
spread = 5
means = spread * np.random.randn(K_true, D)
z = random.categorical(np.ones(K_true), size=N)
data = np.empty((N,D))
for n in range(N):
data[n] = means[z[n]] + np.random.randn(D)
#
# Standard VB-EM algorithm
#
# Full model
mu = Gaussian(np.zeros(D), np.identity(D),
plates=(K,),
name='means')
alpha = Dirichlet(np.ones(K),
name='class probabilities')
Z = Categorical(alpha,
plates=(N,),
name='classes')
Y = Mixture(Z, Gaussian, mu, np.identity(D),
name='observations')
# Break symmetry with random initialization of the means
mu.initialize_from_random()
# Put the data in
Y.observe(data)
# Run inference
Q = VB(Y, Z, mu, alpha)
Q.save(mu)
Q.update(repeat=maxiter)
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
max_cputime = np.sum(Q.cputime[~np.isnan(Q.cputime)])
#
# Stochastic variational inference
#
# Construct smaller model (size of the mini-batch)
mu = Gaussian(np.zeros(D), np.identity(D),
plates=(K,),
name='means')
alpha = Dirichlet(np.ones(K),
name='class probabilities')
Z = Categorical(alpha,
plates=(N_batch,),
plates_multiplier=(N/N_batch,),
name='classes')
Y = Mixture(Z, Gaussian, mu, np.identity(D),
name='observations')
# Break symmetry with random initialization of the means
mu.initialize_from_random()
# Inference engine
Q = VB(Y, Z, mu, alpha, autosave_filename=Q.autosave_filename)
Q.load(mu)
# Because using mini-batches, messages need to be multiplied appropriately
print("Stochastic variational inference...")
Q.ignore_bound_checks = True
maxiter *= int(N/N_batch)
delay = 1
forgetting_rate = 0.7
for n in range(maxiter):
# Observe a mini-batch
subset = np.random.choice(N, N_batch)
Y.observe(data[subset,:])
# Learn intermediate variables
Q.update(Z)
# Set step length
step = (n + delay) ** (-forgetting_rate)
# Stochastic gradient for the global variables
Q.gradient_step(mu, alpha, scale=step)
if np.sum(Q.cputime[:n]) > max_cputime:
break
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.xlabel('CPU time (in seconds)')
bpplt.pyplot.ylabel('VB lower bound')
bpplt.pyplot.legend(['VB-EM', 'Stochastic inference'], loc='lower right')
bpplt.pyplot.title('VB for Gaussian mixture model')
return
def run_dlssm(y, f, mask, D, K, maxiter):
"""
Run VB inference for linear state space model with drifting dynamics.
"""
(M, N) = np.shape(y)
# Dynamics matrix with ARD
# alpha : (D) x ()
alpha = Gamma(1e-5,
1e-5,
plates=(K,),
name='alpha')
# A : (K) x (K)
A = Gaussian(np.zeros(K),
#np.identity(K),
diagonal(alpha),
plates=(K,),
name='A_S')
A.initialize_from_value(np.identity(K))
# rho
## rho = Gamma(1e-5,
## 1e-5,
## plates=(K,),
## name="rho")
# S : () x (N-1,K)
S = GaussianMarkovChain(np.ones(K),
1e-6*np.identity(K),
A,
np.ones(K),
n=N-1,
name='S')
S.initialize_from_value(1*np.ones((N-1,K)))
# Projection matrix of the dynamics matrix
# beta : (K) x ()
beta = Gamma(1e-5,
1e-5,
plates=(K,),
name='beta')
# B : (D) x (D*K)
B = Gaussian(np.zeros(D*K),
diagonal(tile(beta, D)),
plates=(D,),
name='B')
b = np.zeros((D,D,K))
b[np.arange(D),np.arange(D),np.zeros(D,dtype=int)] = 1
B.initialize_from_value(np.reshape(1*b, (D,D*K)))
# A : (N-1,D) x (D)
BS = MatrixDot(B,
S.as_gaussian().add_plate_axis(-1),
name='BS')
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(np.zeros(D), # mean of x0
1e-3*np.identity(D), # prec of x0
BS, # dynamics
np.ones(D), # innovation
n=N, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N,D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5,
1e-5,
plates=(D,),
name='gamma')
# C : (M,1) x (D)
C = Gaussian(np.zeros(D),
diagonal(gamma),
plates=(M,1),
name='C')
C.initialize_from_value(np.random.randn(M,1,D))
# Observation noise
# tau : () x ()
tau = Gamma(1e-5,
1e-5,
name='tau')
# Observations
# Y : (M,N) x ()
CX = Dot(C, X.as_gaussian())
Y = Normal(CX,
tau,
name='Y')
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)
#
# Run inference with rotations.
#
# Rotate the D-dimensional state space (C, X)
rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='rows')
rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
rotC = transformations.RotateGaussianARD(C, gamma)
R_X = transformations.RotationOptimizer(rotX, rotC, D)
# Rotate the K-dimensional latent dynamics space (B, S)
rotA = transformations.RotateGaussianARD(A, alpha)
rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='cols')
R_S = transformations.RotationOptimizer(rotS, rotB, K)
# Iterate
for ind in range(int(maxiter/5)):
Q.update(repeat=5)
#Q.update(X, S, A, alpha, rho, B, beta, C, gamma, tau, repeat=maxiter)
R_X.rotate()
R_S.rotate()
## R_X.rotate(
## check_bound=Q.compute_lowerbound,
## check_bound_terms=Q.compute_lowerbound_terms,
## check_gradient=True
## )
## R_S.rotate(
## check_bound=Q.compute_lowerbound,
## check_bound_terms=Q.compute_lowerbound_terms,
## check_gradient=True
## )
#
# SHOW RESULTS
#
# Mean and standard deviation of the posterior
(f_mean, f_squared) = CX.get_moments()
f_std = np.sqrt(f_squared - f_mean**2)
# Plot observations space
for m in range(M):
plt.subplot(M,1,m+1)
plt.plot(y[m,:], 'r.')
plt.plot(f[m,:], 'b-')
bpplt.errorplot(y=f_mean[m,:], error=2*f_std[m,:])
def run_dlssm(y, f, mask, D, K, maxiter):
"""
Run VB inference for linear state space model with drifting dynamics.
"""
(M, N) = np.shape(y)
# Dynamics matrix with ARD
# alpha : (K) x ()
alpha = Gamma(1e-5,
1e-5,
plates=(K,),
name='alpha')
# A : (K) x (K)
A = GaussianArrayARD(np.identity(K),
alpha,
shape=(K,),
plates=(K,),
name='A_S',
initialize=False)
A.initialize_from_value(np.identity(K))
# State of the drift
# S : () x (N,K)
S = GaussianMarkovChain(np.ones(K),
1e-6*np.identity(K),
A,
np.ones(K),
n=N,
name='S',
initialize=False)
S.initialize_from_value(np.ones((N,K)))
# Projection matrix of the dynamics matrix
# Initialize S and B such that BS is identity matrix
# beta : (K) x ()
beta = Gamma(1e-5,
1e-5,
plates=(D,K),
name='beta')
# B : (D) x (D,K)
b = np.zeros((D,D,K))
b[np.arange(D),np.arange(D),np.zeros(D,dtype=int)] = 1
B = GaussianArrayARD(0,
beta,
plates=(D,),
name='B',
initialize=False)
B.initialize_from_value(np.reshape(1*b, (D,D,K)))
# BS : (N-1,D) x (D)
# TODO/FIXME: Implement __getitem__ method
BS = SumMultiply('dk,k->d',
B,
S.as_gaussian()[...,np.newaxis],
# iterator_axis=0,
name='BS')
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(np.zeros(D), # mean of x0
1e-3*np.identity(D), # prec of x0
BS, # dynamics
np.ones(D), # innovation
n=N+1, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N+1,D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5,
1e-5,
plates=(D,K),
name='gamma')
# C : (M,1) x (D,K)
C = GaussianArrayARD(0,
gamma,
plates=(M,1),
name='C',
initialize=False)
C.initialize_from_random()
# Observation noise
# tau : () x ()
tau = Gamma(1e-5,
1e-5,
name='tau')
# Observations
# Y : (M,N) x ()
F = SumMultiply('dk,d,k',
C,
X.as_gaussian()[1:],
S.as_gaussian(),
name='F')
Y = GaussianArrayARD(F,
tau,
name='Y')
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)
#
# Run inference with rotations.
#
rotate = False
if rotate:
# Rotate the D-dimensional state space (C, X)
rotB = transformations.RotateGaussianMatrixARD(B, beta, D, K, axis='rows')
rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
rotC = transformations.RotateGaussianARD(C, gamma)
R_X = transformations.RotationOptimizer(rotX, rotC, D)
# Rotate the K-dimensional latent dynamics space (B, S)
rotA = transformations.RotateGaussianARD(A, alpha)
rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
rotB = transformations.RotateGaussianMatrixARD(B, beta, D, K, axis='cols')
R_S = transformations.RotationOptimizer(rotS, rotB, K)
# Iterate
for ind in range(maxiter):
print("X update")
Q.update(X)
print("S update")
Q.update(S)
print("A update")
Q.update(A)
print("alpha update")
Q.update(alpha)
print("B update")
Q.update(B)
print("beta update")
Q.update(beta)
print("C update")
Q.update(C)
print("gamma update")
Q.update(gamma)
print("tau update")
Q.update(tau)
if rotate:
if ind >= 0:
R_X.rotate()
if ind >= 0:
R_S.rotate()
Q.plot_iteration_by_nodes()
#
# SHOW RESULTS
#
# Plot observations space
plt.figure()
bpplt.timeseries_normal(F)
bpplt.timeseries(f, 'b-')
bpplt.timeseries(y, 'r.')
# Plot latent space
plt.figure()
bpplt.timeseries_gaussian_mc(X, scale=2)
# Plot drift space
plt.figure()
bpplt.timeseries_gaussian_mc(S, scale=2)
def run(M=10, N=100, D_y=3, D=5):
seed = 45
print('seed =', seed)
np.random.seed(seed)
# Check HDF5 version.
if h5py.version.hdf5_version_tuple < (1, 8, 7):
print(
"WARNING! Your HDF5 version is %s. HDF5 versions <1.8.7 are not "
"able to save empty arrays, thus you may experience problems if "
"you for instance try to save before running any iteration steps."
% str(h5py.version.hdf5_version_tuple))
# Generate data
w = np.random.normal(0, 1, size=(M, 1, D_y))
x = np.random.normal(0, 1, size=(1, N, D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.5, size=(M, N))
# Construct model
(Y, WX, W, X, tau, alpha) = pca_model(M, N, D)
# Data with missing values
mask = random.mask(M, N, p=0.9) # randomly missing
mask[:, 20:40] = False # gap missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha, autosave_iterations=5)
# Initialize some nodes randomly
X.initialize_from_value(X.random())
W.initialize_from_value(W.random())
# Save the state into a HDF5 file
filename = tempfile.NamedTemporaryFile(suffix='hdf5').name
Q.update(X, W, alpha, tau, repeat=1)
Q.save(filename=filename)
# Inference loop.
Q.update(X, W, alpha, tau, repeat=10)
# Reload the state from the HDF5 file
Q.load(filename=filename)
# Inference loop again.
Q.update(X, W, alpha, tau, repeat=10)
# NOTE: Saving and loading requires that you have the model
# constructed. "Save" does not store the model structure nor does "load"
# read it. They are just used for reading and writing the contents of the
# nodes. Thus, if you want to load, you first need to construct the same
# model that was used for saving and then use load to set the states of the
# nodes.
plt.clf()
WX_params = WX.get_parameters()
fh = WX_params[0] * np.ones(y.shape)
err_fh = 2 * np.sqrt(WX_params[1] + 1 / tau.get_moments()[0]) * np.ones(
y.shape)
for m in range(M):
plt.subplot(M, 1, m + 1)
#errorplot(y, error=None, x=None, lower=None, upper=None):
bpplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
plt.plot(np.arange(N), f[m], 'g')
plt.plot(np.arange(N), y[m], 'r+')
plt.figure()
Q.plot_iteration_by_nodes()
plt.figure()
plt.subplot(2, 2, 1)
bpplt.binary_matrix(W.mask)
plt.subplot(2, 2, 2)
bpplt.binary_matrix(X.mask)
plt.subplot(2, 2, 3)
#bpplt.binary_matrix(WX.get_mask())
plt.subplot(2, 2, 4)
bpplt.binary_matrix(Y.mask)
def run(M=40,
N=100,
D_y=6,
D=8,
seed=42,
rotate=False,
maxiter=1000,
debug=False,
plot=True):
"""
Run pattern search demo for PCA.
"""
if seed is not None:
np.random.seed(seed)
# Generate data
w = np.random.normal(0, 1, size=(M, 1, D_y))
x = np.random.normal(0, 1, size=(1, N, D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.2, size=(M, N))
# Construct model
Q = VB(*(pca.model(M, N, D)))
# Data with missing values
mask = random.mask(M, N, p=0.5) # randomly missing
y[~mask] = np.nan
Q['Y'].observe(y, mask=mask)
# Initialize some nodes randomly
Q['X'].initialize_from_random()
Q['W'].initialize_from_random()
# Use a few VB-EM updates at the beginning
Q.update(repeat=10)
Q.save()
# Standard VB-EM as a baseline
Q.update(repeat=maxiter)
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
# Restore initial state
Q.load()
# Pattern search method for comparison
for n in range(maxiter):
Q.pattern_search('W', 'tau', maxiter=3, collapsed=['X', 'alpha'])
Q.update(repeat=20)
if Q.has_converged():
break
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.xlabel('CPU time (in seconds)')
bpplt.pyplot.ylabel('VB lower bound')
bpplt.pyplot.legend(['VB-EM', 'Pattern search'], loc='lower right')
def run(M=10, N=100, D_y=3, D=5):
seed = 45
print('seed =', seed)
np.random.seed(seed)
# Check HDF5 version.
if h5py.version.hdf5_version_tuple < (1,8,7):
print("WARNING! Your HDF5 version is %s. HDF5 versions <1.8.7 are not "
"able to save empty arrays, thus you may experience problems if "
"you for instance try to save before running any iteration steps."
% str(h5py.version.hdf5_version_tuple))
# Generate data
w = np.random.normal(0, 1, size=(M,1,D_y))
x = np.random.normal(0, 1, size=(1,N,D_y))
f = misc.sum_product(w, x, axes_to_sum=[-1])
y = f + np.random.normal(0, 0.5, size=(M,N))
# Construct model
(Y, WX, W, X, tau, alpha) = pca_model(M, N, D)
# Data with missing values
mask = random.mask(M, N, p=0.9) # randomly missing
mask[:,20:40] = False # gap missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha, autosave_iterations=5)
# Initialize some nodes randomly
X.initialize_from_value(X.random())
W.initialize_from_value(W.random())
# Save the state into a HDF5 file
filename = tempfile.NamedTemporaryFile(suffix='hdf5').name
Q.update(X, W, alpha, tau, repeat=1)
Q.save(filename=filename)
# Inference loop.
Q.update(X, W, alpha, tau, repeat=10)
# Reload the state from the HDF5 file
Q.load(filename=filename)
# Inference loop again.
Q.update(X, W, alpha, tau, repeat=10)
# NOTE: Saving and loading requires that you have the model
# constructed. "Save" does not store the model structure nor does "load"
# read it. They are just used for reading and writing the contents of the
# nodes. Thus, if you want to load, you first need to construct the same
# model that was used for saving and then use load to set the states of the
# nodes.
plt.clf()
WX_params = WX.get_parameters()
fh = WX_params[0] * np.ones(y.shape)
err_fh = 2*np.sqrt(WX_params[1] + 1/tau.get_moments()[0]) * np.ones(y.shape)
for m in range(M):
plt.subplot(M,1,m+1)
#errorplot(y, error=None, x=None, lower=None, upper=None):
bpplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
plt.plot(np.arange(N), f[m], 'g')
plt.plot(np.arange(N), y[m], 'r+')
plt.figure()
Q.plot_iteration_by_nodes()
plt.figure()
plt.subplot(2,2,1)
bpplt.binary_matrix(W.mask)
plt.subplot(2,2,2)
bpplt.binary_matrix(X.mask)
plt.subplot(2,2,3)
#bpplt.binary_matrix(WX.get_mask())
plt.subplot(2,2,4)
bpplt.binary_matrix(Y.mask)
def run_dlssm(y, f, mask, D, K, maxiter):
"""
Run VB inference for linear state space model with drifting dynamics.
"""
(M, N) = np.shape(y)
# Dynamics matrix with ARD
# alpha : (D) x ()
alpha = Gamma(1e-5, 1e-5, plates=(K, ), name='alpha')
# A : (K) x (K)
A = Gaussian(
np.zeros(K),
#np.identity(K),
diagonal(alpha),
plates=(K, ),
name='A_S')
A.initialize_from_value(np.identity(K))
# rho
## rho = Gamma(1e-5,
## 1e-5,
## plates=(K,),
## name="rho")
# S : () x (N-1,K)
S = GaussianMarkovChain(np.ones(K),
1e-6 * np.identity(K),
A,
np.ones(K),
n=N - 1,
name='S')
S.initialize_from_value(1 * np.ones((N - 1, K)))
# Projection matrix of the dynamics matrix
# beta : (K) x ()
beta = Gamma(1e-5, 1e-5, plates=(K, ), name='beta')
# B : (D) x (D*K)
B = Gaussian(np.zeros(D * K),
diagonal(tile(beta, D)),
plates=(D, ),
name='B')
b = np.zeros((D, D, K))
b[np.arange(D), np.arange(D), np.zeros(D, dtype=int)] = 1
B.initialize_from_value(np.reshape(1 * b, (D, D * K)))
# A : (N-1,D) x (D)
BS = MatrixDot(B, S.as_gaussian().add_plate_axis(-1), name='BS')
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(
np.zeros(D), # mean of x0
1e-3 * np.identity(D), # prec of x0
BS, # dynamics
np.ones(D), # innovation
n=N, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N, D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
# C : (M,1) x (D)
C = Gaussian(np.zeros(D), diagonal(gamma), plates=(M, 1), name='C')
C.initialize_from_value(np.random.randn(M, 1, D))
# Observation noise
# tau : () x ()
tau = Gamma(1e-5, 1e-5, name='tau')
# Observations
# Y : (M,N) x ()
CX = Dot(C, X.as_gaussian())
Y = Normal(CX, tau, name='Y')
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)
#
# Run inference with rotations.
#
# Rotate the D-dimensional state space (C, X)
rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='rows')
rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
rotC = transformations.RotateGaussianARD(C, gamma)
R_X = transformations.RotationOptimizer(rotX, rotC, D)
# Rotate the K-dimensional latent dynamics space (B, S)
rotA = transformations.RotateGaussianARD(A, alpha)
rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='cols')
R_S = transformations.RotationOptimizer(rotS, rotB, K)
# Iterate
for ind in range(int(maxiter / 5)):
Q.update(repeat=5)
#Q.update(X, S, A, alpha, rho, B, beta, C, gamma, tau, repeat=maxiter)
R_X.rotate()
R_S.rotate()
## R_X.rotate(
## check_bound=Q.compute_lowerbound,
## check_bound_terms=Q.compute_lowerbound_terms,
## check_gradient=True
## )
## R_S.rotate(
## check_bound=Q.compute_lowerbound,
## check_bound_terms=Q.compute_lowerbound_terms,
## check_gradient=True
## )
#
# SHOW RESULTS
#
# Mean and standard deviation of the posterior
(f_mean, f_squared) = CX.get_moments()
f_std = np.sqrt(f_squared - f_mean**2)
# Plot observations space
for m in range(M):
plt.subplot(M, 1, m + 1)
plt.plot(y[m, :], 'r.')
plt.plot(f[m, :], 'b-')
bpplt.errorplot(y=f_mean[m, :], error=2 * f_std[m, :])
def run_lssm(y, f, mask, D, maxiter):
"""
Run VB inference for linear state space model.
"""
(M, N) = np.shape(y)
#
# CONSTRUCT THE MODEL
#
# Dynamic matrix
# alpha: (D) x ()
alpha = Gamma(1e-5, 1e-5, plates=(D, ), name='alpha')
# A : (D) x (D)
A = Gaussian(np.zeros(D), diagonal(alpha), plates=(D, ), name='A')
A.initialize_from_value(np.identity(D))
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(
np.zeros(D), # mean of x0
1e-3 * np.identity(D), # prec of x0
A, # dynamics
np.ones(D), # innovation
n=N, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N, D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
# C : (M,1) x (D)
C = Gaussian(np.zeros(D), diagonal(gamma), plates=(M, 1), name='C')
C.initialize_from_value(np.random.randn(M, 1, D))
# Observation noise
# tau : () x ()
tau = Gamma(1e-5, 1e-5, name='tau')
# Observations
# Y : (M,N) x ()
CX = Dot(C, X.as_gaussian())
Y = Normal(CX, tau, name='Y')
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, A, alpha, C, gamma, tau)
# Iterate
Q.update(X, A, alpha, C, gamma, tau, repeat=maxiter)
#
# SHOW RESULTS
#
# Mean and standard deviation of the posterior
(f_mean, f_squared) = CX.get_moments()
f_std = np.sqrt(f_squared - f_mean**2)
# Plot observations space
#plt.figure()
for m in range(M):
plt.subplot(M, 1, m + 1)
plt.plot(y[m, :], 'r.')
plt.plot(f[m, :], 'b-')
bpplt.errorplot(y=f_mean[m, :], error=2 * f_std[m, :])
def run(N=100000, N_batch=50, seed=42, maxiter=100, plot=True):
"""
Run deterministic annealing demo for 1-D Gaussian mixture.
"""
if seed is not None:
np.random.seed(seed)
# Number of clusters in the model
K = 20
# Dimensionality of the data
D = 5
# Generate data
K_true = 10
spread = 5
means = spread * np.random.randn(K_true, D)
z = random.categorical(np.ones(K_true), size=N)
data = np.empty((N, D))
for n in range(N):
data[n] = means[z[n]] + np.random.randn(D)
#
# Standard VB-EM algorithm
#
# Full model
mu = Gaussian(np.zeros(D), np.identity(D), plates=(K, ), name='means')
alpha = Dirichlet(np.ones(K), name='class probabilities')
Z = Categorical(alpha, plates=(N, ), name='classes')
Y = Mixture(Z, Gaussian, mu, np.identity(D), name='observations')
# Break symmetry with random initialization of the means
mu.initialize_from_random()
# Put the data in
Y.observe(data)
# Run inference
Q = VB(Y, Z, mu, alpha)
Q.save(mu)
Q.update(repeat=maxiter)
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
max_cputime = np.sum(Q.cputime[~np.isnan(Q.cputime)])
#
# Stochastic variational inference
#
# Construct smaller model (size of the mini-batch)
mu = Gaussian(np.zeros(D), np.identity(D), plates=(K, ), name='means')
alpha = Dirichlet(np.ones(K), name='class probabilities')
Z = Categorical(alpha,
plates=(N_batch, ),
plates_multiplier=(N / N_batch, ),
name='classes')
Y = Mixture(Z, Gaussian, mu, np.identity(D), name='observations')
# Break symmetry with random initialization of the means
mu.initialize_from_random()
# Inference engine
Q = VB(Y, Z, mu, alpha, autosave_filename=Q.autosave_filename)
Q.load(mu)
# Because using mini-batches, messages need to be multiplied appropriately
print("Stochastic variational inference...")
Q.ignore_bound_checks = True
maxiter *= int(N / N_batch)
delay = 1
forgetting_rate = 0.7
for n in range(maxiter):
# Observe a mini-batch
subset = np.random.choice(N, N_batch)
Y.observe(data[subset, :])
# Learn intermediate variables
Q.update(Z)
# Set step length
step = (n + delay)**(-forgetting_rate)
# Stochastic gradient for the global variables
Q.gradient_step(mu, alpha, scale=step)
if np.sum(Q.cputime[:n]) > max_cputime:
break
if plot:
bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')
bpplt.pyplot.xlabel('CPU time (in seconds)')
bpplt.pyplot.ylabel('VB lower bound')
bpplt.pyplot.legend(['VB-EM', 'Stochastic inference'],
loc='lower right')
bpplt.pyplot.title('VB for Gaussian mixture model')
return
def run(N=500, seed=42, maxiter=100, plot=True):
"""
Run deterministic annealing demo for 1-D Gaussian mixture.
"""
if seed is not None:
np.random.seed(seed)
mu = GaussianARD(0, 1,
plates=(2,),
name='means')
Z = Categorical([0.3, 0.7],
plates=(N,),
name='classes')
Y = Mixture(Z, GaussianARD, mu, 1,
name='observations')
# Generate data
z = Z.random()
data = np.empty(N)
for n in range(N):
data[n] = [4, -4][z[n]]
Y.observe(data)
# Initialize means closer to the inferior local optimum in which the
# cluster means are swapped
mu.initialize_from_value([0, 6])
Q = VB(Y, Z, mu)
Q.save()
#
# Standard VB-EM algorithm
#
Q.update(repeat=maxiter)
mu_vbem = mu.u[0].copy()
L_vbem = Q.compute_lowerbound()
#
# VB-EM with deterministic annealing
#
Q.load()
beta = 0.01
while beta < 1.0:
beta = min(beta*1.2, 1.0)
print("Set annealing to %.2f" % beta)
Q.set_annealing(beta)
Q.update(repeat=maxiter, tol=1e-4)
mu_anneal = mu.u[0].copy()
L_anneal = Q.compute_lowerbound()
print("==============================")
print("RESULTS FOR VB-EM vs ANNEALING")
print("Fixed component probabilities:", np.array([0.3, 0.7]))
print("True component means:", np.array([4, -4]))
print("VB-EM component means:", mu_vbem)
print("VB-EM lower bound:", L_vbem)
print("Annealed VB-EM component means:", mu_anneal)
print("Annealed VB-EM lower bound:", L_anneal)
return
def run_lssm(y, f, mask, D, maxiter):
"""
Run VB inference for linear state space model.
"""
(M, N) = np.shape(y)
#
# CONSTRUCT THE MODEL
#
# Dynamic matrix
# alpha: (D) x ()
alpha = Gamma(1e-5,
1e-5,
plates=(D,),
name='alpha')
# A : (D) x (D)
A = Gaussian(np.zeros(D),
diagonal(alpha),
plates=(D,),
name='A')
A.initialize_from_value(np.identity(D))
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(np.zeros(D), # mean of x0
1e-3*np.identity(D), # prec of x0
A, # dynamics
np.ones(D), # innovation
n=N, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N,D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5,
1e-5,
plates=(D,),
name='gamma')
# C : (M,1) x (D)
C = Gaussian(np.zeros(D),
diagonal(gamma),
plates=(M,1),
name='C')
C.initialize_from_value(np.random.randn(M,1,D))
# Observation noise
# tau : () x ()
tau = Gamma(1e-5,
1e-5,
name='tau')
# Observations
# Y : (M,N) x ()
CX = Dot(C, X.as_gaussian())
Y = Normal(CX,
tau,
name='Y')
# Rotate the D-dimensional latent space
rotA = transformations.RotateGaussianARD(A, alpha)
rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
rotC = transformations.RotateGaussianARD(C, gamma)
R = transformations.RotationOptimizer(rotX, rotC, D)
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, A, alpha, C, gamma, tau)
# Iterate
for ind in range(maxiter):
Q.update(X, A, alpha, C, gamma, tau)
R.rotate()
#
# SHOW RESULTS
#
plt.figure()
bpplt.timeseries_normal(CX)
bpplt.timeseries(f, 'b-')
bpplt.timeseries(y, 'r.')
def run(maxiter=100):
seed = 496 #np.random.randint(1000)
print("seed = ", seed)
np.random.seed(seed)
# Simulate some data
D = 3
M = 6
N = 200
c = np.random.randn(M, D)
w = 0.3
a = np.array([[np.cos(w), -np.sin(w), 0], [np.sin(w),
np.cos(w), 0], [0, 0, 1]])
x = np.empty((N, D))
f = np.empty((M, N))
y = np.empty((M, N))
x[0] = 10 * np.random.randn(D)
f[:, 0] = np.dot(c, x[0])
y[:, 0] = f[:, 0] + 3 * np.random.randn(M)
for n in range(N - 1):
x[n + 1] = np.dot(a, x[n]) + np.random.randn(D)
f[:, n + 1] = np.dot(c, x[n + 1])
y[:, n + 1] = f[:, n + 1] + 3 * np.random.randn(M)
# Create the model
(Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D, N, M)
# Add missing values randomly
mask = random.mask(M, N, p=0.3)
# Add missing values to a period of time
mask[:, 30:80] = False
y[~mask] = np.nan # BayesPy doesn't require this. Just for plotting.
# Observe the data
Y.observe(y, mask=mask)
# Initialize nodes (must use some randomness for C)
C.initialize_from_random()
# Run inference
Q = VB(Y, X, C, gamma, A, alpha, tau)
#
# Run inference with rotations.
#
rotA = transformations.RotateGaussianARD(A, alpha)
rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
rotC = transformations.RotateGaussianARD(C, gamma)
R = transformations.RotationOptimizer(rotX, rotC, D)
#maxiter = 84
for ind in range(maxiter):
Q.update()
#print('C term', C.lower_bound_contribution())
R.rotate(
maxiter=10,
check_gradient=True,
verbose=False,
check_bound=Q.compute_lowerbound,
#check_bound=None,
check_bound_terms=Q.compute_lowerbound_terms)
#check_bound_terms=None)
X_vb = X.u[0]
varX_vb = utils.diagonal(X.u[1] - X_vb[..., np.newaxis, :] *
X_vb[..., :, np.newaxis])
u_CX = CX.get_moments()
CX_vb = u_CX[0]
varCX_vb = u_CX[1] - CX_vb**2
# Show results
plt.figure(3)
plt.clf()
for m in range(M):
plt.subplot(M, 1, m + 1)
plt.plot(y[m, :], 'r.')
plt.plot(f[m, :], 'b-')
bpplt.errorplot(y=CX_vb[m, :], error=2 * np.sqrt(varCX_vb[m, :]))
plt.figure()
Q.plot_iteration_by_nodes()
def run(M=50, N=200, D_y=10, D=20, maxiter=100):
seed = 45
print('seed =', seed)
np.random.seed(seed)
# Generate data (covariance eigenvalues: 1,1,...,1,2^2,3^2,...,(D_y+1)^2
(q, r) = scipy.linalg.qr(np.random.randn(M, M))
C = np.diag(np.arange(2, 2 + D_y))
C = np.ones(M)
C[:D_y] += np.arange(1, 1 + D_y)
y = C[:, np.newaxis] * np.random.randn(M, N)
y = np.dot(q, y)
# Construct model
(Y, WX, W, X, tau, alpha) = pca_model(M, N, D)
# Data with missing values
mask = utils.random.mask(M, N, p=0.9) # randomly missing
mask[:, 20:40] = False # gap missing
y[~mask] = np.nan
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, W, X, tau, alpha, autosave_filename=utils.utils.tempfile())
# Initialize nodes (from prior and randomly)
X.initialize_from_value(X.random())
W.initialize_from_value(W.random())
Q.update(repeat=1)
Q.save()
#
# Run inference with rotations.
#
rotX = transformations.RotateGaussian(X)
rotW = transformations.RotateGaussianARD(W, alpha)
R = transformations.RotationOptimizer(rotX, rotW, D)
for ind in range(maxiter):
Q.update()
R.rotate(check_gradient=False,
maxiter=10,
verbose=False,
check_bound=Q.compute_lowerbound,
check_bound_terms=Q.compute_lowerbound_terms)
L_rot = Q.L
#
# Re-run inference without rotations.
#
Q.load()
Q.update(repeat=maxiter)
L_norot = Q.L
#
# Plot comparison
#
plt.plot(L_rot)
plt.plot(L_norot)
plt.legend(['With rotations', 'Without rotations'], loc='lower right')
plt.show()
#
# In[4]:
from bayespy.utils import random
# Add missing values randomly (keep only 20%)
mask = random.mask(M, N, p=0.2)
Y.observe(y, mask=mask)
# Then inference (100 iterations) can be run simply as
# In[5]:
Q.update(repeat=10)
### Speeding up with parameter expansion
# VB inference can converge extremely slowly if the variables are strongly coupled. Because VMP updates one variable at a time, it may lead to slow zigzagging. This can be solved by using parameter expansion which reduces the coupling. In state-space models, the states $\mathbf{x}_n$ and the loadings $\mathbf{C}$ are coupled through a dot product $\mathbf{Cx}_n$, which is unaltered if the latent space is rotated arbitrarily:
#
# $$
# \mathbf{y}_n = \mathbf{C}\mathbf{x}_n = \mathbf{C}\mathbf{R}^{-1}\mathbf{R}\mathbf{x}_n \,.
# $$
#
# Thus, one intuitive transformation would be $\mathbf{C}\rightarrow\mathbf{C}\mathbf{R}^{-1}$ and $\mathbf{X}\rightarrow\mathbf{R}\mathbf{X}$. In order to keep the dynamics of the latent states unaffected by the transformation, the state dynamics matrix $\mathbf{A}$ must be transformed accordingly:
#
# $$
# \mathbf{R}\mathbf{x}_n = \mathbf{R}\mathbf{A}\mathbf{R}^{-1} \mathbf{R}\mathbf{x}_{n-1} \,,
# $$
def run(M=6, N=200, D=3, maxiter=100, debug=False, seed=42, rotate=False, precompute=False):
# Use deterministic random numbers
if seed is not None:
np.random.seed(seed)
# Simulate some data
K = 3
c = np.random.randn(M,K)
w = 0.3
a = np.array([[np.cos(w), -np.sin(w), 0],
[np.sin(w), np.cos(w), 0],
[0, 0, 1]])
x = np.empty((N,K))
f = np.empty((M,N))
y = np.empty((M,N))
x[0] = 10*np.random.randn(K)
f[:,0] = np.dot(c,x[0])
y[:,0] = f[:,0] + 3*np.random.randn(M)
for n in range(N-1):
x[n+1] = np.dot(a,x[n]) + np.random.randn(K)
f[:,n+1] = np.dot(c,x[n+1])
y[:,n+1] = f[:,n+1] + 3*np.random.randn(M)
# Create the model
(Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D=D,
N=N,
M=M)
# Add missing values randomly
mask = random.mask(M, N, p=0.3)
# Add missing values to a period of time
mask[:,30:80] = False
y[~mask] = np.nan # BayesPy doesn't require this. Just for plotting.
# Observe the data
Y.observe(y, mask=mask)
# Initialize nodes (must use some randomness for C)
C.initialize_from_random()
# Run inference
Q = VB(Y, X, C, gamma, A, alpha, tau)
#
# Run inference with rotations.
#
if rotate:
rotA = transformations.RotateGaussianArrayARD(A, alpha, precompute=precompute)
rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
rotC = transformations.RotateGaussianArrayARD(C, gamma)
R = transformations.RotationOptimizer(rotX, rotC, D)
for ind in range(maxiter):
Q.update()
if debug:
R.rotate(maxiter=10,
check_gradient=True,
check_bound=True)
else:
R.rotate()
else:
Q.update(repeat=maxiter)
# Show results
plt.figure()
bpplt.timeseries_normal(CX, scale=2)
bpplt.timeseries(f, 'b-')
bpplt.timeseries(y, 'r.')
plt.show()
def run_lssm(y, f, mask, D, maxiter):
"""
Run VB inference for linear state space model.
"""
(M, N) = np.shape(y)
#
# CONSTRUCT THE MODEL
#
# Dynamic matrix
# alpha: (D) x ()
alpha = Gamma(1e-5,
1e-5,
plates=(D,),
name='alpha')
# A : (D) x (D)
A = Gaussian(np.zeros(D),
diagonal(alpha),
plates=(D,),
name='A')
A.initialize_from_value(np.identity(D))
# Latent states with dynamics
# X : () x (N,D)
X = GaussianMarkovChain(np.zeros(D), # mean of x0
1e-3*np.identity(D), # prec of x0
A, # dynamics
np.ones(D), # innovation
n=N, # time instances
name='X',
initialize=False)
X.initialize_from_value(np.random.randn(N,D))
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5,
1e-5,
plates=(D,),
name='gamma')
# C : (M,1) x (D)
C = Gaussian(np.zeros(D),
diagonal(gamma),
plates=(M,1),
name='C')
C.initialize_from_value(np.random.randn(M,1,D))
# Observation noise
# tau : () x ()
tau = Gamma(1e-5,
1e-5,
name='tau')
# Observations
# Y : (M,N) x ()
CX = Dot(C, X.as_gaussian())
Y = Normal(CX,
tau,
name='Y')
#
# RUN INFERENCE
#
# Observe data
Y.observe(y, mask=mask)
# Construct inference machine
Q = VB(Y, X, A, alpha, C, gamma, tau)
# Iterate
Q.update(X, A, alpha, C, gamma, tau, repeat=maxiter)
#
# SHOW RESULTS
#
# Mean and standard deviation of the posterior
(f_mean, f_squared) = CX.get_moments()
f_std = np.sqrt(f_squared - f_mean**2)
# Plot observations space
#plt.figure()
for m in range(M):
plt.subplot(M,1,m+1)
plt.plot(y[m,:], 'r.')
plt.plot(f[m,:], 'b-')
bpplt.errorplot(y=f_mean[m,:], error=2*f_std[m,:])
