def gaussianmix_model(N, K, D):
# N = number of data vectors
# K = number of clusters
# D = dimensionality
# Construct the Gaussian mixture model
# K prior weights (for components)
alpha = nodes.Dirichlet(1e-3 * np.ones(K), name='alpha')
# N K-dimensional cluster assignments (for data)
z = nodes.Categorical(alpha, plates=(N, ), name='z')
# K D-dimensional component means
X = nodes.Gaussian(np.zeros(D),
0.01 * np.identity(D),
plates=(K, ),
name='X')
# K D-dimensional component covariances
Lambda = nodes.Wishart(D,
0.01 * np.identity(D),
plates=(K, ),
name='Lambda')
# N D-dimensional observation vectors
Y = nodes.Mixture(z, nodes.Gaussian, X, Lambda, plates=(N, ), name='Y')
# TODO: Plates should be learned automatically if not given (it
# would be the smallest shape broadcasted from the shapes of the
# parents)
z.initialize_from_random()
return VB(Y, X, Lambda, z, alpha)
def model(M, N, D):
# Construct the PCA model with ARD
# ARD
alpha = nodes.Gamma(1e-2, 1e-2, plates=(D, ), name='alpha')
# Loadings
W = nodes.GaussianARD(0, alpha, shape=(D, ), plates=(M, 1), name='W')
# States
X = nodes.GaussianARD(0, 1, shape=(D, ), plates=(1, N), name='X')
# PCA
F = nodes.SumMultiply('i,i', W, X, name='F')
# Noise
tau = nodes.Gamma(1e-2, 1e-2, name='tau')
# Noisy observations
Y = nodes.GaussianARD(F, tau, name='Y')
# Initialize some nodes randomly
X.initialize_from_random()
W.initialize_from_random()
return VB(Y, F, W, X, tau, alpha)
def model(M=10, N=100, D=3):
"""
Construct linear state-space model.
See, for instance, the following publication:
"Fast variational Bayesian linear state-space model"
Luttinen (ECML 2013)
"""
# Dynamics matrix with ARD
alpha = Gamma(1e-5, 1e-5, plates=(D, ), name='alpha')
A = GaussianARD(0,
alpha,
shape=(D, ),
plates=(D, ),
plotter=bpplt.GaussianHintonPlotter(rows=0,
cols=1,
scale=0),
name='A')
A.initialize_from_value(np.identity(D))
# Latent states with dynamics
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
plotter=bpplt.GaussianMarkovChainPlotter(scale=2),
name='X')
X.initialize_from_value(np.random.randn(N, D))
# Mixing matrix from latent space to observation space using ARD
gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
gamma.initialize_from_value(1e-2 * np.ones(D))
C = GaussianARD(0,
gamma,
shape=(D, ),
plates=(M, 1),
plotter=bpplt.GaussianHintonPlotter(rows=0,
cols=2,
scale=0),
name='C')
C.initialize_from_value(np.random.randn(M, 1, D))
# Observation noise
tau = Gamma(1e-5, 1e-5, name='tau')
tau.initialize_from_value(1e2)
# Underlying noiseless function
F = SumMultiply('i,i', C, X, name='F')
# Noisy observations
Y = GaussianARD(F, tau, name='Y')
Q = VB(Y, F, C, gamma, X, A, alpha, tau, C)
return Q
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 mixture_model(distribution, *args, K=3, N=100):
# Prior for state probabilities
alpha = Dirichlet(1e-3 * np.ones(K), name='alpha')
# Cluster assignments
Z = Categorical(alpha, plates=(N, ), name='Z')
# Observation distribution
Y = Mixture(Z, distribution, *args, name='Y')
Q = VB(Y, Z, alpha)
return Q
def hidden_markov_model(distribution, *args, K=3, N=100):
# Prior for initial state probabilities
alpha = Dirichlet(1e-3 * np.ones(K), name='alpha')
# Prior for state transition probabilities
A = Dirichlet(1e-3 * np.ones(K), plates=(K, ), name='A')
# Hidden states (with unknown initial state probabilities and state
# transition probabilities)
Z = CategoricalMarkovChain(alpha, A, states=N, name='Z')
# Emission/observation distribution
Y = Mixture(Z, distribution, *args, name='Y')
Q = VB(Y, Z, alpha, A)
return Q
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=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 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 gaussianmix_model(N, K, D, covariance='full'):
# N = number of data vectors
# K = number of clusters
# D = dimensionality
# Construct the Gaussian mixture model
# K prior weights (for components)
alpha = nodes.Dirichlet(1e-3*np.ones(K),
name='alpha')
# N K-dimensional cluster assignments (for data)
z = nodes.Categorical(alpha,
plates=(N,),
name='z')
# K D-dimensional component means
X = nodes.GaussianARD(0, 1e-3,
shape=(D,),
plates=(K,),
name='X')
if covariance.lower() == 'full':
# K D-dimensional component covariances
Lambda = nodes.Wishart(D, 0.01*np.identity(D),
plates=(K,),
name='Lambda')
# N D-dimensional observation vectors
Y = nodes.Mixture(z, nodes.Gaussian, X, Lambda, plates=(N,), name='Y')
elif covariance.lower() == 'diagonal':
# Inverse variances
Lambda = nodes.Gamma(1e-3, 1e-3, plates=(K, D), name='Lambda')
# N D-dimensional observation vectors
Y = nodes.Mixture(z, nodes.GaussianARD, X, Lambda, plates=(N,), name='Y')
elif covariance.lower() == 'isotropic':
# Inverse variances
Lambda = nodes.Gamma(1e-3, 1e-3, plates=(K, 1), name='Lambda')
# N D-dimensional observation vectors
Y = nodes.Mixture(z, nodes.GaussianARD, X, Lambda, plates=(N,), name='Y')
z.initialize_from_random()
return VB(Y, X, Lambda, z, alpha)
def model(M, N, D, K):
"""
Construct the linear state-space model with time-varying dynamics
For reference, see the following publication:
(TODO)
"""
#
# The model block for the latent mixing weight process
#
# Dynamics matrix with ARD
# beta : (K) x ()
beta = Gamma(1e-5, 1e-5, plates=(K, ), name='beta')
# B : (K) x (K)
B = GaussianARD(np.identity(K),
beta,
shape=(K, ),
plates=(K, ),
name='B',
plotter=bpplt.GaussianHintonPlotter(rows=0,
cols=1,
scale=0),
initialize=False)
B.initialize_from_value(np.identity(K))
# Mixing weight process, that is, the weights in the linear combination of
# state dynamics matrices
# S : () x (N,K)
S = GaussianMarkovChain(np.ones(K),
1e-6 * np.identity(K),
B,
np.ones(K),
n=N,
name='S',
plotter=bpplt.GaussianMarkovChainPlotter(scale=2),
initialize=False)
s = 10 * np.random.randn(N, K)
s[:, 0] = 10
S.initialize_from_value(s)
#
# The model block for the latent states
#
# Projection matrix of the dynamics matrix
# alpha : (K) x ()
alpha = Gamma(1e-5, 1e-5, plates=(D, K), name='alpha')
alpha.initialize_from_value(1 * np.ones((D, K)))
# A : (D) x (D,K)
A = GaussianARD(0,
alpha,
shape=(D, K),
plates=(D, ),
name='A',
plotter=bpplt.GaussianHintonPlotter(rows=0,
cols=1,
scale=0),
initialize=False)
# Initialize S and A such that A*S is almost an identity matrix
a = np.zeros((D, D, K))
a[np.arange(D), np.arange(D), np.zeros(D, dtype=int)] = 1
a[:, :, 0] = np.identity(D) / s[0, 0]
a[:, :, 1:] = 0.1 / s[0, 0] * np.random.randn(D, D, K - 1)
A.initialize_from_value(a)
# Latent states with dynamics
# X : () x (N,D)
X = VaryingGaussianMarkovChain(
np.zeros(D), # mean of x0
1e-3 * np.identity(D), # prec of x0
A, # dynamics matrices
S._convert(GaussianMoments)[1:], # temporal weights
np.ones(D), # innovation
n=N, # time instances
name='X',
plotter=bpplt.GaussianMarkovChainPlotter(scale=2),
initialize=False)
X.initialize_from_value(np.random.randn(N, D))
#
# The model block for observations
#
# Mixing matrix from latent space to observation space using ARD
# gamma : (D) x ()
gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
gamma.initialize_from_value(1e-2 * np.ones(D))
# C : (M,1) x (D)
C = GaussianARD(0,
gamma,
shape=(D, ),
plates=(M, 1),
name='C',
plotter=bpplt.GaussianHintonPlotter(rows=0,
cols=2,
scale=0))
C.initialize_from_value(np.random.randn(M, 1, D))
# Noiseless process
# F : (M,N) x ()
F = SumMultiply('d,d', C, X, name='F')
# Observation noise
# tau : () x ()
tau = Gamma(1e-5, 1e-5, name='tau')
tau.initialize_from_value(1e2)
# Observations
# Y: (M,N) x ()
Y = GaussianARD(F, tau, name='Y')
# Construct inference machine
Q = VB(Y, F, C, gamma, X, A, alpha, tau, S, B, beta)
return Q
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(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(M=10, N=100, D=5, seed=42, maxiter=100, plot=True):
"""
Run deterministic annealing demo for 1-D Gaussian mixture.
"""
raise NotImplementedError("Black box variational inference not yet implemented, sorry")
if seed is not None:
np.random.seed(seed)
# Generate data
data = np.dot(np.random.randn(M,D),
np.random.randn(D,N))
# Construct model
C = GaussianARD(0, 1, shape=(2,), plates=(M,1), name='C')
X = GaussianARD(0, 1, shape=(2,), plates=(1,N), name='X')
F = Dot(C, X)
# Some arbitrary log likelihood
def logpdf(y, f):
"""
exp(f) / (1 + exp(f)) = 1/(1+exp(-f))
-log(1+exp(-f)) = -log(exp(0)+exp(-f))
also:
1 - exp(f) / (1 + exp(f)) = (1 + exp(f) - exp(f)) / (1 + exp(f))
= 1 / (1 + exp(f))
= -log(1+exp(f)) = -log(exp(0)+exp(f))
"""
return -np.logaddexp(0, -f * np.where(y, -1, +1))
Y = LogPDF(logpdf, F, samples=10, shape=())
#Y = GaussianARD(F, 1)
Y.observe(data)
Q = VB(Y, C, X)
Q.ignore_bound_checks = True
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(C, X, scale=step)
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')
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_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(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(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 model(M=20, N=100, D=10, K=3):
"""
Construct the linear state-space model with switching dynamics.
"""
#
# Switching dynamics (HMM)
#
# Prior for initial state probabilities
rho = Dirichlet(1e-3 * np.ones(K), name='rho')
# Prior for state transition probabilities
V = Dirichlet(1e-3 * np.ones(K), plates=(K, ), name='V')
v = 10 * np.identity(K) + 1 * np.ones((K, K))
v /= np.sum(v, axis=-1, keepdims=True)
V.initialize_from_value(v)
# Hidden states (with unknown initial state probabilities and state
# transition probabilities)
Z = CategoricalMarkovChain(rho,
V,
states=N - 1,
name='Z',
plotter=bpplt.CategoricalMarkovChainPlotter(),
initialize=False)
Z.u[0] = np.random.dirichlet(np.ones(K))
Z.u[1] = np.reshape(
np.random.dirichlet(0.5 * np.ones(K * K), size=(N - 2)), (N - 2, K, K))
#
# Linear state-space models
#
# Dynamics matrix with ARD
# (K,D) x ()
alpha = Gamma(1e-5, 1e-5, plates=(K, 1, D), name='alpha')
# (K,1,1,D) x (D)
A = GaussianARD(0,
alpha,
shape=(D, ),
plates=(K, D),
name='A',
plotter=bpplt.GaussianHintonPlotter())
A.initialize_from_value(
np.identity(D) * np.ones((K, D, D)) + 0.1 * np.random.randn(K, D, D))
# Latent states with dynamics
# (K,1) x (N,D)
X = SwitchingGaussianMarkovChain(
np.zeros(D), # mean of x0
1e-3 * np.identity(D), # prec of x0
A, # dynamics
Z, # dynamics selection
np.ones(D), # innovation
n=N, # time instances
name='X',
plotter=bpplt.GaussianMarkovChainPlotter())
X.initialize_from_value(10 * np.random.randn(N, D))
# Mixing matrix from latent space to observation space using ARD
# (K,1,1,D) x ()
gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
# (K,M,1) x (D)
C = GaussianARD(0,
gamma,
shape=(D, ),
plates=(M, 1),
name='C',
plotter=bpplt.GaussianHintonPlotter(rows=-3, cols=-1))
C.initialize_from_value(np.random.randn(M, 1, D))
# Underlying noiseless function
# (K,M,N) x ()
F = SumMultiply('i,i', C, X, name='F')
#
# Mixing the models
#
# Observation noise
tau = Gamma(1e-5, 1e-5, name='tau')
tau.initialize_from_value(1e2)
# Emission/observation distribution
Y = GaussianARD(F, tau, name='Y')
Q = VB(Y, F, Z, rho, V, C, gamma, X, A, alpha, tau)
return Q
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()