npr.seed(0)
import matplotlib
import matplotlib.pyplot as plt
from ssm.models import HMM
# Set the parameters of the HMM
T = 500 # number of time bins
K = 5 # number of discrete states
D = 2 # number of observed dimensions
# Make an HMM with the true parameters
true_hmm = HMM(K, D, observations="gaussian")
z, y = true_hmm.sample(T)
z_test, y_test = true_hmm.sample(T)
true_ll = true_hmm.log_probability(y)
# Fit models
N_sgd_iters = 1000
N_em_iters = 100
print("Fitting HMM with SGD")
hmm = HMM(K, D, observations="gaussian")
hmm_sgd_lls = hmm.fit(y, method="sgd", num_iters=N_sgd_iters)
hmm_sgd_test_ll = hmm.log_probability(y_test)
hmm_sgd_smooth = hmm.smooth(y)
print("Fitting HMM with EM")
hmm = HMM(K, D, observations="gaussian")
def test_sample(T=10, K=4, D=3, M=2):
"""
Test that we can construct and sample an HMM
with or withou, prefixes, noise, and noise.
"""
transition_names = [
"standard", "sticky", "inputdriven", "recurrent", "recurrent_only",
"rbf_recurrent", "nn_recurrent"
]
observation_names = [
"gaussian", "diagonal_gaussian", "t", "diagonal_t", "bernoulli",
"categorical", "poisson", "vonmises", "ar", "diagonal_ar",
"independent_ar", "robust_ar", "diagonal_robust_ar"
]
# Sample basic (no prefix, inputs, etc.)
for transitions in transition_names:
for observations in observation_names:
hmm = HMM(K,
D,
M=0,
transitions=transitions,
observations=observations)
zsmpl, xsmpl = hmm.sample(T)
# Sample with prefix
for transitions in transition_names:
for observations in observation_names:
hmm = HMM(K,
D,
M=0,
transitions=transitions,
observations=observations)
zpre, xpre = hmm.sample(3)
zsmpl, xsmpl = hmm.sample(T, prefix=(zpre, xpre))
# Sample with inputs
for transitions in transition_names:
for observations in observation_names:
hmm = HMM(K,
D,
M=M,
transitions=transitions,
observations=observations)
zpre, xpre = hmm.sample(3, input=npr.randn(3, M))
zsmpl, xsmpl = hmm.sample(T,
prefix=(zpre, xpre),
input=npr.randn(T, M))
# Sample without noise
for transitions in transition_names:
for observations in observation_names:
hmm = HMM(K,
D,
M=M,
transitions=transitions,
observations=observations)
zpre, xpre = hmm.sample(3, input=npr.randn(3, M))
zsmpl, xsmpl = hmm.sample(T,
prefix=(zpre, xpre),
input=npr.randn(T, M),
with_noise=False)
T = 200 # number of time bins
K = 5 # number of discrete states
D = 2 # data dimension
# Make an HMM
true_hmm = HMM(K, D, observations="gaussian")
# Manually tweak the means to make them farther apart
thetas = np.linspace(0, 2 * np.pi, K, endpoint=False)
true_hmm.observations.mus = 3 * np.column_stack(
(np.cos(thetas), np.sin(thetas)))
# In[3]:
# Sample some data from the HMM
z, y = true_hmm.sample(T)
true_ll = true_hmm.log_probability(y)
# In[4]:
# Plot the observation distributions
lim = .85 * abs(y).max()
XX, YY = np.meshgrid(np.linspace(-lim, lim, 100), np.linspace(-lim, lim, 100))
data = np.column_stack((XX.ravel(), YY.ravel()))
input = np.zeros((data.shape[0], 0))
mask = np.ones_like(data, dtype=bool)
tag = None
lls = true_hmm.observations.log_likelihoods(data, input, mask, tag)
# In[36]:
plt.plot(x[z == k, 0], x[z == k, 1], 'o', mfc=colors[k], mec='none', ms=4)
plt.plot(x[:, 0], x[:, 1], '-k', lw=2, alpha=.5)
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")
plt.title("Observation Distributions")
plt.tight_layout()
if save_figures:
plt.savefig("lds_3.pdf")
# In[16]:
# Simulate from the HMM fit
smpls = [hmm.sample(T - 1, prefix=(z[:1], x[:1])) for _ in range(1)]
# In[17]:
plt.figure(figsize=(8, 6))
lim = abs(x).max()
for d in range(D):
plt.plot(x[:, d] - d * lim, '-k', lw=4)
for i, (_, x_smpl) in enumerate(smpls):
x_smpl = np.concatenate((x[:1], x_smpl))
plt.plot(x_smpl[:, d] - d * lim, '-', lw=1, color=colors[i])
plt.yticks(-np.arange(D) * lim, ["$x_{}$".format(d + 1) for d in range(D)])
plt.xlabel("time")
plt.xlim(0, T)
plt.title("True LDS States and Fitted HMM Simulations")
M = 1 # input dimension
C = 3 # number of output types/categories
# Make an HMM
true_hmm = HMM(K, D, M,
observations="categorical", observation_kwargs=dict(C=C),
transitions="inputdriven")
# Optionally, turn up the input weights to exaggerate the effect
# true_hmm.transitions.Ws *= 3
# Create an exogenous input
inpt = np.sin(2 * np.pi * np.arange(T) / 50)[:, None] + 1e-1 * npr.randn(T, M)
# Sample some data from the HMM
z, y = true_hmm.sample(T, input=inpt)
# Compute the true log probability of the data, summing out the discrete states
true_lp = true_hmm.log_probability(y, inputs=inpt)
# In[3]:
# Plot the data
plt.figure(figsize=(8, 5))
plt.subplot(311)
plt.plot(inpt)
plt.xticks([])
plt.xlim(0, T)
mog.fit(x)
# In[8]:
plt.plot(x[:, 0], x[:, 1])
for mu in mog.observations.mus:
plt.plot(mu[0], mu[1], 'o')
# In[9]:
arhmm = HMM(K=8, D=D, observations="ar")
arhmm.fit(x)
# In[10]:
z_smpl, x_smpl = arhmm.sample(T=3000)
plt.plot(x_smpl[:, 0], x_smpl[:, 1])
# In[11]:
def sample(T=3000,
num_samples=25,
num_burnin=100,
schedule=None,
filename="samples.mp4"):
print("Running Gibbs sampler")
z_smpls, x_smpls, i_smpls = gibbs_sample_constrained_arhmm(
T,
arhmm,