def backward_algorithm():
wordHMMs = {}
isolated = get_isolated(prondict)
plot_p_color_mesh(example['logbeta'], 'example beta matrix')
# verify implementation
wordHMMs['o'] = concatHMMs(phoneHMMsAll, isolated['o'])
log_st_prob = np.log(wordHMMs['o']['startprob'])
log_transmat = np.log(wordHMMs['o']['transmat'])
beta_matrix = backward(example['obsloglik'], log_st_prob, log_transmat)
plot_p_color_mesh(beta_matrix, "hmms all output example beta matrix")
def EM(lmfcc, obsloglik, model):
log_st_prob = np.log(model['startprob'])
log_transmat = np.log(model['transmat'])
# E step
beta_matrix = backward(obsloglik, log_st_prob, log_transmat)
alpha_matrix = forward(obsloglik, log_st_prob, log_transmat)
posterior = statePosteriors(alpha_matrix, beta_matrix)
#M step
new_mean, new_var = updateMeanAndVar(lmfcc, posterior)
model['means'] = new_mean
model['covars'] = new_var
return model
def retrain(feature, model, num_iters=20, threshold=1):
# extract params
means = model['means']
covars = model['covars']
transmat = model['transmat'][:-1, :-1]
startprob = model['startprob'][:-1]
log_pi = np.log(startprob)
log_trans = np.log(transmat)
# calculate the emission
obsloglik = log_multivariate_normal_density_diag(feature, means, covars)
# EM algorithm
loglik_old = -np.inf
for iter_ in range(num_iters):
# E-step
log_alpha = forward(obsloglik, log_pi, log_trans)
log_beta = backward(obsloglik, log_pi, log_trans)
log_gamma = statePosteriors(log_alpha, log_beta)
# M-step
means, covars = updateMeanAndVar(feature, log_gamma)
# update
obsloglik = log_multivariate_normal_density_diag(
feature, means, covars)
loglik = logsumexp(log_alpha[-1])
print("Iter {}: The log likelihood in EM:".format(iter_), loglik)
# check if terminate EM
if (loglik - loglik_old) < threshold or np.isnan(loglik):
print("Terminating the EM")
break
else:
loglik_old = loglik
# load data
data = np.load('data/lab2_data.npz')['data']
example = np.load('data/lab2_example.npz')['example'].item()
phoneHMMs = np.load('data/lab2_models_onespkr.npz')['phoneHMMs'].item()
# Build hmm
wordHMMs = {}
wordHMMs['o'] = concatHMMs(phoneHMMs, isolated['o'])
trans_mat = wordHMMs['o']['transmat'][:-1, :-1]
pi_vec = wordHMMs['o']['startprob'][:-1]
log_startprob = np.log(pi_vec)
log_transmat = np.log(trans_mat)
log_emlik = example['obsloglik']
# =====================================================
log_beta = backward(log_emlik, log_startprob, log_transmat)
log_alpha = forward(log_emlik, log_startprob, log_transmat)
# caculate the log gamma
log_gamma = statePosteriors(log_alpha, log_beta)
# print(np.allclose(example['loggamma'], log_gamma))
# check if sum to one in linear / sum to zero in log domain
ntime_steps = log_gamma.shape[0]
zeros = np.zeros((ntime_steps))
# print(np.allclose(logsumexp(log_gamma, axis=1), zeros))
# ====================================================================
# See notes, just do the normalization to get the state posterior of GMM
logZ = logsumexp(log_emlik,
axis=1) # the normalization factor in log space