def statePosteriors(log_alpha, log_beta):
N = len(log_alpha)
M = len(log_alpha[0])
sum = 0
y = [[0 for j in range(M)] for i in range(N)]
sum = tools2.logsumexp(log_alpha[N - 1])
for n in range(N):
for j in range(M):
y[n][j] = log_alpha[n][j] + log_beta[n][j] - sum
log_gamma = y
return log_gamma
def gmmloglik(log_emlik, weights):
"""Log Likelihood for a GMM model based on Multivariate Normal Distribution.
Args:
log_emlik: array like, shape (N, K).
contains the log likelihoods for each of N observations and
each of K distributions
weights: weight vector for the K components in the mixture
Output:
gmmloglik: scalar, log likelihood of data given the GMM model.
"""
log_lik_gmm = 0
for obs in range(len(log_emlik)):
log_lik_gmm += tools2.logsumexp(log_emlik[obs, :] + np.log(weights))
return log_lik_gmm
def forward(log_emlik, log_startprob, log_transmat):
N = len(log_emlik)
M = len(log_emlik[0])
logAlpha = [[0 for x in range(M)] for y in range(N)]
for j in range(M):
logAlpha[0][j] = log_startprob[j] + log_emlik[0][j]
for n in range(1, N):
for j in range(M):
logAlpha[n][j] = log_emlik[n][j]
# building the array of the log sum
sumArray = []
for i in range(M):
sumArray += [logAlpha[n - 1][i] + log_transmat[i][j]]
logAlpha[n][j] += tools2.logsumexp(np.array(sumArray))
return logAlpha
def backward(log_emlik, log_startprob, log_transmat):
N = len(log_emlik)
M = len(log_emlik[0])
logBeta = [[0 for x in range(M)] for y in range(N)]
for i in range(M):
logBeta[N - 1][i] = 0
for n in range(N - 2, -1, -1):
for i in range(M):
# building the array of the log sum
sumArray = []
for j in range(M):
sumArray += [
log_transmat[i][j] + log_emlik[n + 1][j] +
logBeta[n + 1][j]
]
logBeta[n][i] = tools2.logsumexp(np.array(sumArray))
return logBeta
def forward(log_emlik, log_startprob, log_transmat):
"""Forward probabilities in log domain.
Args:
log_emlik: NxM array of emission log likelihoods, N frames, M states
log_startprob: log probability to start in state i
log_transmat: log transition probability from state i to j
Output:
forward_prob: NxM array of forward log probabilities for each of the M states in the model
"""
log_a = np.zeros((log_emlik.shape[0], log_transmat.shape[0]))
for j in range(log_transmat.shape[0]):
log_a[0, j] = log_startprob[j] + log_emlik[0, j]
for i in range(1, log_emlik.shape[0]):
for j in range(log_transmat.shape[0]):
log_a[i, j] = tools2.logsumexp(
(log_a[i - 1, :] + log_transmat[:, j])) + log_emlik[i, j]
return log_a
for j in range(len(models)):
model = skm.log_multivariate_normal_density(tidigits[i]['mfcc'],
models[j]['gmm']['means'],
models[j]['gmm']['covars'], 'diag')
loglik[j] = proto2.gmmloglik(model, models[j]['gmm']['weights'])
gmm_class.append(loglik)
gmm_class = np.array(gmm_class)
# Find bigger values for each utterances
gmm_ret_labels = np.argmax(gmm_class, axis=1)
# Compute 6: HMM Likelihood and Recognition
log_alpha = proto2.forward(hmm_obsloglik, np.log(models[0]['hmm']['startprob']), np.log(models[0]['hmm']['transmat']) )
# Compute marginalization:
hmm_loglik = tools2.logsumexp( log_alpha[-1, :] )
# Try every models on each utterances (tidigits) and find the best --> conclude
hmm_class = []
# For each utterances
for i in range(len(tidigits)):
loglik = np.zeros(len(models))
for j in range(len(models)):
model = skm.log_multivariate_normal_density(tidigits[i]['mfcc'],
models[j]['hmm']['means'],
models[j]['hmm']['covars'], 'diag')
log_alpha_class = proto2.forward(model, np.log(models[j]['hmm']['startprob']), np.log(models[j]['hmm']['transmat']) )
loglik[j] = tools2.logsumexp( log_alpha_class[-1, :] )
hmm_class.append(loglik)
hmm_class = np.array(hmm_class)
def gmmloglik(logAlpha):
return tools2.logsumexp(np.array(logAlpha[-1]))
def quest_6_1(flag):
#--- Quest 6.1: ---->>> Forward Algorithm
# --->>> check
hmm_log_emlik = tools2.log_multivariate_normal_density_diag(
X, models[0]['hmm']['means'], models[0]['hmm']['covars'])
log_a = forward(hmm_log_emlik, np.log(models[0]['hmm']['startprob']),
np.log(models[0]['hmm']['transmat']))
check_log_a = example['hmm_logalpha']
# print("log_a =", log_a)
# print("check_log_a =", check_log_a)
fig3 = plt.figure()
plt.subplot(2, 1, 1)
plt.imshow(log_a.T, cmap='jet')
plt.title('log_a')
plt.xticks([], [])
plt.gca().invert_yaxis()
plt.subplot(2, 1, 2)
plt.imshow(example['hmm_logalpha'].T, cmap='jet')
plt.title('ckeck log_a')
plt.xticks([], [])
plt.gca().invert_yaxis()
plt.savefig("../figs/quest_6_1.png", bbox_inches='tight')
# plt.show()
#Convert the formula you have derived into log domain
log_lik_a = tools2.logsumexp(log_a[-1, :])
check_log_lik_a = example['hmm_loglik']
if log_lik_a == check_log_lik_a:
print('True')
if flag == 'HMM':
utters = len(tidigits)
models_len = len(models)
log_lik_a = np.zeros((utters, models_len))
for utter in range(utters):
for digit in range(models_len):
hmm_log_emlik = tools2.log_multivariate_normal_density_diag(
tidigits[utter]['mfcc'], models[digit]['hmm']['means'],
models[digit]['hmm']['covars'])
log_a = forward(hmm_log_emlik,
np.log(models[digit]['hmm']['startprob']),
np.log(models[digit]['hmm']['transmat']))
log_lik_a[utter, digit] = tools2.logsumexp(log_a[-1, :])
#---->>> Check for misrecognized utterances
miss = 0
print("------ HMM a-pass ------")
for utter in range(utters):
best_score = np.argmax(log_lik_a[utter, :])
print('tid digit, mod digit: ---> ' +
str(tidigits[utter]['digit']) + " - " +
str(models[best_score]['digit'])
) # Uncomment to see the results analytically!
if models[best_score]['digit'] != tidigits[utter]['digit']:
miss += 1
accuracy = ((utters - miss) / utters) * 100
print()
print("Misrecognized %d out of %d utterances." % (miss, len(tidigits)))
print("Accuracy = " + str("%.2f" % round(accuracy, 2)) + '%')
print()
elif flag == "HMM as GMM":
utters = len(tidigits)
models_len = len(models)
log_like_hmm = np.zeros((utters, models_len))
for utter in range(utters):
for digit in range(models_len):
hmm_log_emlik = tools2.log_multivariate_normal_density_diag(
tidigits[utter]['mfcc'], models[digit]['hmm']['means'],
models[digit]['hmm']['covars'])
weights_hmm = np.ones(
models[digit]['hmm']['startprob'].shape[0]
) / models[digit]['hmm']['startprob'].shape[0]
log_like_hmm[utter, digit] = gmmloglik(hmm_log_emlik,
weights_hmm)
#---->>> Check for misrecognized utterances
miss = 0
print("------ HMM as GMM ------")
for utter in range(utters):
best_score = np.argmax(log_like_hmm[utter, :])
print('tid digit, mod digit: ---> ' +
str(tidigits[utter]['digit']) + " - " +
str(models[best_score]['digit'])
) # Uncomment to see the results analytically!
if models[best_score]['digit'] != tidigits[utter]['digit']:
miss += 1
accuracy = ((utters - miss) / utters) * 100
print()
print("Misrecognized %d out of %d utterances." % (miss, len(tidigits)))
print("Accuracy = " + str("%.2f" % round(accuracy, 2)) + '%')
print()