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hmmc_scaled_diag.py
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# https://udemy.com/unsupervised-machine-learning-hidden-markov-models-in-python
# http://lazyprogrammer.me
# Continuous-observation HMM with scaling and diagonal covariance.
import wave
import numpy as np
import matplotlib.pyplot as plt
from generate_c import get_signals, big_init, simple_init
def random_normalized(d1, d2):
x = np.random.random((d1, d2))
return x / x.sum(axis=1, keepdims=True)
def mvn_pdf(x, mu, sigma):
D = len(x)
c = 1 / np.sqrt(((2*np.pi)**D) * sigma.prod())
e = np.exp( -0.5*((x - mu) * (x - mu) / sigma).sum() )
return c * e
class HMM:
def __init__(self, M, K):
self.M = M # number of hidden states
self.K = K # number of Gaussians
def fit(self, X, max_iter=30, eps=10e-2):
# train the HMM model using the Baum-Welch algorithm
# a specific instance of the expectation-maximization algorithm
N = len(X)
D = X[0].shape[1] # assume each x is organized (T, D)
self.pi = np.ones(self.M) / self.M # initial state distribution
self.A = random_normalized(self.M, self.M) # state transition matrix
self.R = np.ones((self.M, self.K)) / self.K # mixture proportions
self.mu = np.zeros((self.M, self.K, D))
for i in xrange(self.M):
for k in xrange(self.K):
random_idx = np.random.choice(N)
x = X[random_idx]
random_time_idx = np.random.choice(len(x))
self.mu[i,k] = x[random_time_idx]
self.sigma = np.ones((self.M, self.K, D))
costs = []
for it in xrange(max_iter):
if it % 1 == 0:
print "it:", it
alphas = []
betas = []
scales = []
gammas = []
Bs = []
# components = []
logP = np.zeros(N)
for n in xrange(N):
x = X[n]
T = len(x)
scale = np.zeros(T)
# calculate B so we can lookup when updating alpha and beta
B = np.zeros((self.M, T))
component = np.zeros((self.M, self.K, T)) # we'll need these later
C = np.zeros(T)
for j in xrange(self.M):
for t in xrange(T):
for k in xrange(self.K):
# print "sigma:", self.sigma[j,k]
# p = self.R[j,k] * mvn.pdf(x[t], self.mu[j,k], self.sigma[j,k])
tmp = self.R[j,k] * mvn_pdf(x[t], self.mu[j,k], self.sigma[j,k])
# print "p - tmp:", (p - tmp)
# assert(np.abs(tmp - p).sum() < 10e-10)
p = tmp
component[j,k,t] = p
B[j,t] += p
# components.append(component)
Bs.append(B)
# assert(np.all(B <= 1))
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*B[:,0]
scale[0] = alpha[0].sum()
alpha[0] /= scale[0]
for t in xrange(1, T):
alpha_t_prime = alpha[t-1].dot(self.A) * B[:,t]
scale[t] = alpha_t_prime.sum()
alpha[t] = alpha_t_prime / scale[t]
logP[n] = np.log(scale).sum()
alphas.append(alpha)
scales.append(scale)
beta = np.zeros((T, self.M))
beta[-1] = 1
for t in xrange(T - 2, -1, -1):
beta[t] = self.A.dot(B[:,t+1] * beta[t+1]) / scale[t+1]
betas.append(beta)
# update for Gaussians
gamma = np.zeros((T, self.M, self.K))
for t in xrange(T):
alphabeta = alphas[n][t,:].dot(betas[n][t,:])
# print "alpabeta:", alphabeta
for j in xrange(self.M):
factor = alphas[n][t,j] * betas[n][t,j] / alphabeta
# mixture_j = component[j,:,t].sum()
for k in xrange(self.K):
gamma[t,j,k] = factor * component[j,k,t] / B[j,t]
gammas.append(gamma)
cost = logP.sum()
costs.append(cost)
# now re-estimate pi, A, R, mu, sigma
self.pi = np.sum((alphas[n][0] * betas[n][0]) for n in xrange(N)) / N
a_den = np.zeros((self.M, 1))
a_num = np.zeros((self.M, self.M))
r_num = np.zeros((self.M, self.K))
r_den = np.zeros(self.M)
mu_num = np.zeros((self.M, self.K, D))
sigma_num = np.zeros((self.M, self.K, D))
for n in xrange(N):
x = X[n]
T = len(x)
B = Bs[n]
gamma = gammas[n]
a_den += (alphas[n][:-1] * betas[n][:-1]).sum(axis=0, keepdims=True).T
# numerator for A
for i in xrange(self.M):
for j in xrange(self.M):
for t in xrange(T-1):
a_num[i,j] += alphas[n][t,i] * betas[n][t+1,j] * self.A[i,j] * B[j,t+1] / scales[n][t+1]
# update mixture components
r_num_n = np.zeros((self.M, self.K))
r_den_n = np.zeros(self.M)
for j in xrange(self.M):
for k in xrange(self.K):
for t in xrange(T):
r_num_n[j,k] += gamma[t,j,k]
r_den_n[j] += gamma[t,j,k]
r_num += r_num_n * (-logP[n])
r_den += r_den_n * (-logP[n])
# r_num += r_num_n
# r_den += r_den_n
mu_num_n = np.zeros((self.M, self.K, D))
sigma_num_n = np.zeros((self.M, self.K, D))
for j in xrange(self.M):
for k in xrange(self.K):
for t in xrange(T):
# update means
mu_num_n[j,k] += gamma[t,j,k] * x[t]
# update covariances
sigma_num_n[j,k] += gamma[t,j,k] * (x[t] - self.mu[j,k]) * (x[t] - self.mu[j,k])
mu_num += mu_num_n * (-logP[n])
sigma_num += sigma_num_n * (-logP[n])
# mu_num += mu_num_n
# sigma_num += sigma_num_n
self.A = a_num / a_den
# update R, mu, sigma
for j in xrange(self.M):
for k in xrange(self.K):
self.R[j,k] = r_num[j,k] / r_den[j]
self.mu[j,k] = mu_num[j,k] / r_num[j,k]
self.sigma[j,k] = sigma_num[j,k] / r_num[j,k]
self.sigma += eps
assert(np.all(self.R <= 1))
assert(np.all(self.A <= 1))
print "A:", self.A
print "mu:", self.mu
print "sigma:", self.sigma
print "R:", self.R
print "pi:", self.pi
plt.plot(costs)
plt.show()
def log_likelihood(self, x):
# returns log P(x | model)
# using the forward part of the forward-backward algorithm
T = len(x)
scale = np.zeros(T)
B = np.zeros((self.M, T))
for j in xrange(self.M):
for t in xrange(T):
for k in xrange(self.K):
# p = self.R[j,k] * mvn.pdf(x[t], self.mu[j,k], self.sigma[j,k])
p = self.R[j,k] * mvn_pdf(x[t], self.mu[j,k], self.sigma[j,k])
B[j,t] += p
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*B[:,0]
scale[0] = alpha[0].sum()
alpha[0] /= scale[0]
for t in xrange(1, T):
alpha_t_prime = alpha[t-1].dot(self.A) * B[:,t]
scale[t] = alpha_t_prime.sum()
alpha[t] = alpha_t_prime / scale[t]
return np.log(scale).sum()
def log_likelihood_multi(self, X):
return np.array([self.log_likelihood(x) for x in X])
def set(self, pi, A, R, mu, sigma):
self.pi = pi
self.A = A
self.R = R
self.mu = mu
self.sigma = sigma
M, K = R.shape
self.M = M
self.K = K
def real_signal():
spf = wave.open('helloworld.wav', 'r')
#Extract Raw Audio from Wav File
# If you right-click on the file and go to "Get Info", you can see:
# sampling rate = 16000 Hz
# bits per sample = 16
# The first is quantization in time
# The second is quantization in amplitude
# We also do this for images!
# 2^16 = 65536 is how many different sound levels we have
signal = spf.readframes(-1)
signal = np.fromstring(signal, 'Int16')
T = len(signal)
signal = (signal - signal.mean()) / signal.std()
hmm = HMM(5, 3)
hmm.fit(signal.reshape(1, T, 1))
def fake_signal(init=big_init):
signals = get_signals(init=init)
# for signal in signals:
# print "mean:", signal.mean(axis=0), "var:", signal.var(axis=0)
hmm = HMM(5, 3)
hmm.fit(signals)
L = hmm.log_likelihood_multi(signals).sum()
print "LL for fitted params:", L
# test in actual params
_, _, _, pi, A, R, mu, sigma = init()
M, K, D = mu.shape
diag_sigma = np.zeros((M, K, D))
for j in xrange(M):
for k in xrange(K):
diag_sigma[j,k] = np.diag(sigma[j,k])
hmm.set(pi, A, R, mu, diag_sigma)
L = hmm.log_likelihood_multi(signals).sum()
print "LL for actual params:", L
if __name__ == '__main__':
real_signal()
# fake_signal()