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hmmd_scaled.py
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# https://deeplearningcourses.com/c/unsupervised-machine-learning-hidden-markov-models-in-python
# https://udemy.com/unsupervised-machine-learning-hidden-markov-models-in-python
# http://lazyprogrammer.me
# Discrete Hidden Markov Model (HMM) with scaling
import numpy as np
import matplotlib.pyplot as plt
def random_normalized(d1, d2):
x = np.random.random((d1, d2))
return x / x.sum(axis=1, keepdims=True)
class HMM:
def __init__(self, M):
self.M = M # number of hidden states
def fit(self, X, max_iter=30):
np.random.seed(123)
# train the HMM model using the Baum-Welch algorithm
# a specific instance of the expectation-maximization algorithm
# determine V, the vocabulary size
# assume observables are already integers from 0..V-1
# X is a jagged array of observed sequences
V = max(max(x) for x in X) + 1
N = len(X)
self.pi = np.ones(self.M) / self.M # initial state distribution
self.A = random_normalized(self.M, self.M) # state transition matrix
self.B = random_normalized(self.M, V) # output distribution
print "initial A:", self.A
print "initial B:", self.B
costs = []
for it in xrange(max_iter):
if it % 10 == 0:
print "it:", it
# alpha1 = np.zeros((N, self.M))
alphas = []
betas = []
scales = []
logP = np.zeros(N)
for n in xrange(N):
x = X[n]
T = len(x)
scale = np.zeros(T)
# alpha1[n] = self.pi*self.B[:,x[0]]
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*self.B[:,x[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) * self.B[:, x[t]]
scale[t] = alpha_t_prime.sum()
alpha[t] = alpha_t_prime / scale[t]
# P[n] = alpha[-1].sum()
# print "alpha[-1].sum():", alpha[-1].sum()
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(self.B[:, x[t+1]] * beta[t+1]) / scale[t+1]
betas.append(beta)
cost = np.sum(logP)
costs.append(cost)
# now re-estimate pi, A, B
self.pi = np.sum((alphas[n][0] * betas[n][0]) for n in xrange(N)) / N
# print "self.pi:", self.pi
# break
den1 = np.zeros((self.M, 1))
den2 = np.zeros((self.M, 1))
a_num = np.zeros((self.M, self.M))
b_num = np.zeros((self.M, V))
for n in xrange(N):
x = X[n]
T = len(x)
# print "den shape:", den.shape
# test = (alphas[n][:-1] * betas[n][:-1]).sum(axis=0, keepdims=True).T
# print "shape (alphas[n][:-1] * betas[n][:-1]).sum(axis=0): ", test.shape
den1 += (alphas[n][:-1] * betas[n][:-1]).sum(axis=0, keepdims=True).T
den2 += (alphas[n] * betas[n]).sum(axis=0, keepdims=True).T
# numerator for A
# a_num_n = np.zeros((self.M, self.M))
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] * self.B[j, x[t+1]] / scales[n][t+1]
# a_num += a_num_n
# numerator for B
# for i in xrange(self.M):
# for j in xrange(V):
# for t in xrange(T):
# if x[t] == j:
# b_num[i,j] += alphas[n][t][i] * betas[n][t][i]
for i in xrange(self.M):
for t in xrange(T):
b_num[i,x[t]] += alphas[n][t,i] * betas[n][t,i]
self.A = a_num / den1
self.B = b_num / den2
# print "new A:", self.A
# break
# print "P:", P
print "A:", self.A
print "B:", self.B
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)
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*self.B[:,x[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) * self.B[:, x[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 get_state_sequence(self, x):
# returns the most likely state sequence given observed sequence x
# using the Viterbi algorithm
T = len(x)
delta = np.zeros((T, self.M))
psi = np.zeros((T, self.M))
delta[0] = np.log(self.pi) + np.log(self.B[:,x[0]])
for t in xrange(1, T):
for j in xrange(self.M):
delta[t,j] = np.max(delta[t-1] + np.log(self.A[:,j])) + np.log(self.B[j, x[t]])
psi[t,j] = np.argmax(delta[t-1] + np.log(self.A[:,j]))
# backtrack
states = np.zeros(T, dtype=np.int32)
states[T-1] = np.argmax(delta[T-1])
for t in xrange(T-2, -1, -1):
states[t] = psi[t+1, states[t+1]]
return states
def fit_coin():
X = []
for line in open('coin_data.txt'):
# 1 for H, 0 for T
x = [1 if e == 'H' else 0 for e in line.rstrip()]
X.append(x)
hmm = HMM(2)
hmm.fit(X)
L = hmm.log_likelihood_multi(X).sum()
print "LL with fitted params:", L
# try true values
hmm.pi = np.array([0.5, 0.5])
hmm.A = np.array([[0.1, 0.9], [0.8, 0.2]])
hmm.B = np.array([[0.6, 0.4], [0.3, 0.7]])
L = hmm.log_likelihood_multi(X).sum()
print "LL with true params:", L
# try viterbi
print "Best state sequence for:", X[0]
print hmm.get_state_sequence(X[0])
if __name__ == '__main__':
fit_coin()