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hmmd.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)
from __future__ import print_function, division
from builtins import range
# Note: you may need to update your version of future
# sudo pip install -U future
import numpy as np
import matplotlib.pyplot as plt
from datetime import datetime
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):
t0 = datetime.now()
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 range(max_iter):
if it % 10 == 0:
print("it:", it)
alphas = []
betas = []
P = np.zeros(N)
for n in range(N):
x = X[n]
T = len(x)
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*self.B[:,x[0]]
for t in range(1, T):
tmp1 = alpha[t-1].dot(self.A) * self.B[:, x[t]]
# tmp2 = np.zeros(self.M)
# for i in range(self.M):
# for j in range(self.M):
# tmp2[j] += alpha[t-1,i] * self.A[i,j] * self.B[j, x[t]]
# print "diff:", np.abs(tmp1 - tmp2).sum()
alpha[t] = tmp1
P[n] = alpha[-1].sum()
alphas.append(alpha)
beta = np.zeros((T, self.M))
beta[-1] = 1
for t in range(T - 2, -1, -1):
beta[t] = self.A.dot(self.B[:, x[t+1]] * beta[t+1])
betas.append(beta)
# print "P:", P
# break
assert(np.all(P > 0))
cost = np.sum(np.log(P))
costs.append(cost)
# now re-estimate pi, A, B
self.pi = np.sum((alphas[n][0] * betas[n][0])/P[n] for n in range(N)) / N
# print "self.pi:", self.pi
# break
den1 = np.zeros((self.M, 1))
den2 = np.zeros((self.M, 1))
a_num = 0
b_num = 0
for n in range(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 / P[n]
den2 += (alphas[n] * betas[n]).sum(axis=0, keepdims=True).T / P[n]
# tmp2 = np.zeros((self.M, 1))
# for i in range(self.M):
# for t in range(T-1):
# tmp2[i] += alphas[n][t,i] * betas[n][t,i]
# tmp2 /= P[n]
# # print "diff:", np.abs(tmp1 - tmp2).sum()
# den += tmp1
# numerator for A
a_num_n = np.zeros((self.M, self.M))
for i in range(self.M):
for j in range(self.M):
for t in range(T-1):
a_num_n[i,j] += alphas[n][t,i] * self.A[i,j] * self.B[j, x[t+1]] * betas[n][t+1,j]
a_num += a_num_n / P[n]
# numerator for B
# b_num_n = np.zeros((self.M, V))
# for i in range(self.M):
# for j in range(V):
# for t in range(T):
# if x[t] == j:
# b_num_n[i,j] += alphas[n][t][i] * betas[n][t][i]
b_num_n2 = np.zeros((self.M, V))
for i in range(self.M):
for t in range(T):
b_num_n2[i,x[t]] += alphas[n][t,i] * betas[n][t,i]
b_num += b_num_n2 / P[n]
# tmp1 = a_num / den1
# tmp2 = np.zeros(a_num.shape)
# for i in range(self.M):
# for j in range(self.M):
# tmp2[i,j] = a_num[i,j] / den1[i]
# print "diff:", np.abs(tmp1 - tmp2).sum()
# print "tmp1:", tmp1
# print "tmp2:", tmp2
self.A = a_num / den1
self.B = b_num / den2
# print "P:", P
# break
print("A:", self.A)
print("B:", self.B)
print("pi:", self.pi)
print("Fit duration:", (datetime.now() - t0))
plt.plot(costs)
plt.show()
def likelihood(self, x):
# returns log P(x | model)
# using the forward part of the forward-backward algorithm
T = len(x)
alpha = np.zeros((T, self.M))
alpha[0] = self.pi*self.B[:,x[0]]
for t in range(1, T):
alpha[t] = alpha[t-1].dot(self.A) * self.B[:, x[t]]
return alpha[-1].sum()
def likelihood_multi(self, X):
return np.array([self.likelihood(x) for x in X])
def log_likelihood_multi(self, X):
return np.log(self.likelihood_multi(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] = self.pi*self.B[:,x[0]]
for t in range(1, T):
for j in range(self.M):
delta[t,j] = np.max(delta[t-1]*self.A[:,j]) * self.B[j, x[t]]
psi[t,j] = np.argmax(delta[t-1]*self.A[:,j])
# backtrack
states = np.zeros(T, dtype=np.int32)
states[T-1] = np.argmax(delta[T-1])
for t in range(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()