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import numpy as np
from copy import copy
import matplotlib.pyplot as plt
class HMM:
def __init__(self):
pass
def simulate(self,nSteps):
def drawFrom(probs):
return np.where(np.random.multinomial(1,probs) == 1)[0][0]
observations = np.zeros(nSteps)
states = np.zeros(nSteps)
states[0] = drawFrom(self.pi)
observations[0] = drawFrom(self.B[states[0],:])
for t in range(1,nSteps):
states[t] = drawFrom(self.A[states[t-1],:])
observations[t] = drawFrom(self.B[states[t],:])
return observations,states
def train(self,observations,criterion,graphics=False):
if graphics:
plt.ion()
nStates = self.A.shape[0]
nSamples = len(observations)
A = self.A
B = self.B
pi = copy(self.pi)
done = False
while not done:
# alpha_t(i) = P(O_1 O_2 ... O_t, q_t = S_i | hmm)
# Initialize alpha
alpha = np.zeros((nStates,nSamples))
c = np.zeros(nSamples) #scale factors
alpha[:,0] = pi.T * self.B[:,observations[0]]
c[0] = 1.0/np.sum(alpha[:,0])
alpha[:,0] = c[0] * alpha[:,0]
# Update alpha for each observation step
for t in range(1,nSamples):
alpha[:,t] = np.dot(alpha[:,t-1].T, self.A).T * self.B[:,observations[t]]
c[t] = 1.0/np.sum(alpha[:,t])
alpha[:,t] = c[t] * alpha[:,t]
# beta_t(i) = P(O_t+1 O_t+2 ... O_T | q_t = S_i , hmm)
# Initialize beta
beta = np.zeros((nStates,nSamples))
beta[:,nSamples-1] = 1
beta[:,nSamples-1] = c[nSamples-1] * beta[:,nSamples-1]
# Update beta backwards from end of sequence
for t in range(len(observations)-1,0,-1):
beta[:,t-1] = np.dot(self.A, (self.B[:,observations[t]] * beta[:,t]))
beta[:,t-1] = c[t-1] * beta[:,t-1]
xi = np.zeros((nStates,nStates,nSamples-1));
for t in range(nSamples-1):
denom = np.dot(np.dot(alpha[:,t].T, self.A) * self.B[:,observations[t+1]].T,
beta[:,t+1])
for i in range(nStates):
numer = alpha[i,t] * self.A[i,:] * self.B[:,observations[t+1]].T * \
beta[:,t+1].T
xi[i,:,t] = numer / denom
# gamma_t(i) = P(q_t = S_i | O, hmm)
gamma = np.squeeze(np.sum(xi,axis=1))
# Need final gamma element for new B
prod = (alpha[:,nSamples-1] * beta[:,nSamples-1]).reshape((-1,1))
gamma = np.hstack((gamma, prod / np.sum(prod))) #append one more to gamma!!!
newpi = gamma[:,0]
newA = np.sum(xi,2) / np.sum(gamma[:,:-1],axis=1).reshape((-1,1))
newB = copy(B)
if graphics:
plt.subplot(2,1,1)
plt.cla()
#plt.plot(gamma.T)
plt.plot(gamma[1])
plt.ylim(-0.1,1.1)
plt.legend(('Probability State=1'))
plt.xlabel('Time')
plt.draw()
numLevels = self.B.shape[1]
sumgamma = np.sum(gamma,axis=1)
for lev in range(numLevels):
mask = observations == lev
newB[:,lev] = np.sum(gamma[:,mask],axis=1) / sumgamma
if np.max(abs(pi - newpi)) < criterion and \
np.max(abs(A - newA)) < criterion and \
np.max(abs(B - newB)) < criterion:
done = 1;
A[:],B[:],pi[:] = newA,newB,newpi
self.A[:] = newA
self.B[:] = newB
self.pi[:] = newpi
self.gamma = gamma
if __name__ == '__main__':
np.set_printoptions(precision=3,suppress=True)
if True:
#'Two states, three possible observations in a state'
hmm = HMM()
hmm.pi = np.array([0.5, 0.5])
hmm.A = np.array([[0.85, 0.15],
[0.12, 0.88]])
hmm.B = np.array([[0.8, 0.1, 0.1],
[0.0, 0.0, 1]])
hmmguess = HMM()
hmmguess.pi = np.array([0.5, 0.5])
hmmguess.A = np.array([[0.5, 0.5],
[0.5, 0.5]])
hmmguess.B = np.array([[0.3, 0.3, 0.4],
[0.2, 0.5, 0.3]])
else:
#three states
print "Error....this example with three states is not working correctly."
hmm = HMM()
hmm.pi = np.array([0.1, 0.4, 0.5])
hmm.A = np.array([[0.7, 0.2, 0.1],
[0.1, 0.6, 0.3],
[0.4, 0.2, 0.4]])
hmm.B = np.array([[0.5, 0.3, 0.2],
[0.1, 0.6, 0.3],
[0.0, 0.3, 0.7]])
hmmguess = HMM()
hmmguess.pi = np.array([0.333, 0.333, 0.333])
hmmguess.A = np.array([[0.3333, 0.3333, 0.3333],
[0.3333, 0.3333, 0.3333],
[0.3333, 0.3333, 0.3333]])
hmmguess.B = np.array([[0.3, 0.3, 0.4],
[0.2, 0.5, 0.3],
[0.3, 0.3, 0.4]])
o,s = hmm.simulate(1000)
hmmguess.train(o,0.0001,graphics=True)
print 'Actual probabilities\n',hmm.pi
print 'Estimated initial probabilities\n',hmmguess.pi
print 'Actual state transition probabililities\n',hmm.A
print 'Estimated state transition probabililities\n',hmmguess.A
print 'Actual observation probabililities\n',hmm.B
print 'Estimated observation probabililities\n',hmmguess.B
plt.subplot(2,1,2)
plt.cla()
plt.plot(np.vstack((s*0.9+0.05,hmmguess.gamma[1,:])).T,'-o',alpha=0.7)
plt.legend(('True State','Guessed Probability of State=1'))
plt.ylim(-0.1,1.1)
plt.xlabel('Time')
plt.draw()
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