I'm doing a Python project in which I'd like to use the Viterbi Algorithm. Does anyone know of a complete Python implementation of the Viterbi algorithm? The correctness of the one on Wikipedia seems to be in question on the talk page. Does anyone have a pointer?
Hmm I can post mine. Its not pretty though, please let me know if you need clarification. I wrote this relatively recently for specifically part of speech tagging.
class Trellis:
trell = []
def __init__(self, hmm, words):
self.trell = []
temp = {}
for label in hmm.labels:
temp[label] = [0,None]
for word in words:
self.trell.append([word,copy.deepcopy(temp)])
self.fill_in(hmm)
def fill_in(self,hmm):
for i in range(len(self.trell)):
for token in self.trell[i][1]:
word = self.trell[i][0]
if i == 0:
self.trell[i][1][token][0] = hmm.e(token,word)
else:
max = None
guess = None
c = None
for k in self.trell[i-1][1]:
c = self.trell[i-1][1][k][0] + hmm.t(k,token)
if max == None or c > max:
max = c
guess = k
max += hmm.e(token,word)
self.trell[i][1][token][0] = max
self.trell[i][1][token][1] = guess
def return_max(self):
tokens = []
token = None
for i in range(len(self.trell)-1,-1,-1):
if token == None:
max = None
guess = None
for k in self.trell[i][1]:
if max == None or self.trell[i][1][k][0] > max:
max = self.trell[i][1][k][0]
token = self.trell[i][1][k][1]
guess = k
tokens.append(guess)
else:
tokens.append(token)
token = self.trell[i][1][token][1]
tokens.reverse()
return tokens
I found the following code in the example repository of Artificial Intelligence: A Modern Approach. Is something like this what you're looking for?
def viterbi_segment(text, P):
"""Find the best segmentation of the string of characters, given the
UnigramTextModel P."""
# best[i] = best probability for text[0:i]
# words[i] = best word ending at position i
n = len(text)
words = [''] + list(text)
best = [1.0] + [0.0] * n
## Fill in the vectors best, words via dynamic programming
for i in range(n+1):
for j in range(0, i):
w = text[j:i]
if P[w] * best[i - len(w)] >= best[i]:
best[i] = P[w] * best[i - len(w)]
words[i] = w
## Now recover the sequence of best words
sequence = []; i = len(words)-1
while i > 0:
sequence[0:0] = [words[i]]
i = i - len(words[i])
## Return sequence of best words and overall probability
return sequence, best[-1]
Here's mine. Its paraphrased directly from the psuedocode implemenation from wikipedia. It uses numpy for conveince of their ndarray but is otherwise a pure python3 implementation.
import numpy as np
def viterbi(y, A, B, Pi=None):
"""
Return the MAP estimate of state trajectory of Hidden Markov Model.
Parameters
----------
y : array (T,)
Observation state sequence. int dtype.
A : array (K, K)
State transition matrix. See HiddenMarkovModel.state_transition for
details.
B : array (K, M)
Emission matrix. See HiddenMarkovModel.emission for details.
Pi: optional, (K,)
Initial state probabilities: Pi[i] is the probability x[0] == i. If
None, uniform initial distribution is assumed (Pi[:] == 1/K).
Returns
-------
x : array (T,)
Maximum a posteriori probability estimate of hidden state trajectory,
conditioned on observation sequence y under the model parameters A, B,
Pi.
T1: array (K, T)
the probability of the most likely path so far
T2: array (K, T)
the x_j-1 of the most likely path so far
"""
# Cardinality of the state space
K = A.shape[0]
# Initialize the priors with default (uniform dist) if not given by caller
Pi = Pi if Pi is not None else np.full(K, 1 / K)
T = len(y)
T1 = np.empty((K, T), 'd')
T2 = np.empty((K, T), 'B')
# Initilaize the tracking tables from first observation
T1[:, 0] = Pi * B[:, y[0]]
T2[:, 0] = 0
# Iterate throught the observations updating the tracking tables
for i in range(1, T):
T1[:, i] = np.max(T1[:, i - 1] * A.T * B[np.newaxis, :, y[i]].T, 1)
T2[:, i] = np.argmax(T1[:, i - 1] * A.T, 1)
# Build the output, optimal model trajectory
x = np.empty(T, 'B')
x[-1] = np.argmax(T1[:, T - 1])
for i in reversed(range(1, T)):
x[i - 1] = T2[x[i], i]
return x, T1, T2
I have just corrected the pseudo implementation of Viterbi in Wikipedia. From the initial (incorrect) version, it took me a while to figure out where I was going wrong but I finally managed it, thanks partly to Kevin Murphy's implementation of the viterbi_path.m in the MatLab HMM toolbox.
In the context of an HMM object with variables as shown:
hmm = HMM()
hmm.priors = np.array([0.5, 0.5]) # pi = prior probs
hmm.transition = np.array([[0.75, 0.25], # A = transition probs. / 2 states
[0.32, 0.68]])
hmm.emission = np.array([[0.8, 0.1, 0.1], # B = emission (observation) probs. / 3 obs modes
[0.1, 0.2, 0.7]])
The Python function to run Viterbi (best-path) algorithm is below:
def viterbi (self,observations):
"""Return the best path, given an HMM model and a sequence of observations"""
# A - initialise stuff
nSamples = len(observations[0])
nStates = self.transition.shape[0] # number of states
c = np.zeros(nSamples) #scale factors (necessary to prevent underflow)
viterbi = np.zeros((nStates,nSamples)) # initialise viterbi table
psi = np.zeros((nStates,nSamples)) # initialise the best path table
best_path = np.zeros(nSamples); # this will be your output
# B- appoint initial values for viterbi and best path (bp) tables - Eq (32a-32b)
viterbi[:,0] = self.priors.T * self.emission[:,observations(0)]
c[0] = 1.0/np.sum(viterbi[:,0])
viterbi[:,0] = c[0] * viterbi[:,0] # apply the scaling factor
psi[0] = 0;
# C- Do the iterations for viterbi and psi for time>0 until T
for t in range(1,nSamples): # loop through time
for s in range (0,nStates): # loop through the states @(t-1)
trans_p = viterbi[:,t-1] * self.transition[:,s]
psi[s,t], viterbi[s,t] = max(enumerate(trans_p), key=operator.itemgetter(1))
viterbi[s,t] = viterbi[s,t]*self.emission[s,observations(t)]
c[t] = 1.0/np.sum(viterbi[:,t]) # scaling factor
viterbi[:,t] = c[t] * viterbi[:,t]
# D - Back-tracking
best_path[nSamples-1] = viterbi[:,nSamples-1].argmax() # last state
for t in range(nSamples-1,0,-1): # states of (last-1)th to 0th time step
best_path[t-1] = psi[best_path[t],t]
return best_path
This is an old question, but none of the other answers were quite what I needed because my application doesn't have specific observed states.
Taking after @Rhubarb, I've also re-implemented Kevin Murphey's Matlab implementation (see viterbi_path.m), but I've kept it closer to the original. I've included a simple test case as well.
import numpy as np
def viterbi_path(prior, transmat, obslik, scaled=True, ret_loglik=False):
'''Finds the most-probable (Viterbi) path through the HMM state trellis
Notation:
Z[t] := Observation at time t
Q[t] := Hidden state at time t
Inputs:
prior: np.array(num_hid)
prior[i] := Pr(Q[0] == i)
transmat: np.ndarray((num_hid,num_hid))
transmat[i,j] := Pr(Q[t+1] == j | Q[t] == i)
obslik: np.ndarray((num_hid,num_obs))
obslik[i,t] := Pr(Z[t] | Q[t] == i)
scaled: bool
whether or not to normalize the probability trellis along the way
doing so prevents underflow by repeated multiplications of probabilities
ret_loglik: bool
whether or not to return the log-likelihood of the best path
Outputs:
path: np.array(num_obs)
path[t] := Q[t]
'''
num_hid = obslik.shape[0] # number of hidden states
num_obs = obslik.shape[1] # number of observations (not observation *states*)
# trellis_prob[i,t] := Pr((best sequence of length t-1 goes to state i), Z[1:(t+1)])
trellis_prob = np.zeros((num_hid,num_obs))
# trellis_state[i,t] := best predecessor state given that we ended up in state i at t
trellis_state = np.zeros((num_hid,num_obs), dtype=int) # int because its elements will be used as indicies
path = np.zeros(num_obs, dtype=int) # int because its elements will be used as indicies
trellis_prob[:,0] = prior * obslik[:,0] # element-wise mult
if scaled:
scale = np.ones(num_obs) # only instantiated if necessary to save memory
scale[0] = 1.0 / np.sum(trellis_prob[:,0])
trellis_prob[:,0] *= scale[0]
trellis_state[:,0] = 0 # arbitrary value since t == 0 has no predecessor
for t in xrange(1, num_obs):
for j in xrange(num_hid):
trans_probs = trellis_prob[:,t-1] * transmat[:,j] # element-wise mult
trellis_state[j,t] = trans_probs.argmax()
trellis_prob[j,t] = trans_probs[trellis_state[j,t]] # max of trans_probs
trellis_prob[j,t] *= obslik[j,t]
if scaled:
scale[t] = 1.0 / np.sum(trellis_prob[:,t])
trellis_prob[:,t] *= scale[t]
path[-1] = trellis_prob[:,-1].argmax()
for t in range(num_obs-2, -1, -1):
path[t] = trellis_state[(path[t+1]), t+1]
if not ret_loglik:
return path
else:
if scaled:
loglik = -np.sum(np.log(scale))
else:
p = trellis_prob[path[-1],-1]
loglik = np.log(p)
return path, loglik
if __name__=='__main__':
# Assume there are 3 observation states, 2 hidden states, and 5 observations
priors = np.array([0.5, 0.5])
transmat = np.array([
[0.75, 0.25],
[0.32, 0.68]])
emmat = np.array([
[0.8, 0.1, 0.1],
[0.1, 0.2, 0.7]])
observations = np.array([0, 1, 2, 1, 0], dtype=int)
obslik = np.array([emmat[:,z] for z in observations]).T
print viterbi_path(priors, transmat, obslik) #=> [0 1 1 1 0]
print viterbi_path(priors, transmat, obslik, scaled=False) #=> [0 1 1 1 0]
print viterbi_path(priors, transmat, obslik, ret_loglik=True) #=> (array([0, 1, 1, 1, 0]), -7.776472586614755)
print viterbi_path(priors, transmat, obslik, scaled=False, ret_loglik=True) #=> (array([0, 1, 1, 1, 0]), -8.0120386579275227)
Note that this implementation does not use emission probabilities directly but uses a variable obslik. Generally, emissions[i,j] := Pr(observed_state == j | hidden_state == i) for a particular observed state i, making emissions.shape == (num_hidden_states, num_obs_states).
However, given a sequence observations[t] := observation at time t, all the Viterbi Algorithm requires is the likelihood of that observation for each hidden state. Hence, obslik[i,t] := Pr(observations[t] | hidden_state == i). The actual value the of the observed state isn't necessary.
I have modified @Rhubarb's answer for the condition where the marginal probabilities are already known (e.g by computing the Forward Backward algorithm).
def viterbi (transition_probabilities, conditional_probabilities):
# Initialise everything
num_samples = conditional_probabilities.shape[1]
num_states = transition_probabilities.shape[0] # number of states
c = np.zeros(num_samples) #scale factors (necessary to prevent underflow)
viterbi = np.zeros((num_states,num_samples)) # initialise viterbi table
best_path_table = np.zeros((num_states,num_samples)) # initialise the best path table
best_path = np.zeros(num_samples).astype(np.int32) # this will be your output
# B- appoint initial values for viterbi and best path (bp) tables - Eq (32a-32b)
viterbi[:,0] = conditional_probabilities[:,0]
c[0] = 1.0/np.sum(viterbi[:,0])
viterbi[:,0] = c[0] * viterbi[:,0] # apply the scaling factor
# C- Do the iterations for viterbi and psi for time>0 until T
for t in range(1, num_samples): # loop through time
for s in range (0,num_states): # loop through the states @(t-1)
trans_p = viterbi[:, t-1] * transition_probabilities[:,s] # transition probs of each state transitioning
best_path_table[s,t], viterbi[s,t] = max(enumerate(trans_p), key=operator.itemgetter(1))
viterbi[s,t] = viterbi[s,t] * conditional_probabilities[s][t]
c[t] = 1.0/np.sum(viterbi[:,t]) # scaling factor
viterbi[:,t] = c[t] * viterbi[:,t]
## D - Back-tracking
best_path[num_samples-1] = viterbi[:,num_samples-1].argmax() # last state
for t in range(num_samples-1,0,-1): # states of (last-1)th to 0th time step
best_path[t-1] = best_path_table[best_path[t],t]
return best_path
来源：https://stackoverflow.com/questions/9729968/python-implementation-of-viterbi-algorithm