```# Natural Language Toolkit: Hidden Markov Model
#
# Copyright (C) 2001-2006 University of Pennsylvania
# Author: Trevor Cohn <tacohn@csse.unimelb.edu.au>
#         Philip Blunsom <pcbl@csse.unimelb.edu.au>
# URL: <http://nltk.sf.net>
#
# \$Id: hmm.py 3672 2006-11-07 16:27:02Z stevenbird \$

"""
Hidden Markov Models (HMMs) largely used to assign the correct label sequence
to sequential data or assess the probability of a given label and data
sequence. These models are finite state machines characterised by a number of
states, transitions between these states, and output symbols emitted while in
each state. The HMM is an extension to the Markov chain, where each state
corresponds deterministically to a given event. In the HMM the observation is
a probabilistic function of the state. HMMs share the Markov chain's
assumption, being that the probability of transition from one state to another
only depends on the current state - i.e. the series of states that led to the
current state are not used. They are also time invariant.

The HMM is a directed graph, with probability weighted edges (representing the
probability of a transition between the source and sink states) where each
vertex emits an output symbol when entered. The symbol (or observation) is
non-deterministically generated. For this reason, knowing that a sequence of
output observations was generated by a given HMM does not mean that the
corresponding sequence of states (and what the current state is) is known.
This is the 'hidden' in the hidden markov model.

Formally, a HMM can be characterised by:
- the output observation alphabet. This is the set of symbols which may be
observed as output of the system.
- the set of states.
- the transition probabilities M{a_{ij} = P(s_t = j | s_{t-1} = i)}. These
represent the probability of transition to each state from a given
state.
- the output probability matrix M{b_i(k) = P(X_t = o_k | s_t = i)}. These
represent the probability of observing each symbol in a given state.
- the initial state distribution. This gives the probability of starting
in each state.

To ground this discussion, take a common NLP application, part-of-speech (POS)
tagging. An HMM is desirable for this task as the highest probability tag
sequence can be calculated for a given sequence of word forms. This differs
from other tagging techniques which often tag each word individually, seeking
to optimise each individual tagging greedily without regard to the optimal
combination of tags for a larger unit, such as a sentence. The HMM does this
with the Viterbi algorithm, which efficiently computes the optimal path
through the graph given the sequence of words forms.

In POS tagging the states usually have a 1:1 correspondence with the tag
alphabet - i.e. each state represents a single tag. The output observation
alphabet is the set of word forms (the lexicon), and the remaining three
parameters are derived by a training regime. With this information the
probability of a given sentence can be easily derived, by simply summing the
probability of each distinct path through the model. Similarly, the highest
probability tagging sequence can be derived with the Viterbi algorithm,
yielding a state sequence which can be mapped into a tag sequence.

This discussion assumes that the HMM has been trained. This is probably the
most difficult task with the model, and requires either MLE estimates of the
parameters or unsupervised learning using the Baum-Welch algorithm, a variant
of EM.
"""

from en.parser.nltk_lite.probability import *
from numpy import *
import re

# _NINF = float('-inf')  # won't work on Windows
_NINF = float('-1e300')

_TEXT = 0  # index of text in a tuple
_TAG = 1   # index of tag in a tuple

class HiddenMarkovModel(object):
"""
Hidden Markov model class, a generative model for labelling sequence data.
These models define the joint probability of a sequence of symbols and
their labels (state transitions) as the product of the starting state
probability, the probability of each state transition, and the probability
of each observation being generated from each state. This is described in
more detail in the module documentation.

This implementation is based on the HMM description in Chapter 8, Huang,
Acero and Hon, Spoken Language Processing.
"""
def __init__(self, symbols, states, transitions, outputs, priors):
"""
Creates a hidden markov model parametised by the the states,
transition probabilities, output probabilities and priors.

@param  symbols:        the set of output symbols (alphabet)
@type   symbols:        (seq) of any
@param  states:         a set of states representing state space
@type   states:         seq of any
@param  transitions:    transition probabilities; Pr(s_i | s_j)
is the probability of transition from state i
given the model is in state_j
@type   transitions:    C{ConditionalProbDistI}
@param  outputs:        output probabilities; Pr(o_k | s_i) is the
probability of emitting symbol k when entering
state i
@type   outputs:        C{ConditionalProbDistI}
@param  priors:         initial state distribution; Pr(s_i) is the
probability of starting in state i
@type   priors:         C{ProbDistI}
"""

self._states = states
self._transitions = transitions
self._symbols = symbols
self._outputs = outputs
self._priors = priors

def probability(self, sequence):
"""
Returns the probability of the given symbol sequence. If the sequence
is labelled, then returns the joint probability of the symbol, state
sequence. Otherwise, uses the forward algorithm to find the
probability over all label sequences.

@return: the probability of the sequence
@rtype: float
@param sequence: the sequence of symbols which must contain the TEXT
property, and optionally the TAG property
@type sequence:  Token
"""
return exp(self.log_probability(sequence))

def log_probability(self, sequence):
"""
Returns the log-probability of the given symbol sequence. If the
sequence is labelled, then returns the joint log-probability of the
symbol, state sequence. Otherwise, uses the forward algorithm to find
the log-probability over all label sequences.

@return: the log-probability of the sequence
@rtype: float
@param sequence: the sequence of symbols which must contain the TEXT
property, and optionally the TAG property
@type sequence:  Token
"""

T = len(sequence)
N = len(self._states)

if T > 0 and sequence[_TAG]:
last_state = sequence[_TAG]
p = self._priors.logprob(last_state) + \
self._outputs[last_state].logprob(sequence[_TEXT])
for t in range(1, T):
state = sequence[t][_TAG]
p += self._transitions[last_state].logprob(state) + \
self._outputs[state].logprob(sequence[t][_TEXT])
return p
else:
alpha = self._forward_probability(sequence)
return p

def tag(self, unlabelled_sequence):
"""
Tags the sequence with the highest probability state sequence. This
uses the best_path method to find the Viterbi path.

@return: a labelled sequence of symbols
@rtype: list
@param unlabelled_sequence: the sequence of unlabelled symbols
@type unlabelled_sequence: list
"""

path = self.best_path(unlabelled_sequence)
for i in range(len(path)):
unlabelled_sequence[i] = (unlabelled_sequence[i][_TEXT], path[i])
return unlabelled_sequence

def _output_logprob(self, state, symbol):
"""
@return: the log probability of the symbol being observed in the given
state
@rtype: float
"""
return self._outputs[state].logprob(symbol)

def best_path(self, unlabelled_sequence):
"""
Returns the state sequence of the optimal (most probable) path through
the HMM. Uses the Viterbi algorithm to calculate this part by dynamic
programming.

@return: the state sequence
@rtype: sequence of any
@param unlabelled_sequence: the sequence of unlabelled symbols
@type unlabelled_sequence: list
"""

T = len(unlabelled_sequence)
N = len(self._states)
V = zeros((T, N), float64)
B = {}

# find the starting log probabilities for each state
symbol = unlabelled_sequence[_TEXT]
for i, state in enumerate(self._states):
V[0, i] = self._priors.logprob(state) + \
self._output_logprob(state, symbol)
B[0, state] = None

# find the maximum log probabilities for reaching each state at time t
for t in range(1, T):
symbol = unlabelled_sequence[t][_TEXT]
for j in range(N):
sj = self._states[j]
best = None
for i in range(N):
si = self._states[i]
va = V[t-1, i] + self._transitions[si].logprob(sj)
if not best or va > best:
best = (va, si)
V[t, j] = best + self._output_logprob(sj, symbol)
B[t, sj] = best

# find the highest probability final state
best = None
for i in range(N):
val = V[T-1, i]
if not best or val > best:
best = (val, self._states[i])

# traverse the back-pointers B to find the state sequence
current = best
sequence = [current]
for t in range(T-1, 0, -1):
last = B[t, current]
sequence.append(last)
current = last

sequence.reverse()
return sequence

def random_sample(self, rng, length):
"""
Randomly sample the HMM to generate a sentence of a given length. This
samples the prior distribution then the observation distribution and
transition distribution for each subsequent observation and state.
This will mostly generate unintelligible garbage, but can provide some
amusement.

@return:        the randomly created state/observation sequence,
generated according to the HMM's probability
distributions. The SUBTOKENS have TEXT and TAG
properties containing the observation and state
respectively.
@rtype:         list
@param rng:     random number generator
@type rng:      Random (or any object with a random() method)
@param length:  desired output length
@type length:   int
"""

# sample the starting state and symbol prob dists
tokens = []
state = self._sample_probdist(self._priors, rng.random(), self._states)
symbol = self._sample_probdist(self._outputs[state],
rng.random(), self._symbols)
tokens.append((symbol, state))

for i in range(1, length):
# sample the state transition and symbol prob dists
state = self._sample_probdist(self._transitions[state],
rng.random(), self._states)
symbol = self._sample_probdist(self._outputs[state],
rng.random(), self._symbols)
tokens.append((symbol, state))

def _sample_probdist(self, probdist, p, samples):
cum_p = 0
for sample in samples:
if cum_p <= p <= cum_p + add_p:
return sample
raise Exception('Invalid probability distribution - does not sum to one')

def entropy(self, unlabelled_sequence):
"""
Returns the entropy over labellings of the given sequence. This is
given by:

H(O) = - sum_S Pr(S | O) log Pr(S | O)

where the summation ranges over all state sequences, S. Let M{Z =
Pr(O) = sum_S Pr(S, O)} where the summation ranges over all state
sequences and O is the observation sequence. As such the entropy can
be re-expressed as:

H = - sum_S Pr(S | O) log [ Pr(S, O) / Z ]
= log Z - sum_S Pr(S | O) log Pr(S, 0)
= log Z - sum_S Pr(S | O) [ log Pr(S_0) + sum_t Pr(S_t | S_{t-1})
+ sum_t Pr(O_t | S_t) ]

The order of summation for the log terms can be flipped, allowing
dynamic programming to be used to calculate the entropy. Specifically,
we use the forward and backward probabilities (alpha, beta) giving:

H = log Z - sum_s0 alpha_0(s0) beta_0(s0) / Z * log Pr(s0)
+ sum_t,si,sj alpha_t(si) Pr(sj | si) Pr(O_t+1 | sj) beta_t(sj)
/ Z * log Pr(sj | si)
+ sum_t,st alpha_t(st) beta_t(st) / Z * log Pr(O_t | st)

This simply uses alpha and beta to find the probabilities of partial
sequences, constrained to include the given state(s) at some point in
time.
"""

T = len(unlabelled_sequence)
N = len(self._states)

alpha = self._forward_probability(unlabelled_sequence)
beta = self._backward_probability(unlabelled_sequence)

entropy = normalisation

# starting state, t = 0
for i, state in enumerate(self._states):
p = exp(alpha[0, i] + beta[0, i] - normalisation)
entropy -= p * self._priors.logprob(state)
#print 'p(s_0 = %s) =' % state, p

# state transitions
for t0 in range(T - 1):
t1 = t0 + 1
for i0, s0 in enumerate(self._states):
for i1, s1 in enumerate(self._states):
p = exp(alpha[t0, i0] + self._transitions[s0].logprob(s1) +
self._outputs[s1].logprob(unlabelled_sequence[t1][_TEXT]) +
beta[t1, i1] - normalisation)
entropy -= p * self._transitions[s0].logprob(s1)
#print 'p(s_%d = %s, s_%d = %s) =' % (t0, s0, t1, s1), p

# symbol emissions
for t in range(T):
for i, state in enumerate(self._states):
p = exp(alpha[t, i] + beta[t, i] - normalisation)
entropy -= p * self._outputs[state].logprob(unlabelled_sequence[t][_TEXT])
#print 'p(s_%d = %s) =' % (t, state), p

return entropy

def point_entropy(self, unlabelled_sequence):
"""
Returns the pointwise entropy over the possible states at each
position in the chain, given the observation sequence.
"""

T = len(unlabelled_sequence)
N = len(self._states)

alpha = self._forward_probability(unlabelled_sequence)
beta = self._backward_probability(unlabelled_sequence)

entropies = zeros(T, float64)
probs = zeros(N, float64)
for t in range(T):
for s in range(N):
probs[s] = alpha[t, s] + beta[t, s] - normalisation

for s in range(N):
entropies[t] -= exp(probs[s]) * probs[s]

return entropies

def _exhaustive_entropy(self, unlabelled_sequence):
T = len(unlabelled_sequence)
N = len(self._states)

labellings = [[state] for state in self._states]
for t in range(T - 1):
current = labellings
labellings = []
for labelling in current:
for state in self._states:
labellings.append(labelling + [state])

log_probs = []
for labelling in labellings:
labelled_sequence = unlabelled_sequence[:]
for t, label in enumerate(labelling):
labelled_sequence[t] = (labelled_sequence[t][_TEXT], label)
lp = self.log_probability(labelled_sequence)
log_probs.append(lp)

#ps = zeros((T, N), float64)
#for labelling, lp in zip(labellings, log_probs):
#for t in range(T):
#ps[t, self._states.index(labelling[t])] += exp(lp - normalisation)

#for t in range(T):
#print 'prob[%d] =' % t, ps[t]

entropy = 0
for lp in log_probs:
lp -= normalisation
entropy -= exp(lp) * lp

return entropy

def _exhaustive_point_entropy(self, unlabelled_sequence):
T = len(unlabelled_sequence)
N = len(self._states)

labellings = [[state] for state in self._states]
for t in range(T - 1):
current = labellings
labellings = []
for labelling in current:
for state in self._states:
labellings.append(labelling + [state])

log_probs = []
for labelling in labellings:
labelled_sequence = unlabelled_sequence[:]
for t, label in enumerate(labelling):
labelled_sequence[t] = (labelled_sequence[t][_TEXT], label)
lp = self.log_probability(labelled_sequence)
log_probs.append(lp)

probabilities = zeros((T, N), float64)
probabilities[:] = _NINF
for labelling, lp in zip(labellings, log_probs):
lp -= normalisation
for t, label in enumerate(labelling):
index = self._states.index(label)
probabilities[t, index] = _log_add(probabilities[t, index], lp)

entropies = zeros(T, float64)
for t in range(T):
for s in range(N):
entropies[t] -= exp(probabilities[t, s]) * probabilities[t, s]

return entropies

def _forward_probability(self, unlabelled_sequence):
"""
Return the forward probability matrix, a T by N array of
log-probabilities, where T is the length of the sequence and N is the
number of states. Each entry (t, s) gives the probability of being in
state s at time t after observing the partial symbol sequence up to
and including t.

@return: the forward log probability matrix
@rtype:  array
@param unlabelled_sequence: the sequence of unlabelled symbols
@type unlabelled_sequence: list
"""
T = len(unlabelled_sequence)
N = len(self._states)
alpha = zeros((T, N), float64)

symbol = unlabelled_sequence[_TEXT]
for i, state in enumerate(self._states):
alpha[0, i] = self._priors.logprob(state) + \
self._outputs[state].logprob(symbol)

for t in range(1, T):
symbol = unlabelled_sequence[t][_TEXT]
for i, si in enumerate(self._states):
alpha[t, i] = _NINF
for j, sj in enumerate(self._states):
alpha[t, i] = _log_add(alpha[t, i], alpha[t-1, j] +
self._transitions[sj].logprob(si))
alpha[t, i] += self._outputs[si].logprob(symbol)

return alpha

def _backward_probability(self, unlabelled_sequence):
"""
Return the backward probability matrix, a T by N array of
log-probabilities, where T is the length of the sequence and N is the
number of states. Each entry (t, s) gives the probability of being in
state s at time t after observing the partial symbol sequence from t
.. T.

@return: the backward log probability matrix
@rtype:  array
@param unlabelled_sequence: the sequence of unlabelled symbols
@type unlabelled_sequence: list
"""
T = len(unlabelled_sequence)
N = len(self._states)
beta = zeros((T, N), float64)

# initialise the backward values
beta[T-1, :] = log(1)

# inductively calculate remaining backward values
for t in range(T-2, -1, -1):
symbol = unlabelled_sequence[t+1][_TEXT]
for i, si in enumerate(self._states):
beta[t, i] = _NINF
for j, sj in enumerate(self._states):
self._transitions[si].logprob(sj) +
self._outputs[sj].logprob(symbol) +
beta[t + 1, j])

return beta

def __repr__(self):
return '<HiddenMarkovModel %d states and %d output symbols>' \
% (len(self._states), len(self._symbols))

class HiddenMarkovModelTrainer(object):
"""
Algorithms for learning HMM parameters from training data. These include
both supervised learning (MLE) and unsupervised learning (Baum-Welch).
"""
def __init__(self, states=None, symbols=None):
"""
Creates an HMM trainer to induce an HMM with the given states and
output symbol alphabet. A supervised and unsupervised training
method may be used. If either of the states or symbols are not given,
these may be derived from supervised training.

@param states:  the set of state labels
@type states:   sequence of any
@param symbols: the set of observation symbols
@type symbols:  sequence of any
"""
if states:
self._states = states
else:
self._states = []
if symbols:
self._symbols = symbols
else:
self._symbols = []

def train(self, labelled_sequences=None, unlabelled_sequences=None,
**kwargs):
"""
Trains the HMM using both (or either of) supervised and unsupervised
techniques.

@return: the trained model
@rtype: HiddenMarkovModel
@param labelled_sequences: the supervised training data, a set of
labelled sequences of observations
@type labelled_sequences: list
@param unlabelled_sequences: the unsupervised training data, a set of
sequences of observations
@type unlabelled_sequences: list
@param kwargs: additional arguments to pass to the training methods
"""
assert labelled_sequences or unlabelled_sequences
model = None
if labelled_sequences:
model = self.train_supervised(labelled_sequences, **kwargs)
if unlabelled_sequences:
if model: kwargs['model'] = model
model = self.train_unsupervised(unlabelled_sequences, **kwargs)
return model

def train_unsupervised(self, unlabelled_sequences, **kwargs):
"""
Trains the HMM using the Baum-Welch algorithm to maximise the
probability of the data sequence. This is a variant of the EM
algorithm, and is unsupervised in that it doesn't need the state
sequences for the symbols. The code is based on 'A Tutorial on Hidden
Markov Models and Selected Applications in Speech Recognition',
Lawrence Rabiner, IEEE, 1989.

@return: the trained model
@rtype: HiddenMarkovModel
@param unlabelled_sequences: the training data, a set of
sequences of observations
@type unlabelled_sequences: list
@param kwargs: may include the following parameters::
model - a HiddenMarkovModel instance used to begin the Baum-Welch
algorithm
max_iterations - the maximum number of EM iterations
convergence_logprob - the maximum change in log probability to
allow convergence
"""

N = len(self._states)
M = len(self._symbols)
symbol_dict = dict([(self._symbols[i], i) for i in range(M)])

# create a uniform HMM, which will be iteratively refined, unless
# given an existing model
model = kwargs.get('model')
if not model:
priors = UniformProbDist(self._states)
transitions = DictionaryConditionalProbDist(
dict([(state, UniformProbDist(self._states))
for state in self._states]))
output = DictionaryConditionalProbDist(
dict([(state, UniformProbDist(self._symbols))
for state in self._states]))
model = HiddenMarkovModel(self._symbols, self._states,
transitions, output, priors)

# update model prob dists so that they can be modified
model._priors = MutableProbDist(model._priors, self._states)
model._transitions = DictionaryConditionalProbDist(
dict([(s, MutableProbDist(model._transitions[s], self._states))
for s in self._states]))
model._outputs = DictionaryConditionalProbDist(
dict([(s, MutableProbDist(model._outputs[s], self._symbols))
for s in self._states]))

# iterate until convergence
converged = False
last_logprob = None
iteration = 0
max_iterations = kwargs.get('max_iterations', 1000)
epsilon = kwargs.get('convergence_logprob', 1e-6)
while not converged and iteration < max_iterations:
A_numer = ones((N, N), float64) * _NINF
B_numer = ones((N, M), float64) * _NINF
A_denom = ones(N, float64) * _NINF
B_denom = ones(N, float64) * _NINF

logprob = 0
for sequence in unlabelled_sequences:
# compute forward and backward probabilities
alpha = model._forward_probability(sequence)
beta = model._backward_probability(sequence)

# find the log probability of the sequence
T = len(sequence)
logprob += lpk

# now update A and B (transition and output probabilities)
# using the alpha and beta values. Please refer to Rabiner's
# paper for details, it's too hard to explain in comments
local_A_numer = ones((N, N), float64) * _NINF
local_B_numer = ones((N, M), float64) * _NINF
local_A_denom = ones(N, float64) * _NINF
local_B_denom = ones(N, float64) * _NINF

# for each position, accumulate sums for A and B
for t in range(T):
if t < T - 1:
xi = symbol_dict[x]
for i in range(N):
si = self._states[i]
if t < T - 1:
for j in range(N):
sj = self._states[j]
local_A_numer[i, j] =  \
alpha[t, i] +
model._transitions[si].logprob(sj) +
model._outputs[sj].logprob(xnext) +
beta[t+1, j])
alpha[t, i] + beta[t, i])
else:
alpha[t, i] + beta[t, i])

alpha[t, i] + beta[t, i])

# add these sums to the global A and B values
for i in range(N):
for j in range(N):
local_A_numer[i, j] - lpk)
for k in range(M):
local_B_numer[i, k] - lpk)

A_denom[i] = _log_add(A_denom[i], local_A_denom[i] - lpk)
B_denom[i] = _log_add(B_denom[i], local_B_denom[i] - lpk)

# use the calculated values to update the transition and output
# probability values
for i in range(N):
si = self._states[i]
for j in range(N):
sj = self._states[j]
model._transitions[si].update(sj, A_numer[i,j] - A_denom[i])
for k in range(M):
ok = self._symbols[k]
model._outputs[si].update(ok, B_numer[i,k] - B_denom[i])
# Rabiner says the priors don't need to be updated. I don't
# believe him. FIXME

# test for convergence
if iteration > 0 and abs(logprob - last_logprob) < epsilon:
converged = True

print 'iteration', iteration, 'logprob', logprob
iteration += 1
last_logprob = logprob

return model

def train_supervised(self, labelled_sequences, **kwargs):
"""
Supervised training maximising the joint probability of the symbol and
state sequences. This is done via collecting frequencies of
transitions between states, symbol observations while within each
state and which states start a sentence. These frequency distributions
are then normalised into probability estimates, which can be
smoothed if desired.

@return: the trained model
@rtype: HiddenMarkovModel
@param labelled_sequences: the training data, a set of
labelled sequences of observations
@type labelled_sequences: list
@param kwargs: may include an 'estimator' parameter, a function taking
a C{FreqDist} and a number of bins and returning a C{ProbDistI};
otherwise a MLE estimate is used
"""

# default to the MLE estimate
estimator = kwargs.get('estimator')
if estimator == None:
estimator = lambda fdist, bins: MLEProbDist(fdist)

# count occurences of starting states, transitions out of each state
# and output symbols observed in each state
starting = FreqDist()
transitions = ConditionalFreqDist()
outputs = ConditionalFreqDist()
for sequence in labelled_sequences:
lasts = None
for token in sequence:
state = token[_TAG]
symbol = token[_TEXT]
if lasts == None:
starting.inc(state)
else:
transitions[lasts].inc(state)
outputs[state].inc(symbol)
lasts = state

# update the state and symbol lists
if state not in self._states:
self._states.append(state)
if symbol not in self._symbols:
self._symbols.append(symbol)

# create probability distributions (with smoothing)
N = len(self._states)
pi = estimator(starting, N)
A = ConditionalProbDist(transitions, estimator, False, N)
B = ConditionalProbDist(outputs, estimator, False, len(self._symbols))

return HiddenMarkovModel(self._symbols, self._states, A, B, pi)

"""
"""
x = max(values)
if x > _NINF:
sum_diffs = 0
for value in values:
sum_diffs += exp(value - x)
return x + log(sum_diffs)
else:
return x

def demo():
# demonstrates HMM probability calculation

# example taken from page 381, Huang et al
symbols = ['up', 'down', 'unchanged']
states = ['bull', 'bear', 'static']

def pd(values, samples):
d = {}
for value, item in zip(values, samples):
d[item] = value
return DictionaryProbDist(d)

def cpd(array, conditions, samples):
d = {}
for values, condition in zip(array, conditions):
d[condition] = pd(values, samples)
return DictionaryConditionalProbDist(d)

A = array([[0.6, 0.2, 0.2], [0.5, 0.3, 0.2], [0.4, 0.1, 0.5]], float64)
A = cpd(A, states, states)
B = array([[0.7, 0.1, 0.2], [0.1, 0.6, 0.3], [0.3, 0.3, 0.4]], float64)
B = cpd(B, states, symbols)
pi = array([0.5, 0.2, 0.3], float64)
pi = pd(pi, states)

model = HiddenMarkovModel(symbols=symbols, states=states,
transitions=A, outputs=B, priors=pi)

print 'Testing', model

for test in [['up', 'up'], ['up', 'down', 'up'],
['down'] * 5, ['unchanged'] * 5 + ['up']]:

sequence = [(t, None) for t in test]

print 'Testing with state sequence', test
print 'probability =', model.probability(sequence)
print 'tagging =    ', model.tag(sequence)
print 'p(tagged) =  ', model.probability(sequence)
print 'H =          ', model.entropy(sequence)
print 'H_exh =      ', model._exhaustive_entropy(sequence)
print 'H(point) =   ', model.point_entropy(sequence)
print 'H_exh(point)=', model._exhaustive_point_entropy(sequence)
print

from en.parser.nltk_lite.corpora import brown
from itertools import islice

sentences = list(islice(brown.tagged(), 100))

tag_set = ["'", "''", '(', ')', '*', ',', '.', ':', '--', '``', 'abl',
'abn', 'abx', 'ap', 'ap\$', 'at', 'be', 'bed', 'bedz', 'beg', 'bem',
'ben', 'ber', 'bez', 'cc', 'cd', 'cd\$', 'cs', 'do', 'dod', 'doz',
'dt', 'dt\$', 'dti', 'dts', 'dtx', 'ex', 'fw', 'hv', 'hvd', 'hvg',
'hvn', 'hvz', 'in', 'jj', 'jjr', 'jjs', 'jjt', 'md', 'nn', 'nn\$',
'nns', 'nns\$', 'np', 'np\$', 'nps', 'nps\$', 'nr', 'nr\$', 'od', 'pn',
'pn\$', 'pp\$', 'ppl', 'ppls', 'ppo', 'pps', 'ppss', 'ql', 'qlp', 'rb',
'rb\$', 'rbr', 'rbt', 'rp', 'to', 'uh', 'vb', 'vbd', 'vbg', 'vbn',
'vbz', 'wdt', 'wp\$', 'wpo', 'wps', 'wql', 'wrb']

sequences = []
sequence = []
symbols = set()
start_re = re.compile(r'[^-*+]*')
for sentence in sentences:
for i in range(len(sentence)):
word, tag = sentence[i]
word = word.lower()  # normalize
m = start_re.match(tag)
# cleanup the tag
tag = m.group(0)
if tag not in tag_set:
tag = '*'
sentence[i] = (word, tag)  # store cleaned-up tagged token

return sentences, tag_set, list(symbols)

def test_pos(model, sentences, display=False):
from sys import stdout

count = correct = 0
for sentence in sentences:
sentence = [(token, None) for token in sentence]
pts = model.best_path(sentence)
if display:
print sentence
print 'HMM >>>'
print pts
print model.entropy(sentences)
print '-' * 60
else:
print '\b.',
stdout.flush()
for token, tag in zip(sentence, pts):
count += 1
if tag == token[TAG]:
correct += 1

print 'accuracy over', count, 'tokens %.1f' % (100.0 * correct / count)

def demo_pos():
# demonstrates POS tagging using supervised training

print 'Training HMM...'
trainer = HiddenMarkovModelTrainer(tag_set, symbols)
hmm = trainer.train_supervised(labelled_sequences[100:],
estimator=lambda fd, bins: LidstoneProbDist(fd, 0.1, bins))

print 'Testing...'
test_pos(hmm, labelled_sequences[:100], True)

def _untag(sentences):
unlabelled = []
for sentence in sentences:
unlabelled.append([(token, None) for token in sentence])
return unlabelled

def demo_pos_bw():
# demonstrates the Baum-Welch algorithm in POS tagging

print 'Training HMM (supervised)...'
symbols = set()
for sentence in sentences:
for token in sentence:

trainer = HiddenMarkovModelTrainer(tag_set, list(symbols))
hmm = trainer.train_supervised(sentences[100:300],
estimator=lambda fd, bins: LidstoneProbDist(fd, 0.1, bins))
print 'Training (unsupervised)...'
# it's rather slow - so only use 10 samples
unlabelled = _untag(sentences[301:311])
hmm = trainer.train_unsupervised(unlabelled, model=hmm, max_iterations=5)
test_pos(hmm, sentences[:100], True)

def demo_bw():
# demo Baum Welch by generating some sequences and then performing
# unsupervised training on them

# example taken from page 381, Huang et al
symbols = ['up', 'down', 'unchanged']
states = ['bull', 'bear', 'static']

def pd(values, samples):
d = {}
for value, item in zip(values, samples):
d[item] = value
return DictionaryProbDist(d)

def cpd(array, conditions, samples):
d = {}
for values, condition in zip(array, conditions):
d[condition] = pd(values, samples)
return DictionaryConditionalProbDist(d)

A = array([[0.6, 0.2, 0.2], [0.5, 0.3, 0.2], [0.4, 0.1, 0.5]], float64)
A = cpd(A, states, states)
B = array([[0.7, 0.1, 0.2], [0.1, 0.6, 0.3], [0.3, 0.3, 0.4]], float64)
B = cpd(B, states, symbols)
pi = array([0.5, 0.2, 0.3], float64)
pi = pd(pi, states)

model = HiddenMarkovModel(symbols=symbols, states=states,
transitions=A, outputs=B, priors=pi)

# generate some random sequences
training = []
import random
rng = random.Random()
for i in range(10):
item = model.random_sample(rng, 5)
training.append([(i, None) for i in item])

# train on those examples, starting with the model that generated them
trainer = HiddenMarkovModelTrainer(states, symbols)
hmm = trainer.train_unsupervised(training, model=model, max_iterations=1000)

if __name__ == '__main__':
demo()
#demo_pos()
#demo_pos_bw()
#demo_bw()

```