# Natural Language Toolkit: Hidden Markov Model
#
# Copyright (C) 2001-2023 NLTK Project
# Author: Trevor Cohn <tacohn@csse.unimelb.edu.au>
# Philip Blunsom <pcbl@csse.unimelb.edu.au>
# Tiago Tresoldi <tiago@tresoldi.pro.br> (fixes)
# Steven Bird <stevenbird1@gmail.com> (fixes)
# Joseph Frazee <jfrazee@mail.utexas.edu> (fixes)
# Steven Xu <xxu@student.unimelb.edu.au> (fixes)
# URL: <https://www.nltk.org/>
# For license information, see LICENSE.TXT
"""
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 *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 *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.
For more information, please consult the source code for this module,
which includes extensive demonstration code.
"""
import itertools
import re
try:
import numpy as np
except ImportError:
pass
from nltk.metrics import accuracy
from nltk.probability import (
ConditionalFreqDist,
ConditionalProbDist,
DictionaryConditionalProbDist,
DictionaryProbDist,
FreqDist,
LidstoneProbDist,
MLEProbDist,
MutableProbDist,
RandomProbDist,
)
from nltk.tag.api import TaggerI
from nltk.util import LazyMap, unique_list
_TEXT = 0 # index of text in a tuple
_TAG = 1 # index of tag in a tuple
def _identity(labeled_symbols):
return labeled_symbols
[docs]class HiddenMarkovModelTagger(TaggerI):
"""
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 and includes an extension for
training shallow HMM parsers or specialized HMMs as in Molina et.
al, 2002. A specialized HMM modifies training data by applying a
specialization function to create a new training set that is more
appropriate for sequential tagging with an HMM. A typical use case is
chunking.
: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: ConditionalProbDistI
:param outputs: output probabilities; Pr(o_k | s_i) is the probability
of emitting symbol k when entering state i
:type outputs: ConditionalProbDistI
:param priors: initial state distribution; Pr(s_i) is the probability
of starting in state i
:type priors: ProbDistI
:param transform: an optional function for transforming training
instances, defaults to the identity function.
:type transform: callable
"""
[docs] def __init__(
self, symbols, states, transitions, outputs, priors, transform=_identity
):
self._symbols = unique_list(symbols)
self._states = unique_list(states)
self._transitions = transitions
self._outputs = outputs
self._priors = priors
self._cache = None
self._transform = transform
@classmethod
def _train(
cls,
labeled_sequence,
test_sequence=None,
unlabeled_sequence=None,
transform=_identity,
estimator=None,
**kwargs,
):
if estimator is None:
def estimator(fd, bins):
return LidstoneProbDist(fd, 0.1, bins)
labeled_sequence = LazyMap(transform, labeled_sequence)
symbols = unique_list(word for sent in labeled_sequence for word, tag in sent)
tag_set = unique_list(tag for sent in labeled_sequence for word, tag in sent)
trainer = HiddenMarkovModelTrainer(tag_set, symbols)
hmm = trainer.train_supervised(labeled_sequence, estimator=estimator)
hmm = cls(
hmm._symbols,
hmm._states,
hmm._transitions,
hmm._outputs,
hmm._priors,
transform=transform,
)
if test_sequence:
hmm.test(test_sequence, verbose=kwargs.get("verbose", False))
if unlabeled_sequence:
max_iterations = kwargs.get("max_iterations", 5)
hmm = trainer.train_unsupervised(
unlabeled_sequence, model=hmm, max_iterations=max_iterations
)
if test_sequence:
hmm.test(test_sequence, verbose=kwargs.get("verbose", False))
return hmm
[docs] @classmethod
def train(
cls, labeled_sequence, test_sequence=None, unlabeled_sequence=None, **kwargs
):
"""
Train a new HiddenMarkovModelTagger using the given labeled and
unlabeled training instances. Testing will be performed if test
instances are provided.
:return: a hidden markov model tagger
:rtype: HiddenMarkovModelTagger
:param labeled_sequence: a sequence of labeled training instances,
i.e. a list of sentences represented as tuples
:type labeled_sequence: list(list)
:param test_sequence: a sequence of labeled test instances
:type test_sequence: list(list)
:param unlabeled_sequence: a sequence of unlabeled training instances,
i.e. a list of sentences represented as words
:type unlabeled_sequence: list(list)
:param transform: an optional function for transforming training
instances, defaults to the identity function, see ``transform()``
:type transform: function
:param estimator: an optional function or class that maps a
condition's frequency distribution to its probability
distribution, defaults to a Lidstone distribution with gamma = 0.1
:type estimator: class or function
:param verbose: boolean flag indicating whether training should be
verbose or include printed output
:type verbose: bool
:param max_iterations: number of Baum-Welch iterations to perform
:type max_iterations: int
"""
return cls._train(labeled_sequence, test_sequence, unlabeled_sequence, **kwargs)
[docs] 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 2 ** (self.log_probability(self._transform(sequence)))
[docs] 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
"""
sequence = self._transform(sequence)
T = len(sequence)
if T > 0 and sequence[0][_TAG]:
last_state = sequence[0][_TAG]
p = self._priors.logprob(last_state) + self._output_logprob(
last_state, sequence[0][_TEXT]
)
for t in range(1, T):
state = sequence[t][_TAG]
p += self._transitions[last_state].logprob(
state
) + self._output_logprob(state, sequence[t][_TEXT])
last_state = state
return p
else:
alpha = self._forward_probability(sequence)
p = logsumexp2(alpha[T - 1])
return p
[docs] def tag(self, unlabeled_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 unlabeled_sequence: the sequence of unlabeled symbols
:type unlabeled_sequence: list
"""
unlabeled_sequence = self._transform(unlabeled_sequence)
return self._tag(unlabeled_sequence)
def _tag(self, unlabeled_sequence):
path = self._best_path(unlabeled_sequence)
return list(zip(unlabeled_sequence, path))
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 _create_cache(self):
"""
The cache is a tuple (P, O, X, S) where:
- S maps symbols to integers. I.e., it is the inverse
mapping from self._symbols; for each symbol s in
self._symbols, the following is true::
self._symbols[S[s]] == s
- O is the log output probabilities::
O[i,k] = log( P(token[t]=sym[k]|tag[t]=state[i]) )
- X is the log transition probabilities::
X[i,j] = log( P(tag[t]=state[j]|tag[t-1]=state[i]) )
- P is the log prior probabilities::
P[i] = log( P(tag[0]=state[i]) )
"""
if not self._cache:
N = len(self._states)
M = len(self._symbols)
P = np.zeros(N, np.float32)
X = np.zeros((N, N), np.float32)
O = np.zeros((N, M), np.float32)
for i in range(N):
si = self._states[i]
P[i] = self._priors.logprob(si)
for j in range(N):
X[i, j] = self._transitions[si].logprob(self._states[j])
for k in range(M):
O[i, k] = self._output_logprob(si, self._symbols[k])
S = {}
for k in range(M):
S[self._symbols[k]] = k
self._cache = (P, O, X, S)
def _update_cache(self, symbols):
# add new symbols to the symbol table and repopulate the output
# probabilities and symbol table mapping
if symbols:
self._create_cache()
P, O, X, S = self._cache
for symbol in symbols:
if symbol not in self._symbols:
self._cache = None
self._symbols.append(symbol)
# don't bother with the work if there aren't any new symbols
if not self._cache:
N = len(self._states)
M = len(self._symbols)
Q = O.shape[1]
# add new columns to the output probability table without
# destroying the old probabilities
O = np.hstack([O, np.zeros((N, M - Q), np.float32)])
for i in range(N):
si = self._states[i]
# only calculate probabilities for new symbols
for k in range(Q, M):
O[i, k] = self._output_logprob(si, self._symbols[k])
# only create symbol mappings for new symbols
for k in range(Q, M):
S[self._symbols[k]] = k
self._cache = (P, O, X, S)
[docs] def reset_cache(self):
self._cache = None
[docs] def best_path(self, unlabeled_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 unlabeled_sequence: the sequence of unlabeled symbols
:type unlabeled_sequence: list
"""
unlabeled_sequence = self._transform(unlabeled_sequence)
return self._best_path(unlabeled_sequence)
def _best_path(self, unlabeled_sequence):
T = len(unlabeled_sequence)
N = len(self._states)
self._create_cache()
self._update_cache(unlabeled_sequence)
P, O, X, S = self._cache
V = np.zeros((T, N), np.float32)
B = -np.ones((T, N), int)
V[0] = P + O[:, S[unlabeled_sequence[0]]]
for t in range(1, T):
for j in range(N):
vs = V[t - 1, :] + X[:, j]
best = np.argmax(vs)
V[t, j] = vs[best] + O[j, S[unlabeled_sequence[t]]]
B[t, j] = best
current = np.argmax(V[T - 1, :])
sequence = [current]
for t in range(T - 1, 0, -1):
last = B[t, current]
sequence.append(last)
current = last
sequence.reverse()
return list(map(self._states.__getitem__, sequence))
[docs] def best_path_simple(self, unlabeled_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. This uses a simple, direct method, and is included for
teaching purposes.
:return: the state sequence
:rtype: sequence of any
:param unlabeled_sequence: the sequence of unlabeled symbols
:type unlabeled_sequence: list
"""
unlabeled_sequence = self._transform(unlabeled_sequence)
return self._best_path_simple(unlabeled_sequence)
def _best_path_simple(self, unlabeled_sequence):
T = len(unlabeled_sequence)
N = len(self._states)
V = np.zeros((T, N), np.float64)
B = {}
# find the starting log probabilities for each state
symbol = unlabeled_sequence[0]
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 = unlabeled_sequence[t]
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[0]:
best = (va, si)
V[t, j] = best[0] + self._output_logprob(sj, symbol)
B[t, sj] = best[1]
# find the highest probability final state
best = None
for i in range(N):
val = V[T - 1, i]
if not best or val > best[0]:
best = (val, self._states[i])
# traverse the back-pointers B to find the state sequence
current = best[1]
sequence = [current]
for t in range(T - 1, 0, -1):
last = B[t, current]
sequence.append(last)
current = last
sequence.reverse()
return sequence
[docs] 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))
return tokens
def _sample_probdist(self, probdist, p, samples):
cum_p = 0
for sample in samples:
add_p = probdist.prob(sample)
if cum_p <= p <= cum_p + add_p:
return sample
cum_p += add_p
raise Exception("Invalid probability distribution - " "does not sum to one")
[docs] def entropy(self, unlabeled_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
*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.
"""
unlabeled_sequence = self._transform(unlabeled_sequence)
T = len(unlabeled_sequence)
N = len(self._states)
alpha = self._forward_probability(unlabeled_sequence)
beta = self._backward_probability(unlabeled_sequence)
normalisation = logsumexp2(alpha[T - 1])
entropy = normalisation
# starting state, t = 0
for i, state in enumerate(self._states):
p = 2 ** (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 = 2 ** (
alpha[t0, i0]
+ self._transitions[s0].logprob(s1)
+ self._outputs[s1].logprob(unlabeled_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 = 2 ** (alpha[t, i] + beta[t, i] - normalisation)
entropy -= p * self._outputs[state].logprob(
unlabeled_sequence[t][_TEXT]
)
# print('p(s_%d = %s) =' % (t, state), p)
return entropy
[docs] def point_entropy(self, unlabeled_sequence):
"""
Returns the pointwise entropy over the possible states at each
position in the chain, given the observation sequence.
"""
unlabeled_sequence = self._transform(unlabeled_sequence)
T = len(unlabeled_sequence)
N = len(self._states)
alpha = self._forward_probability(unlabeled_sequence)
beta = self._backward_probability(unlabeled_sequence)
normalisation = logsumexp2(alpha[T - 1])
entropies = np.zeros(T, np.float64)
probs = np.zeros(N, np.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] -= 2 ** (probs[s]) * probs[s]
return entropies
def _exhaustive_entropy(self, unlabeled_sequence):
unlabeled_sequence = self._transform(unlabeled_sequence)
T = len(unlabeled_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:
labeled_sequence = unlabeled_sequence[:]
for t, label in enumerate(labelling):
labeled_sequence[t] = (labeled_sequence[t][_TEXT], label)
lp = self.log_probability(labeled_sequence)
log_probs.append(lp)
normalisation = _log_add(*log_probs)
entropy = 0
for lp in log_probs:
lp -= normalisation
entropy -= 2 ** (lp) * lp
return entropy
def _exhaustive_point_entropy(self, unlabeled_sequence):
unlabeled_sequence = self._transform(unlabeled_sequence)
T = len(unlabeled_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 = unlabeled_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)
normalisation = _log_add(*log_probs)
probabilities = _ninf_array((T, N))
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 = np.zeros(T, np.float64)
for t in range(T):
for s in range(N):
entropies[t] -= 2 ** (probabilities[t, s]) * probabilities[t, s]
return entropies
def _transitions_matrix(self):
"""Return a matrix of transition log probabilities."""
trans_iter = (
self._transitions[sj].logprob(si)
for sj in self._states
for si in self._states
)
transitions_logprob = np.fromiter(trans_iter, dtype=np.float64)
N = len(self._states)
return transitions_logprob.reshape((N, N)).T
def _outputs_vector(self, symbol):
"""
Return a vector with log probabilities of emitting a symbol
when entering states.
"""
out_iter = (self._output_logprob(sj, symbol) for sj in self._states)
return np.fromiter(out_iter, dtype=np.float64)
def _forward_probability(self, unlabeled_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.
:param unlabeled_sequence: the sequence of unlabeled symbols
:type unlabeled_sequence: list
:return: the forward log probability matrix
:rtype: array
"""
T = len(unlabeled_sequence)
N = len(self._states)
alpha = _ninf_array((T, N))
transitions_logprob = self._transitions_matrix()
# Initialization
symbol = unlabeled_sequence[0][_TEXT]
for i, state in enumerate(self._states):
alpha[0, i] = self._priors.logprob(state) + self._output_logprob(
state, symbol
)
# Induction
for t in range(1, T):
symbol = unlabeled_sequence[t][_TEXT]
output_logprob = self._outputs_vector(symbol)
for i in range(N):
summand = alpha[t - 1] + transitions_logprob[i]
alpha[t, i] = logsumexp2(summand) + output_logprob[i]
return alpha
def _backward_probability(self, unlabeled_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 unlabeled_sequence: the sequence of unlabeled symbols
:type unlabeled_sequence: list
"""
T = len(unlabeled_sequence)
N = len(self._states)
beta = _ninf_array((T, N))
transitions_logprob = self._transitions_matrix().T
# initialise the backward values;
# "1" is an arbitrarily chosen value from Rabiner tutorial
beta[T - 1, :] = np.log2(1)
# inductively calculate remaining backward values
for t in range(T - 2, -1, -1):
symbol = unlabeled_sequence[t + 1][_TEXT]
outputs = self._outputs_vector(symbol)
for i in range(N):
summand = transitions_logprob[i] + beta[t + 1] + outputs
beta[t, i] = logsumexp2(summand)
return beta
[docs] def test(self, test_sequence, verbose=False, **kwargs):
"""
Tests the HiddenMarkovModelTagger instance.
:param test_sequence: a sequence of labeled test instances
:type test_sequence: list(list)
:param verbose: boolean flag indicating whether training should be
verbose or include printed output
:type verbose: bool
"""
def words(sent):
return [word for (word, tag) in sent]
def tags(sent):
return [tag for (word, tag) in sent]
def flatten(seq):
return list(itertools.chain(*seq))
test_sequence = self._transform(test_sequence)
predicted_sequence = list(map(self._tag, map(words, test_sequence)))
if verbose:
for test_sent, predicted_sent in zip(test_sequence, predicted_sequence):
print(
"Test:",
" ".join(f"{token}/{tag}" for (token, tag) in test_sent),
)
print()
print("Untagged:", " ".join("%s" % token for (token, tag) in test_sent))
print()
print(
"HMM-tagged:",
" ".join(f"{token}/{tag}" for (token, tag) in predicted_sent),
)
print()
print(
"Entropy:",
self.entropy([(token, None) for (token, tag) in predicted_sent]),
)
print()
print("-" * 60)
test_tags = flatten(map(tags, test_sequence))
predicted_tags = flatten(map(tags, predicted_sequence))
acc = accuracy(test_tags, predicted_tags)
count = sum(len(sent) for sent in test_sequence)
print("accuracy over %d tokens: %.2f" % (count, acc * 100))
def __repr__(self):
return "<HiddenMarkovModelTagger %d states and %d output symbols>" % (
len(self._states),
len(self._symbols),
)
[docs]class HiddenMarkovModelTrainer:
"""
Algorithms for learning HMM parameters from training data. These include
both supervised learning (MLE) and unsupervised learning (Baum-Welch).
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
"""
[docs] def __init__(self, states=None, symbols=None):
self._states = states if states else []
self._symbols = symbols if symbols else []
[docs] def train(self, labeled_sequences=None, unlabeled_sequences=None, **kwargs):
"""
Trains the HMM using both (or either of) supervised and unsupervised
techniques.
:return: the trained model
:rtype: HiddenMarkovModelTagger
:param labelled_sequences: the supervised training data, a set of
labelled sequences of observations
ex: [ (word_1, tag_1),...,(word_n,tag_n) ]
:type labelled_sequences: list
:param unlabeled_sequences: the unsupervised training data, a set of
sequences of observations
ex: [ word_1, ..., word_n ]
:type unlabeled_sequences: list
:param kwargs: additional arguments to pass to the training methods
"""
assert labeled_sequences or unlabeled_sequences
model = None
if labeled_sequences:
model = self.train_supervised(labeled_sequences, **kwargs)
if unlabeled_sequences:
if model:
kwargs["model"] = model
model = self.train_unsupervised(unlabeled_sequences, **kwargs)
return model
def _baum_welch_step(self, sequence, model, symbol_to_number):
N = len(model._states)
M = len(model._symbols)
T = len(sequence)
# compute forward and backward probabilities
alpha = model._forward_probability(sequence)
beta = model._backward_probability(sequence)
# find the log probability of the sequence
lpk = logsumexp2(alpha[T - 1])
A_numer = _ninf_array((N, N))
B_numer = _ninf_array((N, M))
A_denom = _ninf_array(N)
B_denom = _ninf_array(N)
transitions_logprob = model._transitions_matrix().T
for t in range(T):
symbol = sequence[t][_TEXT] # not found? FIXME
next_symbol = None
if t < T - 1:
next_symbol = sequence[t + 1][_TEXT] # not found? FIXME
xi = symbol_to_number[symbol]
next_outputs_logprob = model._outputs_vector(next_symbol)
alpha_plus_beta = alpha[t] + beta[t]
if t < T - 1:
numer_add = (
transitions_logprob
+ next_outputs_logprob
+ beta[t + 1]
+ alpha[t].reshape(N, 1)
)
A_numer = np.logaddexp2(A_numer, numer_add)
A_denom = np.logaddexp2(A_denom, alpha_plus_beta)
else:
B_denom = np.logaddexp2(A_denom, alpha_plus_beta)
B_numer[:, xi] = np.logaddexp2(B_numer[:, xi], alpha_plus_beta)
return lpk, A_numer, A_denom, B_numer, B_denom
[docs] def train_unsupervised(self, unlabeled_sequences, update_outputs=True, **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: HiddenMarkovModelTagger
:param unlabeled_sequences: the training data, a set of
sequences of observations
:type unlabeled_sequences: list
kwargs may include following parameters:
:param model: a HiddenMarkovModelTagger instance used to begin
the Baum-Welch algorithm
:param max_iterations: the maximum number of EM iterations
:param convergence_logprob: the maximum change in log probability to
allow convergence
"""
# create a uniform HMM, which will be iteratively refined, unless
# given an existing model
model = kwargs.get("model")
if not model:
priors = RandomProbDist(self._states)
transitions = DictionaryConditionalProbDist(
{state: RandomProbDist(self._states) for state in self._states}
)
outputs = DictionaryConditionalProbDist(
{state: RandomProbDist(self._symbols) for state in self._states}
)
model = HiddenMarkovModelTagger(
self._symbols, self._states, transitions, outputs, priors
)
self._states = model._states
self._symbols = model._symbols
N = len(self._states)
M = len(self._symbols)
symbol_numbers = {sym: i for i, sym in enumerate(self._symbols)}
# update model prob dists so that they can be modified
# model._priors = MutableProbDist(model._priors, self._states)
model._transitions = DictionaryConditionalProbDist(
{
s: MutableProbDist(model._transitions[s], self._states)
for s in self._states
}
)
if update_outputs:
model._outputs = DictionaryConditionalProbDist(
{
s: MutableProbDist(model._outputs[s], self._symbols)
for s in self._states
}
)
model.reset_cache()
# 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 = _ninf_array((N, N))
B_numer = _ninf_array((N, M))
A_denom = _ninf_array(N)
B_denom = _ninf_array(N)
logprob = 0
for sequence in unlabeled_sequences:
sequence = list(sequence)
if not sequence:
continue
(
lpk,
seq_A_numer,
seq_A_denom,
seq_B_numer,
seq_B_denom,
) = self._baum_welch_step(sequence, model, symbol_numbers)
# add these sums to the global A and B values
for i in range(N):
A_numer[i] = np.logaddexp2(A_numer[i], seq_A_numer[i] - lpk)
B_numer[i] = np.logaddexp2(B_numer[i], seq_B_numer[i] - lpk)
A_denom = np.logaddexp2(A_denom, seq_A_denom - lpk)
B_denom = np.logaddexp2(B_denom, seq_B_denom - lpk)
logprob += lpk
# use the calculated values to update the transition and output
# probability values
for i in range(N):
logprob_Ai = A_numer[i] - A_denom[i]
logprob_Bi = B_numer[i] - B_denom[i]
# We should normalize all probabilities (see p.391 Huang et al)
# Let sum(P) be K.
# We can divide each Pi by K to make sum(P) == 1.
# Pi' = Pi/K
# log2(Pi') = log2(Pi) - log2(K)
logprob_Ai -= logsumexp2(logprob_Ai)
logprob_Bi -= logsumexp2(logprob_Bi)
# update output and transition probabilities
si = self._states[i]
for j in range(N):
sj = self._states[j]
model._transitions[si].update(sj, logprob_Ai[j])
if update_outputs:
for k in range(M):
ok = self._symbols[k]
model._outputs[si].update(ok, logprob_Bi[k])
# 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
[docs] def train_supervised(self, labelled_sequences, estimator=None):
"""
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: HiddenMarkovModelTagger
:param labelled_sequences: the training data, a set of
labelled sequences of observations
:type labelled_sequences: list
:param estimator: a function taking
a FreqDist and a number of bins and returning a CProbDistI;
otherwise a MLE estimate is used
"""
# default to the MLE estimate
if estimator is None:
estimator = lambda fdist, bins: MLEProbDist(fdist)
# count occurrences of starting states, transitions out of each state
# and output symbols observed in each state
known_symbols = set(self._symbols)
known_states = set(self._states)
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 is None:
starting[state] += 1
else:
transitions[lasts][state] += 1
outputs[state][symbol] += 1
lasts = state
# update the state and symbol lists
if state not in known_states:
self._states.append(state)
known_states.add(state)
if symbol not in known_symbols:
self._symbols.append(symbol)
known_symbols.add(symbol)
# create probability distributions (with smoothing)
N = len(self._states)
pi = estimator(starting, N)
A = ConditionalProbDist(transitions, estimator, N)
B = ConditionalProbDist(outputs, estimator, len(self._symbols))
return HiddenMarkovModelTagger(self._symbols, self._states, A, B, pi)
def _ninf_array(shape):
res = np.empty(shape, np.float64)
res.fill(-np.inf)
return res
[docs]def logsumexp2(arr):
max_ = arr.max()
return np.log2(np.sum(2 ** (arr - max_))) + max_
def _log_add(*values):
"""
Adds the logged values, returning the logarithm of the addition.
"""
x = max(values)
if x > -np.inf:
sum_diffs = 0
for value in values:
sum_diffs += 2 ** (value - x)
return x + np.log2(sum_diffs)
else:
return x
def _create_hmm_tagger(states, symbols, A, B, pi):
def pd(values, samples):
d = dict(zip(samples, values))
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 = cpd(A, states, states)
B = cpd(B, states, symbols)
pi = pd(pi, states)
return HiddenMarkovModelTagger(
symbols=symbols, states=states, transitions=A, outputs=B, priors=pi
)
def _market_hmm_example():
"""
Return an example HMM (described at page 381, Huang et al)
"""
states = ["bull", "bear", "static"]
symbols = ["up", "down", "unchanged"]
A = np.array([[0.6, 0.2, 0.2], [0.5, 0.3, 0.2], [0.4, 0.1, 0.5]], np.float64)
B = np.array([[0.7, 0.1, 0.2], [0.1, 0.6, 0.3], [0.3, 0.3, 0.4]], np.float64)
pi = np.array([0.5, 0.2, 0.3], np.float64)
model = _create_hmm_tagger(states, symbols, A, B, pi)
return model, states, symbols
[docs]def demo():
# demonstrates HMM probability calculation
print()
print("HMM probability calculation demo")
print()
model, states, symbols = _market_hmm_example()
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([word for (word, tag) in 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()
[docs]def load_pos(num_sents):
from nltk.corpus import brown
sentences = brown.tagged_sents(categories="news")[:num_sents]
tag_re = re.compile(r"[*]|--|[^+*-]+")
tag_set = set()
symbols = set()
cleaned_sentences = []
for sentence in sentences:
for i in range(len(sentence)):
word, tag = sentence[i]
word = word.lower() # normalize
symbols.add(word) # log this word
# Clean up the tag.
tag = tag_re.match(tag).group()
tag_set.add(tag)
sentence[i] = (word, tag) # store cleaned-up tagged token
cleaned_sentences += [sentence]
return cleaned_sentences, list(tag_set), list(symbols)
[docs]def demo_pos():
# demonstrates POS tagging using supervised training
print()
print("HMM POS tagging demo")
print()
print("Training HMM...")
labelled_sequences, tag_set, symbols = load_pos(20000)
trainer = HiddenMarkovModelTrainer(tag_set, symbols)
hmm = trainer.train_supervised(
labelled_sequences[10:],
estimator=lambda fd, bins: LidstoneProbDist(fd, 0.1, bins),
)
print("Testing...")
hmm.test(labelled_sequences[:10], verbose=True)
def _untag(sentences):
unlabeled = []
for sentence in sentences:
unlabeled.append([(token[_TEXT], None) for token in sentence])
return unlabeled
[docs]def demo_pos_bw(
test=10, supervised=20, unsupervised=10, verbose=True, max_iterations=5
):
# demonstrates the Baum-Welch algorithm in POS tagging
print()
print("Baum-Welch demo for POS tagging")
print()
print("Training HMM (supervised, %d sentences)..." % supervised)
sentences, tag_set, symbols = load_pos(test + supervised + unsupervised)
symbols = set()
for sentence in sentences:
for token in sentence:
symbols.add(token[_TEXT])
trainer = HiddenMarkovModelTrainer(tag_set, list(symbols))
hmm = trainer.train_supervised(
sentences[test : test + supervised],
estimator=lambda fd, bins: LidstoneProbDist(fd, 0.1, bins),
)
hmm.test(sentences[:test], verbose=verbose)
print("Training (unsupervised, %d sentences)..." % unsupervised)
# it's rather slow - so only use 10 samples by default
unlabeled = _untag(sentences[test + supervised :])
hmm = trainer.train_unsupervised(
unlabeled, model=hmm, max_iterations=max_iterations
)
hmm.test(sentences[:test], verbose=verbose)
[docs]def demo_bw():
# demo Baum Welch by generating some sequences and then performing
# unsupervised training on them
print()
print("Baum-Welch demo for market example")
print()
model, states, symbols = _market_hmm_example()
# generate some random sequences
training = []
import random
rng = random.Random()
rng.seed(0)
for i in range(10):
item = model.random_sample(rng, 5)
training.append([(i[0], 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)