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# Natural Language Toolkit: Hidden Markov Model # # Copyright (C) 2001-2012 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 <sb@csse.unimelb.edu.au> (fixes) # Joseph Frazee <jfrazee@mail.utexas.edu> (fixes) # URL: <http://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. """
ConditionalProbDist, DictionaryProbDist, DictionaryConditionalProbDist, LidstoneProbDist, MutableProbDist, MLEProbDist, UniformProbDist)
# _NINF = float('-inf') # won't work on Windows
""" 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 specializaed 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: function or HiddenMarkovModelTaggerTransform """ self._symbols = list(set(symbols)) self._states = list(set(states)) self._transitions = transitions self._outputs = outputs self._priors = priors self._cache = None
self._transform = kwargs.get('transform', IdentityTransform()) if isinstance(self._transform, types.FunctionType): self._transform = LambdaTransform(self._transform) elif not isinstance(self._transform, HiddenMarkovModelTaggerTransformI): raise
unlabeled_sequence=None, **kwargs): transform = kwargs.get('transform', IdentityTransform()) if isinstance(transform, types.FunctionType): transform = LambdaTransform(transform) elif \ not isinstance(transform, HiddenMarkovModelTaggerTransformI): raise
estimator = kwargs.get('estimator', lambda fd, bins: \ LidstoneProbDist(fd, 0.1, bins))
labeled_sequence = LazyMap(transform.transform, labeled_sequence) symbols = list(set(word for sent in labeled_sequence for word, tag in sent)) tag_set = list(set(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
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 interations to perform :type max_iterations: int """ return cls._train(labeled_sequence, test_sequence, unlabeled_sequence, **kwargs)
""" 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.transform(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.transform(sequence)
T = len(sequence) N = len(self._states)
if T > 0 and sequence[0][_TAG]: last_state = sequence[0][_TAG] p = self._priors.logprob(last_state) + \ self._outputs[last_state].logprob(sequence[0][_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]) last_state = state return p else: alpha = self._forward_probability(sequence) p = _log_add(*alpha[T-1, :]) return p
""" 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.transform(unlabeled_sequence) return self._tag(unlabeled_sequence)
path = self._best_path(unlabeled_sequence) return list(zip(unlabeled_sequence, path))
""" :return: the log probability of the symbol being observed in the given state :rtype: float """ return self._outputs[state].logprob(symbol)
""" 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 = zeros(N, float32) X = zeros((N, N), float32) O = zeros((N, M), 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._outputs[si].logprob(self._symbols[k]) S = {} for k in range(M): S[self._symbols[k]] = k self._cache = (P, O, X, S)
# 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 = hstack([O, zeros((N, M - Q), 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._outputs[si].logprob(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)
""" 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.transform(unlabeled_sequence) return self._best_path(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 = zeros((T, N), float32) B = ones((T, N), int) * -1
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 = argmax(vs) V[t, j] = vs[best] + O[j, S[unlabeled_sequence[t]]] B[t, j] = best
current = 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))
""" 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.transform(unlabeled_sequence) return self._best_path_simple(unlabeled_sequence)
T = len(unlabeled_sequence) N = len(self._states) V = zeros((T, N), 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
""" 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
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')
""" 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.transform(unlabeled_sequence)
T = len(unlabeled_sequence) N = len(self._states)
alpha = self._forward_probability(unlabeled_sequence) beta = self._backward_probability(unlabeled_sequence) normalisation = _log_add(*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
""" Returns the pointwise entropy over the possible states at each position in the chain, given the observation sequence. """ unlabeled_sequence = self._transform.transform(unlabeled_sequence)
T = len(unlabeled_sequence) N = len(self._states)
alpha = self._forward_probability(unlabeled_sequence) beta = self._backward_probability(unlabeled_sequence) normalisation = _log_add(*alpha[T-1, :])
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] -= 2**(probs[s]) * probs[s]
return entropies
unlabeled_sequence = self._transform.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)
#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])] += \ # 2**(lp - normalisation)
#for t in range(T): #print 'prob[%d] =' % t, ps[t]
entropy = 0 for lp in log_probs: lp -= normalisation entropy -= 2**(lp) * lp
return entropy
unlabeled_sequence = self._transform.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 = 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] -= 2**(probabilities[t, s]) * probabilities[t, s]
return entropies
""" 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 = zeros((T, N), float64)
symbol = unlabeled_sequence[0][_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 = unlabeled_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
""" 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 = zeros((T, N), float64)
# initialise the backward values beta[T-1, :] = log2(1)
# inductively calculate remaining backward values for t in range(T-2, -1, -1): symbol = unlabeled_sequence[t+1][_TEXT] for i, si in enumerate(self._states): beta[t, i] = _NINF for j, sj in enumerate(self._states): beta[t, i] = _log_add(beta[t, i], self._transitions[si].logprob(sj) + self._outputs[sj].logprob(symbol) + beta[t + 1, j])
return beta
""" 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]
test_sequence = LazyMap(self._transform.transform, test_sequence) predicted_sequence = LazyMap(self._tag, LazyMap(words, test_sequence))
if kwargs.get('verbose', False): # This will be used again later for accuracy so there's no sense # in tagging it twice. test_sequence = list(test_sequence) predicted_sequence = list(predicted_sequence)
for test_sent, predicted_sent in zip(test_sequence, predicted_sequence): print('Test:', \ ' '.join(['%s/%s' % (str(token), str(tag)) for (token, tag) in test_sent])) print() print('Untagged:', \ ' '.join([str(token) for (token, tag) in test_sent])) print() print('HMM-tagged:', \ ' '.join(['%s/%s' % (str(token), str(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 = LazyConcatenation(LazyMap(tags, test_sequence)) predicted_tags = LazyConcatenation(LazyMap(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))
return ('<HiddenMarkovModelTagger %d states and %d output symbols>' % (len(self._states), len(self._symbols)))
""" 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 """ if states: self._states = states else: self._states = [] if symbols: self._symbols = symbols else: self._symbols = []
**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 :type labelled_sequences: list :param unlabeled_sequences: the unsupervised training data, a set of sequences of observations :type unlabeled_sequences: list :param kwargs: additional arguments to pass to the training methods """ assert labelled_sequences or unlabeled_sequences model = None if labelled_sequences: model = self.train_supervised(labelled_sequences, **kwargs) if unlabeled_sequences: if model: kwargs['model'] = model model = self.train_unsupervised(unlabeled_sequences, **kwargs) return model
""" 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 """
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 = HiddenMarkovModelTagger(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 unlabeled_sequences: sequence = list(sequence) if not sequence: continue
# 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) lpk = _log_add(*alpha[T-1, :]) 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): x = sequence[t][_TEXT] #not found? FIXME if t < T - 1: xnext = sequence[t+1][_TEXT] #not found? FIXME 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] = \ _log_add(local_A_numer[i, j], alpha[t, i] + model._transitions[si].logprob(sj) + model._outputs[sj].logprob(xnext) + beta[t+1, j]) local_A_denom[i] = _log_add(local_A_denom[i], alpha[t, i] + beta[t, i]) else: local_B_denom[i] = _log_add(local_A_denom[i], alpha[t, i] + beta[t, i])
local_B_numer[i, xi] = _log_add(local_B_numer[i, xi], 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): A_numer[i, j] = _log_add(A_numer[i, j], local_A_numer[i, j] - lpk) for k in range(M): B_numer[i, k] = _log_add(B_numer[i, k], 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
""" 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 kwargs: may include an 'estimator' parameter, 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 estimator = kwargs.get('estimator') 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 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.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, N) B = ConditionalProbDist(outputs, estimator, len(self._symbols))
return HiddenMarkovModelTagger(self._symbols, self._states, A, B, pi)
""" An abstract subclass of HiddenMarkovModelTaggerTransformI. """ if self.__class__ == HiddenMarkovModelTaggerTransform: raise NotImplementedError("Abstract classes can't be instantiated")
""" A subclass of HiddenMarkovModelTaggerTransform that is backed by an arbitrary user-defined function, instance method, or lambda function. """ """ :param func: a user-defined or lambda transform function :type func: function """ self._transform = transform
return self._transform(labeled_symbols)
""" A subclass of HiddenMarkovModelTaggerTransform that implements transform() as the identity function, i.e. symbols passed to transform() are returned unmodified. """ return labeled_symbols
""" Adds the logged values, returning the logarithm of the addition. """ x = max(values) if x > _NINF: sum_diffs = 0 for value in values: sum_diffs += 2**(value - x) return x + log2(sum_diffs) else: return x
# demonstrates HMM probability calculation
print() print("HMM probability calculation demo") print()
# 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 = HiddenMarkovModelTagger(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([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()
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)
# demonstrates POS tagging using supervised training
print() print("HMM POS tagging demo") print()
print('Training HMM...') labelled_sequences, tag_set, symbols = load_pos(200) 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)
unlabeled = [] for sentence in sentences: unlabeled.append((token[_TEXT], None) for token in sentence) return unlabeled
# demonstrates the Baum-Welch algorithm in POS tagging
print() print("Baum-Welch demo for POS tagging") print()
print('Training HMM (supervised)...') sentences, tag_set, symbols = load_pos(210) 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[10:200], estimator=lambda fd, bins: LidstoneProbDist(fd, 0.1, bins)) print('Training (unsupervised)...') # it's rather slow - so only use 10 samples unlabeled = _untag(sentences[200:210]) hmm = trainer.train_unsupervised(unlabeled, model=hmm, max_iterations=5) hmm.test(sentences[:10], verbose=True)
# demo Baum Welch by generating some sequences and then performing # unsupervised training on them
print() print("Baum-Welch demo for market example") print()
# 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 = HiddenMarkovModelTagger(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[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)
import doctest doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE)
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