00001 """Provide trainers which estimate parameters based on training sequences. These should be used to 'train' a Markov Model prior to actually using it to decode state paths. When supplied training sequences and a model to work from, these classes will estimate paramters of the model. This aims to estimate two parameters: * a_{kl} -- the number of times there is a transition from k to l in the training data. * e_{k}(b) -- the number of emissions of the state b from the letter k in the training data. """ # standard modules import math # local stuff from DynamicProgramming import ScaledDPAlgorithms 00021 class TrainingSequence: """Hold a training sequence with emissions and optionally, a state path. """ 00024 def __init__(self, emissions, state_path): """Initialize a training sequence. Arguments: o emissions - A Seq object containing the sequence of emissions in the training sequence, and the alphabet of the sequence. o state_path - A Seq object containing the sequence of states and the alphabet of the states. If there is no known state path, then the sequence of states should be an empty string. """ if len(state_path) > 0: assert len(emissions) == len(state_path), \ "State path does not match associated emissions." self.emissions = emissions self.states = state_path 00042 class AbstractTrainer: """Provide generic functionality needed in all trainers. """ def __init__(self, markov_model): self._markov_model = markov_model 00048 def log_likelihood(self, probabilities): """Calculate the log likelihood of the training seqs. Arguments: o probabilities -- A list of the probabilities of each training sequence under the current paramters, calculated using the forward algorithm. """ total_likelihood = 0 for probability in probabilities: total_likelihood += math.log(probability) return total_likelihood 00063 def estimate_params(self, transition_counts, emission_counts): """Get a maximum likelihood estimation of transition and emmission. Arguments: o transition_counts -- A dictionary with the total number of counts of transitions between two states. o emissions_counts -- A dictionary with the total number of counts of emmissions of a particular emission letter by a state letter. This then returns the maximum likelihood estimators for the transitions and emissions, estimated by formulas 3.18 in Durbin et al: a_{kl} = A_{kl} / sum(A_{kl'}) e_{k}(b) = E_{k}(b) / sum(E_{k}(b')) Returns: Transition and emission dictionaries containing the maximum likelihood estimators. """ # now calculate the information ml_transitions = self.ml_estimator(transition_counts) ml_emissions = self.ml_estimator(emission_counts) return ml_transitions, ml_emissions 00091 def ml_estimator(self, counts): """Calculate the maximum likelihood estimator. This can calculate maximum likelihoods for both transitions and emissions. Arguments: o counts -- A dictionary of the counts for each item. See estimate_params for a description of the formula used for calculation. """ # get an ordered list of all items all_ordered = counts.keys() all_ordered.sort() ml_estimation = {} # the total counts for the current letter we are on cur_letter = None cur_letter_counts = 0 for cur_item in all_ordered: # if we are on a new letter (ie. the first letter of the tuple) if cur_item[0] != cur_letter: # set the new letter we are working with cur_letter = cur_item[0] # count up the total counts for this letter cur_letter_counts = counts[cur_item] # add counts for all other items with the same first letter cur_position = all_ordered.index(cur_item) + 1 # keep adding while we have the same first letter or until # we get to the end of the ordered list while (cur_position < len(all_ordered) and all_ordered[cur_position][0] == cur_item[0]): cur_letter_counts += counts[all_ordered[cur_position]] cur_position += 1 # otherwise we've already got the total counts for this letter else: pass # now calculate the ml and add it to the estimation cur_ml = float(counts[cur_item]) / float(cur_letter_counts) ml_estimation[cur_item] = cur_ml return ml_estimation 00142 class BaumWelchTrainer(AbstractTrainer): """Trainer that uses the Baum-Welch algorithm to estimate parameters. These should be used when a training sequence for an HMM has unknown paths for the actual states, and you need to make an estimation of the model parameters from the observed emissions. This uses the Baum-Welch algorithm, first described in Baum, L.E. 1972. Inequalities. 3:1-8 This is based on the description in 'Biological Sequence Analysis' by Durbin et al. in section 3.3 This algorithm is guaranteed to converge to a local maximum, but not necessarily to the global maxima, so use with care! """ 00157 def __init__(self, markov_model): """Initialize the trainer. Arguments: o markov_model - The model we are going to estimate parameters for. This should have the parameters with some initial estimates, that we can build from. """ AbstractTrainer.__init__(self, markov_model) 00168 def train(self, training_seqs, stopping_criteria, dp_method = ScaledDPAlgorithms): """Estimate the parameters using training sequences. The algorithm for this is taken from Durbin et al. p64, so this is a good place to go for a reference on what is going on. Arguments: o training_seqs -- A list of TrainingSequence objects to be used for estimating the parameters. o stopping_criteria -- A function, that when passed the change in log likelihood and threshold, will indicate if we should stop the estimation iterations. o dp_method -- A class instance specifying the dynamic programming implementation we should use to calculate the forward and backward variables. By default, we use the scaling method. """ prev_log_likelihood = None num_iterations = 1 while 1: transition_count = self._markov_model.get_blank_transitions() emission_count = self._markov_model.get_blank_emissions() # remember all of the sequence probabilities all_probabilities = [] for training_seq in training_seqs: # calculate the forward and backward variables DP = dp_method(self._markov_model, training_seq) forward_var, seq_prob = DP.forward_algorithm() backward_var = DP.backward_algorithm() all_probabilities.append(seq_prob) # update the counts for transitions and emissions transition_count = self.update_transitions(transition_count, training_seq, forward_var, backward_var, seq_prob) emission_count = self.update_emissions(emission_count, training_seq, forward_var, backward_var, seq_prob) # update the markov model with the new probabilities ml_transitions, ml_emissions = \ self.estimate_params(transition_count, emission_count) self._markov_model.transition_prob = ml_transitions self._markov_model.emission_prob = ml_emissions cur_log_likelihood = self.log_likelihood(all_probabilities) # if we have previously calculated the log likelihood (ie. # not the first round), see if we can finish if prev_log_likelihood is not None: # XXX log likelihoods are negatives -- am I calculating # the change properly, or should I use the negatives... # I'm not sure at all if this is right. log_likelihood_change = abs(abs(cur_log_likelihood) - abs(prev_log_likelihood)) # check whether we have completed enough iterations to have # a good estimation if stopping_criteria(log_likelihood_change, num_iterations): break # set up for another round of iterations prev_log_likelihood = cur_log_likelihood num_iterations += 1 return self._markov_model 00246 def update_transitions(self, transition_counts, training_seq, forward_vars, backward_vars, training_seq_prob): """Add the contribution of a new training sequence to the transitions. Arguments: o transition_counts -- A dictionary of the current counts for the transitions o training_seq -- The training sequence we are working with o forward_vars -- Probabilities calculated using the forward algorithm. o backward_vars -- Probabilities calculated using the backwards algorithm. o training_seq_prob - The probability of the current sequence. This calculates A_{kl} (the estimated transition counts from state k to state l) using formula 3.20 in Durbin et al. """ # set up the transition and emission probabilities we are using transitions = self._markov_model.transition_prob emissions = self._markov_model.emission_prob # loop over the possible combinations of state path letters for k in training_seq.states.alphabet.letters: for l in self._markov_model.transitions_from(k): estimated_counts = 0 # now loop over the entire training sequence for i in range(len(training_seq.emissions) - 1): # the forward value of k at the current position forward_value = forward_vars[(k, i)] # the backward value of l in the next position backward_value = backward_vars[(l, i + 1)] # the probability of a transition from k to l trans_value = transitions[(k, l)] # the probability of getting the emission at the next pos emm_value = emissions[(l, training_seq.emissions[i + 1])] estimated_counts += (forward_value * trans_value * emm_value * backward_value) # update the transition approximation transition_counts[(k, l)] += (float(estimated_counts) / training_seq_prob) return transition_counts 00299 def update_emissions(self, emission_counts, training_seq, forward_vars, backward_vars, training_seq_prob): """Add the contribution of a new training sequence to the emissions Arguments: o emission_counts -- A dictionary of the current counts for the emissions o training_seq -- The training sequence we are working with o forward_vars -- Probabilities calculated using the forward algorithm. o backward_vars -- Probabilities calculated using the backwards algorithm. o training_seq_prob - The probability of the current sequence. This calculates E_{k}(b) (the estimated emission probability for emission letter b from state k) using formula 3.21 in Durbin et al. """ # loop over the possible combinations of state path letters for k in training_seq.states.alphabet.letters: # now loop over all of the possible emissions for b in training_seq.emissions.alphabet.letters: expected_times = 0 # finally loop over the entire training sequence for i in range(len(training_seq.emissions)): # only count the forward and backward probability if the # emission at the position is the same as b if training_seq.emissions[i] == b: # f_{k}(i) b_{k}(i) expected_times += (forward_vars[(k, i)] * backward_vars[(k, i)]) # add to E_{k}(b) emission_counts[(k, b)] += (float(expected_times) / training_seq_prob) return emission_counts 00341 class KnownStateTrainer(AbstractTrainer): """Estimate probabilities with known state sequences. This should be used for direct estimation of emission and transition probabilities when both the state path and emission sequence are known for the training examples. """ def __init__(self, markov_model): AbstractTrainer.__init__(self, markov_model) 00351 def train(self, training_seqs): """Estimate the Markov Model parameters with known state paths. This trainer requires that both the state and the emissions are known for all of the training sequences in the list of TrainingSequence objects. This training will then count all of the transitions and emissions, and use this to estimate the parameters of the model. """ # count up all of the transitions and emissions transition_counts = self._markov_model.get_blank_transitions() emission_counts = self._markov_model.get_blank_emissions() for training_seq in training_seqs: emission_counts = self._count_emissions(training_seq, emission_counts) transition_counts = self._count_transitions(training_seq.states, transition_counts) # update the markov model from the counts ml_transitions, ml_emissions = \ self.estimate_params(transition_counts, emission_counts) self._markov_model.transition_prob = ml_transitions self._markov_model.emission_prob = ml_emissions return self._markov_model 00379 def _count_emissions(self, training_seq, emission_counts): """Add emissions from the training sequence to the current counts. Arguments: o training_seq -- A TrainingSequence with states and emissions to get the counts from o emission_counts -- The current emission counts to add to. """ for index in range(len(training_seq.emissions)): cur_state = training_seq.states[index] cur_emission = training_seq.emissions[index] try: emission_counts[(cur_state, cur_emission)] += 1 except KeyError: raise KeyError("Unexpected emission (%s, %s)" % (cur_state, cur_emission)) return emission_counts 00400 def _count_transitions(self, state_seq, transition_counts): """Add transitions from the training sequence to the current counts. Arguments: o state_seq -- A Seq object with the states of the current training sequence. o transition_counts -- The current transition counts to add to. """ for cur_pos in range(len(state_seq) - 1): cur_state = state_seq[cur_pos] next_state = state_seq[cur_pos + 1] try: transition_counts[(cur_state, next_state)] += 1 except KeyError: raise KeyError("Unexpected transition (%s, %s)" % (cur_state, next_state)) return transition_counts

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