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"""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
class TrainingSequence(object):
"""Hold a training sequence with emissions and optionally, a state path.
"""
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
class AbstractTrainer(object):
"""Provide generic functionality needed in all trainers.
"""
def __init__(self, markov_model):
self._markov_model = markov_model
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
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
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
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!
"""
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)
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
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
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
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)
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
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
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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