00001 """Dynamic Programming algorithms for general usage. This module contains classes which implement Dynamic Programming algorithms that can be used generally. """ 00007 class AbstractDPAlgorithms: """An abstract class to calculate forward and backward probabiliies. This class should not be instantiated directly, but should be used through a derived class which implements proper scaling of variables. This class is just meant to encapsulate the basic foward and backward algorithms, and allow derived classes to deal with the problems of multiplying probabilities. Derived class of this must implement: o _forward_recursion -- Calculate the forward values in the recursion using some kind of technique for preventing underflow errors. o _backward_recursion -- Calculate the backward values in the recursion step using some technique to prevent underflow errors. """ 00025 def __init__(self, markov_model, sequence): """Initialize to calculate foward and backward probabilities. Arguments: o markov_model -- The current Markov model we are working with. o sequence -- A training sequence containing a set of emissions. """ self._mm = markov_model self._seq = sequence 00037 def _foward_recursion(self, cur_state, sequence_pos, forward_vars): """Calculate the forward recursion value. """ raise NotImplementedError("Subclasses must implement") 00042 def forward_algorithm(self): """Calculate sequence probability using the forward algorithm. This implements the foward algorithm, as described on p57-58 of Durbin et al. Returns: o A dictionary containing the foward variables. This has keys of the form (state letter, position in the training sequence), and values containing the calculated forward variable. o The calculated probability of the sequence. """ # all of the different letters that the state path can be in state_letters = self._seq.states.alphabet.letters # -- initialize the algorithm # # NOTE: My index numbers are one less than what is given in Durbin # et al, since we are indexing the sequence going from 0 to # (Length - 1) not 1 to Length, like in Durbin et al. # forward_var = {} # f_{0}(0) = 1 forward_var[(state_letters[0], -1)] = 1 # f_{k}(0) = 0, for k > 0 for k in range(1, len(state_letters)): forward_var[(state_letters[k], -1)] = 0 # -- now do the recursion step # loop over the training sequence # Recursion step: (i = 1 .. L) for i in range(len(self._seq.emissions)): # now loop over the letters in the state path for main_state in state_letters: # calculate the forward value using the appropriate # method to prevent underflow errors forward_value = self._forward_recursion(main_state, i, forward_var) if forward_value is not None: forward_var[(main_state, i)] = forward_value # -- termination step - calculate the probability of the sequence first_state = state_letters[0] seq_prob = 0 for state_item in state_letters: # f_{k}(L) forward_value = forward_var[(state_item, len(self._seq.emissions) - 1)] # a_{k0} transition_value = self._mm.transition_prob[(state_item, first_state)] seq_prob += forward_value * transition_value return forward_var, seq_prob 00102 def _backward_recursion(self, cur_state, sequence_pos, forward_vars): """Calculate the backward recursion value. """ raise NotImplementedError("Subclasses must implement") 00107 def backward_algorithm(self): """Calculate sequence probability using the backward algorithm. This implements the backward algorithm, as described on p58-59 of Durbin et al. Returns: o A dictionary containing the backwards variables. This has keys of the form (state letter, position in the training sequence), and values containing the calculated backward variable. """ # all of the different letters that the state path can be in state_letters = self._seq.states.alphabet.letters # -- initialize the algorithm # # NOTE: My index numbers are one less than what is given in Durbin # et al, since we are indexing the sequence going from 0 to # (Length - 1) not 1 to Length, like in Durbin et al. # backward_var = {} first_letter = state_letters[0] # b_{k}(L) = a_{k0} for all k for state in state_letters: backward_var[(state, len(self._seq.emissions) - 1)] = \ self._mm.transition_prob[(state, state_letters[0])] # -- recursion # first loop over the training sequence backwards # Recursion step: (i = L - 1 ... 1) all_indexes = range(len(self._seq.emissions) - 1) all_indexes.reverse() for i in all_indexes: # now loop over the letters in the state path for main_state in state_letters: # calculate the backward value using the appropriate # method to prevent underflow errors backward_value = self._backward_recursion(main_state, i, backward_var) if backward_value is not None: backward_var[(main_state, i)] = backward_value # skip the termination step to avoid recalculations -- you should # get sequence probabilities using the forward algorithm return backward_var 00157 class ScaledDPAlgorithms(AbstractDPAlgorithms): """Implement forward and backward algorithms using a rescaling approach. This scales the f and b variables, so that they remain within a manageable numerical interval during calculations. This approach is described in Durbin et al. on p 78. This approach is a little more straightfoward then log transformation but may still give underflow errors for some types of models. In these cases, the LogDPAlgorithms class should be used. """ 00168 def __init__(self, markov_model, sequence): """Initialize the scaled approach to calculating probabilities. Arguments: o markov_model -- The current Markov model we are working with. o sequence -- A TrainingSequence object that must have a set of emissions to work with. """ AbstractDPAlgorithms.__init__(self, markov_model, sequence) self._s_values = {} 00181 def _calculate_s_value(self, seq_pos, previous_vars): """Calculate the next scaling variable for a sequence position. This utilizes the approach of choosing s values such that the sum of all of the scaled f values is equal to 1. Arguments: o seq_pos -- The current position we are at in the sequence. o previous_vars -- All of the forward or backward variables calculated so far. Returns: o The calculated scaling variable for the sequence item. """ # all of the different letters the state can have state_letters = self._seq.states.alphabet.letters # loop over all of the possible states s_value = 0 for main_state in state_letters: emission = self._mm.emission_prob[(main_state, self._seq.emissions[seq_pos])] # now sum over all of the previous vars and transitions trans_and_var_sum = 0 for second_state in self._mm.transitions_from(main_state): # the value of the previous f or b value var_value = previous_vars[(second_state, seq_pos - 1)] # the transition probability trans_value = self._mm.transition_prob[(second_state, main_state)] trans_and_var_sum += (var_value * trans_value) s_value += (emission * trans_and_var_sum) return s_value 00223 def _forward_recursion(self, cur_state, sequence_pos, forward_vars): """Calculate the value of the forward recursion. Arguments: o cur_state -- The letter of the state we are calculating the forward variable for. o sequence_pos -- The position we are at in the training seq. o forward_vars -- The current set of forward variables """ # calculate the s value, if we haven't done so already (ie. during # a previous forward or backward recursion) if not(self._s_values.has_key(sequence_pos)): self._s_values[sequence_pos] = \ self._calculate_s_value(sequence_pos, forward_vars) # e_{l}(x_{i}) seq_letter = self._seq.emissions[sequence_pos] cur_emission_prob = self._mm.emission_prob[(cur_state, seq_letter)] # divide by the scaling value scale_emission_prob = (float(cur_emission_prob) / float(self._s_values[sequence_pos])) # loop over all of the possible states at the position state_pos_sum = 0 have_transition = 0 for second_state in self._mm.transitions_from(cur_state): have_transition = 1 # get the previous forward_var values # f_{k}(i - 1) prev_forward = forward_vars[(second_state, sequence_pos - 1)] # a_{kl} cur_trans_prob = self._mm.transition_prob[(second_state, cur_state)] state_pos_sum += prev_forward * cur_trans_prob # if we have the possiblity of having a transition # return the recursion value if have_transition: return (scale_emission_prob * state_pos_sum) else: return None 00270 def _backward_recursion(self, cur_state, sequence_pos, backward_vars): """Calculate the value of the backward recursion Arguments: o cur_state -- The letter of the state we are calculating the forward variable for. o sequence_pos -- The position we are at in the training seq. o backward_vars -- The current set of backward variables """ # calculate the s value, if we haven't done so already (ie. during # a previous forward or backward recursion) if not(self._s_values.has_key(sequence_pos)): self._s_values[sequence_pos] = \ self._calculate_s_value(sequence_pos, backward_vars) # loop over all of the possible states at the position state_pos_sum = 0 have_transition = 0 for second_state in self._mm.transitions_from(cur_state): have_transition = 1 # e_{l}(x_{i + 1}) seq_letter = self._seq.emissions[sequence_pos + 1] cur_emission_prob = self._mm.emission_prob[(cur_state, seq_letter)] # get the previous backward_var value # b_{l}(i + 1) prev_backward = backward_vars[(second_state, sequence_pos + 1)] # the transition probability -- a_{kl} cur_transition_prob = self._mm.transition_prob[(cur_state, second_state)] state_pos_sum += (cur_emission_prob * prev_backward * cur_transition_prob) # if we have a probability for a transition, return it if have_transition: return (state_pos_sum / float(self._s_values[sequence_pos])) # otherwise we have no probability (ie. we can't do this transition) # and return None else: return None 00316 class LogDPAlgorithms(AbstractDPAlgorithms): """Implement forward and backward algorithms using a log approach. This uses the approach of calculating the sum of log probabilities using a lookup table for common values. XXX This is not implemented yet! """ 00324 def __init__(self, markov_model, sequence): raise NotImplementedError("Haven't coded this yet...")

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