MarkovModel.py

00001 """Deal with representations of Markov Models. """ # standard modules import copy import math import random #TODO - Take advantage of defaultdict once Python 2.4 is dead? #from collections import defaultdict # biopython from Bio.Seq import MutableSeq 00014 def _gen_random_array(n): """ Return an array of n random numbers, where the elements of the array sum to 1.0""" randArray = [random.random() for i in range(n)] total = sum(randArray) normalizedRandArray = [x/total for x in randArray] return normalizedRandArray 00023 def _calculate_emissions(emission_probs): """Calculate which symbols can be emitted in each state """ # loop over all of the state-symbol duples, mapping states to # lists of emitted symbols emissions = dict() for state, symbol in emission_probs: try: emissions[state].append(symbol) except KeyError: emissions[state] = [symbol] return emissions 00037 def _calculate_from_transitions(trans_probs): """Calculate which 'from transitions' are allowed for each state This looks through all of the trans_probs, and uses this dictionary to determine allowed transitions. It converts this information into a dictionary, whose keys are source states and whose values are lists of destination states reachable from the source state via a transition. """ transitions = dict() for from_state, to_state in trans_probs: try: transitions[from_state].append(to_state) except KeyError: transitions[from_state] = [to_state] return transitions 00055 def _calculate_to_transitions(trans_probs): """Calculate which 'to transitions' are allowed for each state This looks through all of the trans_probs, and uses this dictionary to determine allowed transitions. It converts this information into a dictionary, whose keys are destination states and whose values are lists of source states from which the destination is reachable via a transition. """ transitions = dict() for from_state, to_state in trans_probs: try: transitions[to_state].append(from_state) except KeyError: transitions[to_state] = [from_state] return transitions 00073 class MarkovModelBuilder(object): """Interface to build up a Markov Model. This class is designed to try to separate the task of specifying the Markov Model from the actual model itself. This is in hopes of making the actual Markov Model classes smaller. So, this builder class should be used to create Markov models instead of trying to initiate a Markov Model directly. """ # the default pseudo counts to use DEFAULT_PSEUDO = 1 00086 def __init__(self, state_alphabet, emission_alphabet): """Initialize a builder to create Markov Models. Arguments: o state_alphabet -- An alphabet containing all of the letters that can appear in the states o emission_alphabet -- An alphabet containing all of the letters for states that can be emitted by the HMM. """ self._state_alphabet = state_alphabet self._emission_alphabet = emission_alphabet # probabilities for the initial state, initialized by calling # set_initial_probabilities (required) self.initial_prob = {} # the probabilities for transitions and emissions # by default we have no transitions and all possible emissions self.transition_prob = {} self.emission_prob = self._all_blank(state_alphabet, emission_alphabet) # the default pseudocounts for transition and emission counting self.transition_pseudo = {} self.emission_pseudo = self._all_pseudo(state_alphabet, emission_alphabet) 00115 def _all_blank(self, first_alphabet, second_alphabet): """Return a dictionary with all counts set to zero. This uses the letters in the first and second alphabet to create a dictionary with keys of two tuples organized as (letter of first alphabet, letter of second alphabet). The values are all set to 0. """ all_blank = {} for first_state in first_alphabet.letters: for second_state in second_alphabet.letters: all_blank[(first_state, second_state)] = 0 return all_blank 00130 def _all_pseudo(self, first_alphabet, second_alphabet): """Return a dictionary with all counts set to a default value. This takes the letters in first alphabet and second alphabet and creates a dictionary with keys of two tuples organized as: (letter of first alphabet, letter of second alphabet). The values are all set to the value of the class attribute DEFAULT_PSEUDO. """ all_counts = {} for first_state in first_alphabet.letters: for second_state in second_alphabet.letters: all_counts[(first_state, second_state)] = self.DEFAULT_PSEUDO return all_counts 00145 def get_markov_model(self): """Return the markov model corresponding with the current parameters. Each markov model returned by a call to this function is unique (ie. they don't influence each other). """ # user must set initial probabilities if not self.initial_prob: raise Exception("set_initial_probabilities must be called to " + "fully initialize the Markov model") initial_prob = copy.deepcopy(self.initial_prob) transition_prob = copy.deepcopy(self.transition_prob) emission_prob = copy.deepcopy(self.emission_prob) transition_pseudo = copy.deepcopy(self.transition_pseudo) emission_pseudo = copy.deepcopy(self.emission_pseudo) return HiddenMarkovModel(initial_prob, transition_prob, emission_prob, transition_pseudo, emission_pseudo) 00166 def set_initial_probabilities(self, initial_prob): """Set initial state probabilities. initial_prob is a dictionary mapping states to probabilities. Suppose, for example, that the state alphabet is ['A', 'B']. Call set_initial_prob({'A': 1}) to guarantee that the initial state will be 'A'. Call set_initial_prob({'A': 0.5, 'B': 0.5}) to make each initial state equally probable. This method must now be called in order to use the Markov model because the calculation of initial probabilities has changed incompatibly; the previous calculation was incorrect. If initial probabilities are set for all states, then they should add up to 1. Otherwise the sum should be <= 1. The residual probability is divided up evenly between all the states for which the initial probability has not been set. For example, calling set_initial_prob({}) results in P('A') = 0.5 and P('B') = 0.5, for the above example. """ self.initial_prob = copy.copy(initial_prob) # ensure that all referenced states are valid for state in initial_prob.iterkeys(): assert state in self._state_alphabet.letters, \ "State %s was not found in the sequence alphabet" % state # distribute the residual probability, if any num_states_not_set =\ len(self._state_alphabet.letters) - len(self.initial_prob) if num_states_not_set < 0: raise Exception("Initial probabilities can't exceed # of states") prob_sum = sum(self.initial_prob.values()) if prob_sum > 1.0: raise Exception("Total initial probability cannot exceed 1.0") if num_states_not_set > 0: prob = (1.0 - prob_sum) / num_states_not_set for state in self._state_alphabet.letters: if not state in self.initial_prob: self.initial_prob[state] = prob 00207 def set_equal_probabilities(self): """Reset all probabilities to be an average value. Resets the values of all initial probabilities and all allowed transitions and all allowed emissions to be equal to 1 divided by the number of possible elements. This is useful if you just want to initialize a Markov Model to starting values (ie. if you have no prior notions of what the probabilities should be -- or if you are just feeling too lazy to calculate them :-). Warning 1 -- this will reset all currently set probabilities. Warning 2 -- This just sets all probabilities for transitions and emissions to total up to 1, so it doesn't ensure that the sum of each set of transitions adds up to 1. """ # set initial state probabilities new_initial_prob = float(1) / float(len(self.transition_prob)) for state in self._state_alphabet.letters: self.initial_prob[state] = new_initial_prob # set the transitions new_trans_prob = float(1) / float(len(self.transition_prob)) for key in self.transition_prob: self.transition_prob[key] = new_trans_prob # set the emissions new_emission_prob = float(1) / float(len(self.emission_prob)) for key in self.emission_prob: self.emission_prob[key] = new_emission_prob 00242 def set_random_initial_probabilities(self): """Set all initial state probabilities to a randomly generated distribution. Returns the dictionary containing the initial probabilities. """ initial_freqs = _gen_random_array(len(self._state_alphabet.letters)) for state in self._state_alphabet.letters: self.initial_prob[state] = initial_freqs.pop() return self.initial_prob 00252 def set_random_transition_probabilities(self): """Set all allowed transition probabilities to a randomly generated distribution. Returns the dictionary containing the transition probabilities. """ if not self.transition_prob: raise Exception("No transitions have been allowed yet. " + "Allow some or all transitions by calling " + "allow_transition or allow_all_transitions first.") transitions_from = _calculate_from_transitions(self.transition_prob) for from_state in transitions_from.keys(): freqs = _gen_random_array(len(transitions_from[from_state])) for to_state in transitions_from[from_state]: self.transition_prob[(from_state, to_state)] = freqs.pop() return self.transition_prob 00270 def set_random_emission_probabilities(self): """Set all allowed emission probabilities to a randomly generated distribution. Returns the dictionary containing the emission probabilities. """ if not self.emission_prob: raise Exception("No emissions have been allowed yet. " + "Allow some or all emissions.") emissions = _calculate_emissions(self.emission_prob) for state in emissions.iterkeys(): freqs = _gen_random_array(len(emissions[state])) for symbol in emissions[state]: self.emission_prob[(state, symbol)] = freqs.pop() return self.emission_prob 00289 def set_random_probabilities(self): """Set all probabilities to randomly generated numbers. Resets probabilities of all initial states, transitions, and emissions to random values. """ self.set_random_initial_probabilities() self.set_random_transition_probabilities() self.set_random_emission_probabilities() # --- functions to deal with the transitions in the sequence 00301 def allow_all_transitions(self): """A convenience function to create transitions between all states. By default all transitions within the alphabet are disallowed; this is a way to change this to allow all possible transitions. """ # first get all probabilities and pseudo counts set # to the default values all_probs = self._all_blank(self._state_alphabet, self._state_alphabet) all_pseudo = self._all_pseudo(self._state_alphabet, self._state_alphabet) # now set any probabilities and pseudo counts that # were previously set for set_key in self.transition_prob: all_probs[set_key] = self.transition_prob[set_key] for set_key in self.transition_pseudo: all_pseudo[set_key] = self.transition_pseudo[set_key] # finally reinitialize the transition probs and pseudo counts self.transition_prob = all_probs self.transition_pseudo = all_pseudo 00327 def allow_transition(self, from_state, to_state, probability = None, pseudocount = None): """Set a transition as being possible between the two states. probability and pseudocount are optional arguments specifying the probabilities and pseudo counts for the transition. If these are not supplied, then the values are set to the default values. Raises: KeyError -- if the two states already have an allowed transition. """ # check the sanity of adding these states for state in [from_state, to_state]: assert state in self._state_alphabet.letters, \ "State %s was not found in the sequence alphabet" % state # ensure that the states are not already set if ((from_state, to_state) not in self.transition_prob and (from_state, to_state) not in self.transition_pseudo): # set the initial probability if probability is None: probability = 0 self.transition_prob[(from_state, to_state)] = probability # set the initial pseudocounts if pseudocount is None: pseudcount = self.DEFAULT_PSEUDO self.transition_pseudo[(from_state, to_state)] = pseudocount else: raise KeyError("Transition from %s to %s is already allowed." % (from_state, to_state)) 00360 def destroy_transition(self, from_state, to_state): """Restrict transitions between the two states. Raises: KeyError if the transition is not currently allowed. """ try: del self.transition_prob[(from_state, to_state)] del self.transition_pseudo[(from_state, to_state)] except KeyError: raise KeyError("Transition from %s to %s is already disallowed." % (from_state, to_state)) 00373 def set_transition_score(self, from_state, to_state, probability): """Set the probability of a transition between two states. Raises: KeyError if the transition is not allowed. """ if (from_state, to_state) in self.transition_prob: self.transition_prob[(from_state, to_state)] = probability else: raise KeyError("Transition from %s to %s is not allowed." % (from_state, to_state)) 00385 def set_transition_pseudocount(self, from_state, to_state, count): """Set the default pseudocount for a transition. To avoid computational problems, it is helpful to be able to set a 'default' pseudocount to start with for estimating transition and emission probabilities (see p62 in Durbin et al for more discussion on this. By default, all transitions have a pseudocount of 1. Raises: KeyError if the transition is not allowed. """ if (from_state, to_state) in self.transition_pseudo: self.transition_pseudo[(from_state, to_state)] = count else: raise KeyError("Transition from %s to %s is not allowed." % (from_state, to_state)) # --- functions to deal with emissions from the sequence 00405 def set_emission_score(self, seq_state, emission_state, probability): """Set the probability of a emission from a particular state. Raises: KeyError if the emission from the given state is not allowed. """ if (seq_state, emission_state) in self.emission_prob: self.emission_prob[(seq_state, emission_state)] = probability else: raise KeyError("Emission of %s from %s is not allowed." % (emission_state, seq_state)) 00417 def set_emission_pseudocount(self, seq_state, emission_state, count): """Set the default pseudocount for an emission. To avoid computational problems, it is helpful to be able to set a 'default' pseudocount to start with for estimating transition and emission probabilities (see p62 in Durbin et al for more discussion on this. By default, all emissions have a pseudocount of 1. Raises: KeyError if the emission from the given state is not allowed. """ if (seq_state, emission_state) in self.emission_pseudo: self.emission_pseudo[(seq_state, emission_state)] = count else: raise KeyError("Emission of %s from %s is not allowed." % (emission_state, seq_state)) 00435 class HiddenMarkovModel(object): """Represent a hidden markov model that can be used for state estimation. """ 00438 def __init__(self, initial_prob, transition_prob, emission_prob, transition_pseudo, emission_pseudo): """Initialize a Markov Model. Note: You should use the MarkovModelBuilder class instead of initiating this class directly. Arguments: o initial_prob - A dictionary of initial probabilities for all states. o transition_prob -- A dictionary of transition probabilities for all possible transitions in the sequence. o emission_prob -- A dictionary of emission probabilities for all possible emissions from the sequence states. o transition_pseudo -- Pseudo-counts to be used for the transitions, when counting for purposes of estimating transition probabilities. o emission_pseudo -- Pseudo-counts to be used for the emissions, when counting for purposes of estimating emission probabilities. """ self.initial_prob = initial_prob self._transition_pseudo = transition_pseudo self._emission_pseudo = emission_pseudo self.transition_prob = transition_prob self.emission_prob = emission_prob # a dictionary of the possible transitions from each state # each key is a source state, mapped to a list of the destination states # that are reachable from the source state via a transition self._transitions_from = \ _calculate_from_transitions(self.transition_prob) # a dictionary of the possible transitions to each state # each key is a destination state, mapped to a list of source states # from which the destination is reachable via a transition self._transitions_to = \ _calculate_to_transitions(self.transition_prob) 00483 def get_blank_transitions(self): """Get the default transitions for the model. Returns a dictionary of all of the default transitions between any two letters in the sequence alphabet. The dictionary is structured with keys as (letter1, letter2) and values as the starting number of transitions. """ return self._transition_pseudo 00493 def get_blank_emissions(self): """Get the starting default emmissions for each sequence. This returns a dictionary of the default emmissions for each letter. The dictionary is structured with keys as (seq_letter, emmission_letter) and values as the starting number of emmissions. """ return self._emission_pseudo 00503 def transitions_from(self, state_letter): """Get all destination states to which there are transitions from the state_letter source state. This returns all letters which the given state_letter can transition to. An empty list is returned if state_letter has no outgoing transitions. """ if state_letter in self._transitions_from: return self._transitions_from[state_letter] else: return [] 00516 def transitions_to(self, state_letter): """Get all source states from which there are transitions to the state_letter destination state. This returns all letters which the given state_letter is reachable from. An empty list is returned if state_letter is unreachable. """ if state_letter in self._transitions_to: return self._transitions_to[state_letter] else: return [] 00528 def viterbi(self, sequence, state_alphabet): """Calculate the most probable state path using the Viterbi algorithm. This implements the Viterbi algorithm (see pgs 55-57 in Durbin et al for a full explanation -- this is where I took my implementation ideas from), to allow decoding of the state path, given a sequence of emissions. Arguments: o sequence -- A Seq object with the emission sequence that we want to decode. o state_alphabet -- The alphabet of the possible state sequences that can be generated. """ # calculate logarithms of the initial, transition, and emission probs log_initial = self._log_transform(self.initial_prob) log_trans = self._log_transform(self.transition_prob) log_emission = self._log_transform(self.emission_prob) viterbi_probs = {} pred_state_seq = {} state_letters = state_alphabet.letters # --- recursion # loop over the training squence (i = 1 .. L) # 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. for i in range(0, len(sequence)): # loop over all of the possible i-th states in the state path for cur_state in state_letters: # e_{l}(x_{i}) emission_part = log_emission[(cur_state, sequence[i])] max_prob = 0 if i == 0: # for the first state, use the initial probability rather # than looking back to previous states max_prob = log_initial[cur_state] else: # loop over all possible (i-1)-th previous states possible_state_probs = {} for prev_state in self.transitions_to(cur_state): # a_{kl} trans_part = log_trans[(prev_state, cur_state)] # v_{k}(i - 1) viterbi_part = viterbi_probs[(prev_state, i - 1)] cur_prob = viterbi_part + trans_part possible_state_probs[prev_state] = cur_prob # calculate the viterbi probability using the max max_prob = max(possible_state_probs.values()) # v_{k}(i) viterbi_probs[(cur_state, i)] = (emission_part + max_prob) if i > 0: # get the most likely prev_state leading to cur_state for state in possible_state_probs: if possible_state_probs[state] == max_prob: pred_state_seq[(i - 1, cur_state)] = state break # --- termination # calculate the probability of the state path # loop over all states all_probs = {} for state in state_letters: # v_{k}(L) all_probs[state] = viterbi_probs[(state, len(sequence) - 1)] state_path_prob = max(all_probs.values()) # find the last pointer we need to trace back from last_state = '' for state in all_probs: if all_probs[state] == state_path_prob: last_state = state assert last_state != '', "Didn't find the last state to trace from!" # --- traceback traceback_seq = MutableSeq('', state_alphabet) loop_seq = range(1, len(sequence)) loop_seq.reverse() # last_state is the last state in the most probable state sequence. # Compute that sequence by walking backwards in time. From the i-th # state in the sequence, find the (i-1)-th state as the most # probable state preceding the i-th state. state = last_state traceback_seq.append(state) for i in loop_seq: state = pred_state_seq[(i - 1, state)] traceback_seq.append(state) # put the traceback sequence in the proper orientation traceback_seq.reverse() return traceback_seq.toseq(), state_path_prob 00635 def _log_transform(self, probability): """Return log transform of the given probability dictionary. When calculating the Viterbi equation, add logs of probabilities rather than multiplying probabilities, to avoid underflow errors. This method returns a new dictionary with the same keys as the given dictionary and log-transformed values. """ log_prob = copy.copy(probability) try: neg_inf = float("-inf") except ValueError: #On Python 2.5 or older that was handled in C code, #and failed on Windows XP 32bit neg_inf = - 1E400 for key in log_prob: prob = log_prob[key] if prob > 0: log_prob[key] = math.log(log_prob[key]) else: log_prob[key] = neg_inf return log_prob

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