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MarkovModel.py

00001 """
This is an implementation of a state-emitting MarkovModel.  I am using
terminology similar to Manning and Schutze.



Functions:
train_bw        Train a markov model using the Baum-Welch algorithm.
train_visible   Train a visible markov model using MLE.
find_states     Find the a state sequence that explains some observations.

load            Load a MarkovModel.
save            Save a MarkovModel.

Classes:
MarkovModel     Holds the description of a markov model
"""

import numpy


def itemindex(values):
    d = {}
    entries = enumerate(values[::-1])
    n = len(values)-1
    for index, key in entries: d[key] = n-index
    return d

numpy.random.seed()

VERY_SMALL_NUMBER = 1E-300
LOG0 = numpy.log(VERY_SMALL_NUMBER)

class MarkovModel:
    def __init__(self, states, alphabet,
                 p_initial=None, p_transition=None, p_emission=None):
        self.states = states
        self.alphabet = alphabet
        self.p_initial = p_initial
        self.p_transition = p_transition
        self.p_emission = p_emission
    def __str__(self):
        import StringIO
        handle = StringIO.StringIO()
        save(self, handle)
        handle.seek(0)
        return handle.read()

def _readline_and_check_start(handle, start):
    line = handle.readline()
    if not line.startswith(start):
        raise ValueError("I expected %r but got %r" % (start, line))
    return line

00055 def load(handle):
    """load(handle) -> MarkovModel()"""
    # Load the states.
    line = _readline_and_check_start(handle, "STATES:")
    states = line.split()[1:]

    # Load the alphabet.
    line = _readline_and_check_start(handle, "ALPHABET:")
    alphabet = line.split()[1:]

    mm = MarkovModel(states, alphabet)
    N, M = len(states), len(alphabet)

    # Load the initial probabilities.
    mm.p_initial = numpy.zeros(N)
    line = _readline_and_check_start(handle, "INITIAL:")
    for i in range(len(states)):
        line = _readline_and_check_start(handle, "  %s:" % states[i])
        mm.p_initial[i] = float(line.split()[-1])

    # Load the transition.
    mm.p_transition = numpy.zeros((N, N))
    line = _readline_and_check_start(handle, "TRANSITION:")
    for i in range(len(states)):
        line = _readline_and_check_start(handle, "  %s:" % states[i])
        mm.p_transition[i,:] = map(float, line.split()[1:])

    # Load the emission.
    mm.p_emission = numpy.zeros((N, M))
    line = _readline_and_check_start(handle, "EMISSION:")
    for i in range(len(states)):
        line = _readline_and_check_start(handle, "  %s:" % states[i])
        mm.p_emission[i,:] = map(float, line.split()[1:])

    return mm
        
00091 def save(mm, handle):
    """save(mm, handle)"""
    # This will fail if there are spaces in the states or alphabet.
    w = handle.write
    w("STATES: %s\n" % ' '.join(mm.states))
    w("ALPHABET: %s\n" % ' '.join(mm.alphabet))
    w("INITIAL:\n")
    for i in range(len(mm.p_initial)):
        w("  %s: %g\n" % (mm.states[i], mm.p_initial[i]))
    w("TRANSITION:\n")
    for i in range(len(mm.p_transition)):
        x = map(str, mm.p_transition[i])
        w("  %s: %s\n" % (mm.states[i], ' '.join(x)))
    w("EMISSION:\n")
    for i in range(len(mm.p_emission)):
        x = map(str, mm.p_emission[i])
        w("  %s: %s\n" % (mm.states[i], ' '.join(x)))

# XXX allow them to specify starting points
00110 def train_bw(states, alphabet, training_data, 
             pseudo_initial=None, pseudo_transition=None, pseudo_emission=None,
             update_fn=None,             
             ):
    """train_bw(states, alphabet, training_data[, pseudo_initial]
    [, pseudo_transition][, pseudo_emission][, update_fn]) -> MarkovModel

    Train a MarkovModel using the Baum-Welch algorithm.  states is a list
    of strings that describe the names of each state.  alphabet is a
    list of objects that indicate the allowed outputs.  training_data
    is a list of observations.  Each observation is a list of objects
    from the alphabet.

    pseudo_initial, pseudo_transition, and pseudo_emission are
    optional parameters that you can use to assign pseudo-counts to
    different matrices.  They should be matrices of the appropriate
    size that contain numbers to add to each parameter matrix, before
    normalization.

    update_fn is an optional callback that takes parameters
    (iteration, log_likelihood).  It is called once per iteration.

    """
    N, M = len(states), len(alphabet)
    if not training_data:
        raise ValueError("No training data given.")
    if pseudo_initial!=None:
        pseudo_initial = asarray(pseudo_initial)
        if pseudo_initial.shape != (N,):
            raise ValueError("pseudo_initial not shape len(states)")
    if pseudo_transition!=None:
        pseudo_transition = asarray(pseudo_transition)
        if pseudo_transition.shape != (N,N):
            raise ValueError("pseudo_transition not shape " + \
                             "len(states) X len(states)")
    if pseudo_emission!=None:
        pseudo_emission = asarray(pseudo_emission)
        if pseudo_emission.shape != (N,M):
            raise ValueError("pseudo_emission not shape " + \
                             "len(states) X len(alphabet)")
        
    # Training data is given as a list of members of the alphabet.
    # Replace those with indexes into the alphabet list for easier
    # computation.
    training_outputs = []
    indexes = itemindex(alphabet)
    for outputs in training_data:
        training_outputs.append([indexes[x] for x in outputs])

    # Do some sanity checking on the outputs.
    lengths = map(len, training_outputs)
    if min(lengths) == 0:
        raise ValueError("I got training data with outputs of length 0")

    # Do the training with baum welch.
    x = _baum_welch(N, M, training_outputs,
                    pseudo_initial=pseudo_initial,
                    pseudo_transition=pseudo_transition,
                    pseudo_emission=pseudo_emission,
                    update_fn=update_fn)
    p_initial, p_transition, p_emission = x
    return MarkovModel(states, alphabet, p_initial, p_transition, p_emission)

MAX_ITERATIONS = 1000
def _baum_welch(N, M, training_outputs,
                p_initial=None, p_transition=None, p_emission=None,
                pseudo_initial=None, pseudo_transition=None,
                pseudo_emission=None, update_fn=None):
    # Returns (p_initial, p_transition, p_emission)
    if p_initial==None:
        p_initial = _random_norm(N)
    else:
        p_initial = _copy_and_check(p_initial, (N,))

    if p_transition==None:
        p_transition = _random_norm((N,N))
    else:
        p_transition = _copy_and_check(p_transition, (N,N))
    if p_emission==None:
        p_emission = _random_norm((N,M))
    else:
        p_emission = _copy_and_check(p_emission, (N,M))
    
    # Do all the calculations in log space to avoid underflows.
    lp_initial, lp_transition, lp_emission = map(
        numpy.log, (p_initial, p_transition, p_emission))
    if pseudo_initial!=None:
        lpseudo_initial = numpy.log(pseudo_initial)
    else:
        lpseudo_initial = None
    if pseudo_transition!=None:
        lpseudo_transition = numpy.log(pseudo_transition)
    else:
        lpseudo_transition = None
    if pseudo_emission!=None:
        lpseudo_emission = numpy.log(pseudo_emission)
    else:
        lpseudo_emission = None

    # Iterate through each sequence of output, updating the parameters
    # to the HMM.  Stop when the log likelihoods of the sequences
    # stops varying.
    prev_llik = None
    for i in range(MAX_ITERATIONS):
        llik = LOG0
        for outputs in training_outputs:
            x = _baum_welch_one(
                N, M, outputs,
                lp_initial, lp_transition, lp_emission,
                lpseudo_initial, lpseudo_transition, lpseudo_emission,)
            llik += x
        if update_fn is not None:
            update_fn(i, llik)
        if prev_llik is not None and numpy.fabs(prev_llik-llik) < 0.1:
            break
        prev_llik = llik
    else:
        raise RuntimeError("HMM did not converge in %d iterations" \
                           % MAX_ITERATIONS)

    # Return everything back in normal space.
    return map(numpy.exp, (lp_initial, lp_transition, lp_emission))
    
def _baum_welch_one(N, M, outputs,
                    lp_initial, lp_transition, lp_emission,
                    lpseudo_initial, lpseudo_transition, lpseudo_emission):
    # Do one iteration of Baum-Welch based on a sequence of output.
    # NOTE: This will change the values of lp_initial, lp_transition,
    # and lp_emission in place.
    T = len(outputs)
    fmat = _forward(N, T, lp_initial, lp_transition, lp_emission, outputs)
    bmat = _backward(N, T, lp_transition, lp_emission, outputs)

    # Calculate the probability of traversing each arc for any given
    # transition.
    lp_arc = numpy.zeros((N, N, T))
    for t in range(T):
        k = outputs[t]
        lp_traverse = numpy.zeros((N, N)) # P going over one arc.
        for i in range(N):
            for j in range(N):
                # P(getting to this arc)
                # P(making this transition)
                # P(emitting this character)
                # P(going to the end)
                lp = fmat[i][t] + \
                     lp_transition[i][j] + \
                     lp_emission[i][k] + \
                     bmat[j][t+1]
                lp_traverse[i][j] = lp
        # Normalize the probability for this time step.
        lp_arc[:,:,t] = lp_traverse - _logsum(lp_traverse)


    # Sum of all the transitions out of state i at time t.
    lp_arcout_t = numpy.zeros((N, T))
    for t in range(T):
        for i in range(N):
            lp_arcout_t[i][t] = _logsum(lp_arc[i,:,t])
            
    # Sum of all the transitions out of state i.
    lp_arcout = numpy.zeros(N)
    for i in range(N):
        lp_arcout[i] = _logsum(lp_arcout_t[i,:])

    # UPDATE P_INITIAL.
    lp_initial = lp_arcout_t[:,0]
    if lpseudo_initial!=None:
        lp_initial = _logvecadd(lp_initial, lpseudo_initial)
        lp_initial = lp_initial - _logsum(lp_initial)
    
    # UPDATE P_TRANSITION.  p_transition[i][j] is the sum of all the
    # transitions from i to j, normalized by the sum of the
    # transitions out of i.
    for i in range(N):
        for j in range(N):
            lp_transition[i][j] = _logsum(lp_arc[i,j,:]) - lp_arcout[i]
        if lpseudo_transition!=None:
            lp_transition[i] = _logvecadd(lp_transition[i], lpseudo_transition)
            lp_transition[i] = lp_transition[i] - _logsum(lp_transition[i])
            
    # UPDATE P_EMISSION.  lp_emission[i][k] is the sum of all the
    # transitions out of i when k is observed, divided by the sum of
    # the transitions out of i.
    for i in range(N):
        ksum = numpy.zeros(M)+LOG0    # ksum[k] is the sum of all i with k.
        for t in range(T):
            k = outputs[t]
            for j in range(N):
                ksum[k] = _logadd(ksum[k], lp_arc[i,j,t])
        ksum = ksum - _logsum(ksum)      # Normalize
        if lpseudo_emission!=None:
            ksum = _logvecadd(ksum, lpseudo_emission[i])
            ksum = ksum - _logsum(ksum)  # Renormalize
        lp_emission[i,:] = ksum

    # Calculate the log likelihood of the output based on the forward
    # matrix.  Since the parameters of the HMM has changed, the log
    # likelihoods are going to be a step behind, and we might be doing
    # one extra iteration of training.  The alternative is to rerun
    # the _forward algorithm and calculate from the clean one, but
    # that may be more expensive than overshooting the training by one
    # step.
    return _logsum(fmat[:,T])

def _forward(N, T, lp_initial, lp_transition, lp_emission, outputs):
    # Implement the forward algorithm.  This actually calculates a
    # Nx(T+1) matrix, where the last column is the total probability
    # of the output.
    
    matrix = numpy.zeros((N, T+1))
    
    # Initialize the first column to be the initial values.
    matrix[:,0] = lp_initial
    for t in range(1, T+1):
        k = outputs[t-1]
        for j in range(N):
            # The probability of the state is the sum of the
            # transitions from all the states from time t-1.
            lprob = LOG0
            for i in range(N):
                lp = matrix[i][t-1] + \
                     lp_transition[i][j] + \
                     lp_emission[i][k]
                lprob = _logadd(lprob, lp)
            matrix[j][t] = lprob
    return matrix

def _backward(N, T, lp_transition, lp_emission, outputs):
    matrix = numpy.zeros((N, T+1))
    for t in range(T-1, -1, -1):
        k = outputs[t]
        for i in range(N):
            # The probability of the state is the sum of the
            # transitions from all the states from time t+1.
            lprob = LOG0
            for j in range(N):
                lp = matrix[j][t+1] + \
                     lp_transition[i][j] + \
                     lp_emission[i][k]
                lprob = _logadd(lprob, lp)
            matrix[i][t] = lprob
    return matrix

00354 def train_visible(states, alphabet, training_data,
                  pseudo_initial=None, pseudo_transition=None,
                  pseudo_emission=None):
    """train_visible(states, alphabet, training_data[, pseudo_initial]
    [, pseudo_transition][, pseudo_emission]) -> MarkovModel

    Train a visible MarkovModel using maximum likelihoood estimates
    for each of the parameters.  states is a list of strings that
    describe the names of each state.  alphabet is a list of objects
    that indicate the allowed outputs.  training_data is a list of
    (outputs, observed states) where outputs is a list of the emission
    from the alphabet, and observed states is a list of states from
    states.

    pseudo_initial, pseudo_transition, and pseudo_emission are
    optional parameters that you can use to assign pseudo-counts to
    different matrices.  They should be matrices of the appropriate
    size that contain numbers to add to each parameter matrix

    """
    N, M = len(states), len(alphabet)
    if pseudo_initial!=None:
        pseudo_initial = asarray(pseudo_initial)
        if pseudo_initial.shape != (N,):
            raise ValueError("pseudo_initial not shape len(states)")
    if pseudo_transition!=None:
        pseudo_transition = asarray(pseudo_transition)
        if pseudo_transition.shape != (N,N):
            raise ValueError("pseudo_transition not shape " + \
                             "len(states) X len(states)")
    if pseudo_emission!=None:
        pseudo_emission = asarray(pseudo_emission)
        if pseudo_emission.shape != (N,M):
            raise ValueError("pseudo_emission not shape " + \
                             "len(states) X len(alphabet)")
    
    # Training data is given as a list of members of the alphabet.
    # Replace those with indexes into the alphabet list for easier
    # computation.
    training_states, training_outputs = [], []
    states_indexes = itemindex(states)
    outputs_indexes = itemindex(alphabet)
    for toutputs, tstates in training_data:
        if len(tstates) != len(toutputs):
            raise ValueError("states and outputs not aligned")
        training_states.append([states_indexes[x] for x in tstates])
        training_outputs.append([outputs_indexes[x] for x in toutputs])

    x = _mle(N, M, training_outputs, training_states,
             pseudo_initial, pseudo_transition, pseudo_emission)
    p_initial, p_transition, p_emission = x

    return MarkovModel(states, alphabet, p_initial, p_transition, p_emission)

def _mle(N, M, training_outputs, training_states, pseudo_initial,
         pseudo_transition, pseudo_emission):
    # p_initial is the probability that a sequence of states starts
    # off with a particular one.
    p_initial = numpy.zeros(N)
    if pseudo_initial:
        p_initial = p_initial + pseudo_initial
    for states in training_states:
        p_initial[states[0]] += 1
    p_initial = _normalize(p_initial)
    
    # p_transition is the probability that a state leads to the next
    # one.  C(i,j)/C(i) where i and j are states.
    p_transition = numpy.zeros((N,N))
    if pseudo_transition:
        p_transition = p_transition + pseudo_transition
    for states in training_states:
        for n in range(len(states)-1):
            i, j = states[n], states[n+1]
            p_transition[i, j] += 1
    for i in range(len(p_transition)):
        p_transition[i,:] = p_transition[i,:] / sum(p_transition[i,:])

    # p_emission is the probability of an output given a state.
    # C(s,o)|C(s) where o is an output and s is a state.
    p_emission = numpy.zeros((N,M))
    if pseudo_emission:
        p_emission = p_emission + pseudo_emission
    p_emission = numpy.ones((N,M))
    for outputs, states in zip(training_outputs, training_states):
        for o, s in zip(outputs, states):
            p_emission[s, o] += 1
    for i in range(len(p_emission)):
        p_emission[i,:] = p_emission[i,:] / sum(p_emission[i,:])

    return p_initial, p_transition, p_emission
          
def _argmaxes(vector, allowance=None):
    return [numpy.argmax(vector)]

00448 def find_states(markov_model, output):
    """find_states(markov_model, output) -> list of (states, score)"""
    mm = markov_model
    N = len(mm.states)
    
    # _viterbi does calculations in log space.  Add a tiny bit to the
    # matrices so that the logs will not break.
    x = mm.p_initial + VERY_SMALL_NUMBER
    y = mm.p_transition + VERY_SMALL_NUMBER
    z = mm.p_emission + VERY_SMALL_NUMBER
    lp_initial, lp_transition, lp_emission = map(numpy.log, (x, y, z))
    # Change output into a list of indexes into the alphabet.
    indexes = itemindex(mm.alphabet)
    output = [indexes[x] for x in output]
    
    # Run the viterbi algorithm.
    results = _viterbi(N, lp_initial, lp_transition, lp_emission, output)

    for i in range(len(results)):
        states, score = results[i]
        results[i] = [mm.states[x] for x in states], numpy.exp(score)
    return results

def _viterbi(N, lp_initial, lp_transition, lp_emission, output):
    # The Viterbi algorithm finds the most likely set of states for a
    # given output.  Returns a list of states.

    T = len(output)
    # Store the backtrace in a NxT matrix.
    backtrace = []    # list of indexes of states in previous timestep.
    for i in range(N):
        backtrace.append([None] * T)

    # Store the best scores.
    scores = numpy.zeros((N, T))
    scores[:,0] = lp_initial + lp_emission[:,output[0]]
    for t in range(1, T):
        k = output[t]
        for j in range(N):
            # Find the most likely place it came from.
            i_scores = scores[:,t-1] + \
                       lp_transition[:,j] + \
                       lp_emission[j,k]
            indexes = _argmaxes(i_scores)
            scores[j,t] = i_scores[indexes[0]]
            backtrace[j][t] = indexes

    # Do the backtrace.  First, find a good place to start.  Then,
    # we'll follow the backtrace matrix to find the list of states.
    # In the event of ties, there may be multiple paths back through
    # the matrix, which implies a recursive solution.  We'll simulate
    # it by keeping our own stack.
    in_process = []    # list of (t, states, score)
    results = []       # return values.  list of (states, score)
    indexes = _argmaxes(scores[:,T-1])      # pick the first place
    for i in indexes:
        in_process.append((T-1, [i], scores[i][T-1]))
    while in_process:
        t, states, score = in_process.pop()
        if t == 0:
            results.append((states, score))
        else:
            indexes = backtrace[states[0]][t]
            for i in indexes:
                in_process.append((t-1, [i]+states, score))
    return results

def _normalize(matrix):
    # Make sure numbers add up to 1.0
    if len(matrix.shape) == 1:
        matrix = matrix / float(sum(matrix))
    elif len(matrix.shape) == 2:
        # Normalize by rows.
        for i in range(len(matrix)):
            matrix[i,:] = matrix[i,:] / sum(matrix[i,:])
    else:
        raise ValueError("I cannot handle matrixes of that shape")
    return matrix
    
def _uniform_norm(shape):
    matrix = numpy.ones(shape)
    return _normalize(matrix)

def _random_norm(shape):
    matrix = numpy.random.random(shape)
    return _normalize(matrix)

def _copy_and_check(matrix, desired_shape):
    # Copy the matrix.
    matrix = numpy.array(matrix, copy=1)
    # Check the dimensions.
    if matrix.shape != desired_shape:
        raise ValuError("Incorrect dimension")
    # Make sure it's normalized.
    if len(matrix.shape) == 1:
        if numpy.fabs(sum(matrix)-1.0) > 0.01:
            raise ValueError("matrix not normalized to 1.0")
    elif len(matrix.shape) == 2:
        for i in range(len(matrix)):
            if numpy.fabs(sum(matrix[i])-1.0) > 0.01:
                raise ValueError("matrix %d not normalized to 1.0" % i)
    else:
        raise ValueError("I don't handle matrices > 2 dimensions")
    return matrix

def _logadd(logx, logy):
    if logy - logx > 100:
        return logy
    elif logx - logy > 100:
        return logx
    minxy = min(logx, logy)
    return minxy + numpy.log(numpy.exp(logx-minxy) + numpy.exp(logy-minxy))

def _logsum(matrix):
    if len(matrix.shape) > 1:
        vec = numpy.reshape(matrix, (numpy.product(matrix.shape),))
    else:
        vec = matrix
    sum = LOG0
    for num in vec:
        sum = _logadd(sum, num)
    return sum

def _logvecadd(logvec1, logvec2):
    assert len(logvec1) == len(logvec2), "vectors aren't the same length"
    sumvec = numpy.zeros(len(logvec1))
    for i in range(len(logvec1)):
        sumvec[i] = _logadd(logvec1[i], logvec2[i])
    return sumvec

def _exp_logsum(numbers):
    sum = _logsum(numbers)
    return numpy.exp(sum)

try:
    import cMarkovModel
except ImportError, x:
    pass
else:
    import sys
    this_module = sys.modules[__name__]
    for name in cMarkovModel.__dict__.keys():
        if not name.startswith("__"):
            this_module.__dict__[name] = cMarkovModel.__dict__[name]

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