def sequence_updates(sequence, last_model, empty_arrays, array_ids, update_initial=True):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: usually you want to update all distributions,
including the initial state distribution. If update_initial=False,
the initial state distribution updates won't be made for this sequence.
We want this when the sequence is actually a non-initial fragment of
a longer sequence
"""
try:
trans, ems, trans_denom, ems_denom = empty_arrays
state_ids, em_ids = array_ids
# Compute the forwards with seq_prob=True
fwds,seq_logprob = last_model.normal_forward_probabilities(sequence, seq_prob=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.gamma_probabilities(sequence, forward=fwds)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence)
label_dom = last_model.label_dom
T = len(sequence)
for time in range(T):
for state in label_dom:
state_i = state_ids[state]
if time < T-1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
state_j = state_ids[next_state]
## Transition dist update ##
trans[state_i][state_j] += xi[time][state_i][state_j]
## Emission dist update ##
ems[state_ids[state]][em_ids[sequence[time]]] += \
gamma[time][state_i]
# Calculate the denominators by summing
trans_denom = array_sum(trans, axis=1)
ems_denom = array_sum(ems, axis=1)
# Wrap this all up in a tuple to return to the master
return (trans, ems, trans_denom, ems_denom, seq_logprob)
except KeyboardInterrupt:
return
def normal_backward_probabilities(self, sequence, array=False):
"""
@see: normal_forward_probabilities
(except that this doesn't return the logprob)
@type array: bool
@param array: if True, returns a numpy 2d array instead of a list of
dicts.
"""
T = len(sequence)
N = len(self.label_dom)
beta = numpy.zeros((T, N), numpy.float64)
scale = numpy.zeros(T, numpy.float64)
# Initialize with the probabilities of transitioning to the final state
for i,si in enumerate(self.label_dom):
beta[T-1, i] = self.transition_probability(None, si)
# Normalize
total = array_sum(beta[T-1, :])
beta[T-1,:] /= total
# Initialize
scale[T-1] = total
# Iterate backwards over the other timesteps
for t in range(T-2, -1, -1):
# To speed things up, calculate all the t+1 emission probabilities
# first, instead of calculating them all for every t state
em_probs = [
self.emission_probability(sequence[t+1], sj) \
for sj in self.label_dom]
for i,si in enumerate(self.label_dom):
# Multiply each next state's prob by the transition prob
# from this state to that and the emission prob in that next
# state
beta[t, i] = sum(
(beta[t+1, j] * self.transition_probability(sj, si) * \
em_probs[j] \
for j,sj in enumerate(self.label_dom)), 0.0)
# Normalize by dividing all values by the total probability
total = array_sum(beta[t,:])
beta[t,:] /= total
scale[t] = total
if not array:
# Convert this into a list of dicts
matrix = []
for t in range(T):
timestep = {}
for (i,label) in enumerate(self.label_dom):
timestep[label] = beta[t,i]
matrix.append(timestep)
return matrix,scale
else:
return beta,scale
def f_tt(theta, f, k0T, S, nCoeffs):
TnTT = S[0][:, 1:nCoeffs + 1]
tnTT = S[1][:, 1:nCoeffs + 1]
nfreq = len(f)
n = arange(1, nCoeffs + 1, 1)
n = ones((nfreq, 1)) * n
if theta == 0:
return array_sum((2 * n + 1) * (TnTT + tnTT), axis=1) / (1j * k0T)
elif theta == pi:
return array_sum((2 * n + 1) * (TnTT - tnTT), axis=1) / (1j * k0T)
def calcul_f_sc(theta, f, k0, S, nCoeffs):
n = arange(0, nCoeffs + 1, 1)
TnLL = S[:, 0:nCoeffs + 1]
if theta == 0:
return array_sum((2 * n + 1) * TnLL, axis=1) / (1j * k0)
elif theta == pi:
return array_sum((2 * n + 1) * ((-1)**n) * TnLL, axis=1) / (1j * k0)
nfreq = len(f)
Pncos0 = calculPn(cos(theta), nCoeffs)
Pncos0 = ones((nfreq, 1)) * transpose(Pncos0)
n = ones((nfreq, 1)) * n
fsc_theta_f = array_sum((2 * n + 1) * TnLL * Pncos0, axis=1)
fsc_theta_f = 1 / (1j * k0) * fsc_theta_f
return fsc_theta_f
def compute_decomposed_xi(self, sequence, forward=None, backward=None,
emission_matrix=None, transition_matrix=None):
from numpy import newaxis
if forward is None:
forward = self.normal_forward_probabilities(sequence, decomposed=True)
if backward is None:
backward = self.normal_backward_probabilities(sequence, decomposed=True)
# T is the number of timesteps
# N is the number of states
T = forward.shape[0]
C = len(self.chord_types)
# Create the empty array to fill
xi = numpy.zeros((T-1,12,12,C,12,12,C), numpy.float64)
# Precompute all the emission probabilities
if emission_matrix is None:
emission_matrix = self.get_small_emission_matrix(sequence)
# And transition probabilities: we'll need these many times over
if transition_matrix is None:
transition_matrix = self.get_small_transition_matrix(transpose=True)
# Do it without logs - much faster
for t in range(T-1):
total = 0.0
# Add axies to the forward probabilities to represent the next state
fwd_trans = forward[t,:,:,:, newaxis,newaxis,newaxis]
# Compute the xi values by multiplying the arrays together
xi[t] = transition_matrix * fwd_trans * backward[t+1] * \
emission_matrix[t+1]
# Normalize all the probabilities
# Sum all the probs for the timestep and divide them all by total
xi[t] /= array_sum(xi[t])
return xi
def test_convergence(self):
"""
Test that weights converge to the required value on toy data.
"""
input_batches = [
self.sc.parallelize(self.generateLogisticInput(
0, 1.5, 100, 42 + i)) for i in range(20)
]
input_stream = self.ssc.queueStream(input_batches)
models = []
slr = StreamingLogisticRegressionWithSGD(stepSize=0.2,
numIterations=25)
slr.setInitialWeights([0.0])
slr.trainOn(input_stream)
input_stream.foreachRDD(
lambda x: models.append(slr.latestModel().weights[0]))
t = time()
self.ssc.start()
self._ssc_wait(t, 15.0, 0.01)
t_models = array(models)
diff = t_models[1:] - t_models[:-1]
# Test that weights improve with a small tolerance,
self.assertTrue(all(diff >= -0.1))
self.assertTrue(array_sum(diff > 0) > 1)
def test_convergence(self):
"""
Test that weights converge to the required value on toy data.
"""
input_batches = [
self.sc.parallelize(self.generateLogisticInput(
0, 1.5, 100, 42 + i)) for i in range(20)
]
input_stream = self.ssc.queueStream(input_batches)
models = []
slr = StreamingLogisticRegressionWithSGD(stepSize=0.2,
numIterations=25)
slr.setInitialWeights([0.0])
slr.trainOn(input_stream)
input_stream.foreachRDD(
lambda x: models.append(slr.latestModel().weights[0]))
self.ssc.start()
def condition():
self.assertEqual(len(models), len(input_batches))
return True
# We want all batches to finish for this test.
eventually(condition, 60.0, catch_assertions=True)
t_models = array(models)
diff = t_models[1:] - t_models[:-1]
# Test that weights improve with a small tolerance
self.assertTrue(all(diff >= -0.1))
self.assertTrue(array_sum(diff > 0) > 1)
def test_accuracy_for_single_center(self):
"""Test that parameters obtained are correct for a single center."""
centers, batches = self.streamingKMeansDataGenerator(batches=5,
numPoints=5,
k=1,
d=5,
r=0.1,
seed=0)
stkm = StreamingKMeans(1)
stkm.setInitialCenters([[0., 0., 0., 0., 0.]], [0.])
input_stream = self.ssc.queueStream(
[self.sc.parallelize(batch, 1) for batch in batches])
stkm.trainOn(input_stream)
self.ssc.start()
def condition():
self.assertEqual(stkm.latestModel().clusterWeights, [25.0])
return True
eventually(condition, catch_assertions=True)
realCenters = array_sum(array(centers), axis=0)
for i in range(5):
modelCenters = stkm.latestModel().centers[0][i]
self.assertAlmostEqual(centers[0][i], modelCenters, 1)
self.assertAlmostEqual(realCenters[i], modelCenters, 1)
def compute_gamma(self, sequence, forward=None, backward=None):
"""
Computes the gamma matrix used in Baum-Welch. This is the matrix
of state occupation probabilities for each timestep. It is computed
from the forward and backward matrices.
These can be passed in as
arguments to avoid recomputing if you need to reuse them, but will
be computed from the model if not given. They are assumed to be
the matrices computed by L{normal_forward_probabilities} and
L{normal_backward_probabilities} (i.e. normalized, non-log
probabilities).
"""
if forward is None:
forward = self.normal_forward_probabilities(sequence, array=True)[0]
if backward is None:
backward = self.normal_backward_probabilities(sequence, array=True)[0]
# T is the number of timesteps
# N is the number of states
T,N = forward.shape
# Multiply forward and backward elementwise to get unnormalised gamma
gamma = forward * backward
# Sum the values in each timestep to get the normalizing denominator
denominators = array_sum(gamma, axis=1)
# Divide all the values in each timestep by each denominator
gamma = (gamma.transpose() / denominators).transpose()
return gamma
def test_convergence(self):
"""
Test that weights converge to the required value on toy data.
"""
input_batches = [
self.sc.parallelize(self.generateLogisticInput(0, 1.5, 100, 42 + i))
for i in range(20)]
input_stream = self.ssc.queueStream(input_batches)
models = []
slr = StreamingLogisticRegressionWithSGD(
stepSize=0.2, numIterations=25)
slr.setInitialWeights([0.0])
slr.trainOn(input_stream)
input_stream.foreachRDD(
lambda x: models.append(slr.latestModel().weights[0]))
t = time()
self.ssc.start()
self._ssc_wait(t, 15.0, 0.01)
t_models = array(models)
diff = t_models[1:] - t_models[:-1]
# Test that weights improve with a small tolerance,
self.assertTrue(all(diff >= -0.1))
self.assertTrue(array_sum(diff > 0) > 1)
def compute_gamma(self, sequence, forward=None, backward=None):
"""
Computes the gamma matrix used in Baum-Welch. This is the matrix
of state occupation probabilities for each timestep. It is computed
from the forward and backward matrices.
These can be passed in as
arguments to avoid recomputing if you need to reuse them, but will
be computed from the model if not given. They are assumed to be
the matrices computed by L{normal_forward_probabilities} and
L{normal_backward_probabilities} (i.e. normalized, non-log
probabilities).
"""
if forward is None:
forward = self.normal_forward_probabilities(sequence)
if backward is None:
backward = self.normal_backward_probabilities(sequence)
# T is the number of timesteps
# N is the number of states
T,N = forward.shape
gamma = zeros((T,N), float64)
for t in range(T):
for i in range(N):
gamma[t][i] = forward[t][i]*backward[t][i]
denominator = array_sum(gamma[t])
# Normalize
for i in range(N):
gamma[t][i] /= denominator
return gamma
def compute_gamma(self, sequence, forward=None, backward=None):
"""
Computes the gamma matrix used in Baum-Welch. This is the matrix
of state occupation probabilities for each timestep. It is computed
from the forward and backward matrices.
These can be passed in as
arguments to avoid recomputing if you need to reuse them, but will
be computed from the model if not given. They are assumed to be
the matrices computed by L{normal_forward_probabilities} and
L{normal_backward_probabilities} (i.e. normalized, non-log
probabilities).
"""
if forward is None:
forward = self.normal_forward_probabilities(sequence)
if backward is None:
backward = self.normal_backward_probabilities(sequence)
# T is the number of timesteps
# N is the number of states
T, N = forward.shape
gamma = zeros((T, N), float64)
for t in range(T):
for i in range(N):
gamma[t][i] = forward[t][i] * backward[t][i]
denominator = array_sum(gamma[t])
# Normalize
for i in range(N):
gamma[t][i] /= denominator
return gamma
def test_convergence(self):
"""
Test that weights converge to the required value on toy data.
"""
input_batches = [
self.sc.parallelize(self.generateLogisticInput(0, 1.5, 100, 42 + i))
for i in range(20)]
input_stream = self.ssc.queueStream(input_batches)
models = []
slr = StreamingLogisticRegressionWithSGD(
stepSize=0.2, numIterations=25)
slr.setInitialWeights([0.0])
slr.trainOn(input_stream)
input_stream.foreachRDD(
lambda x: models.append(slr.latestModel().weights[0]))
self.ssc.start()
def condition():
self.assertEqual(len(models), len(input_batches))
return True
# We want all batches to finish for this test.
self._eventually(condition, 60.0, catch_assertions=True)
t_models = array(models)
diff = t_models[1:] - t_models[:-1]
# Test that weights improve with a small tolerance
self.assertTrue(all(diff >= -0.1))
self.assertTrue(array_sum(diff > 0) > 1)
def corr_from_index(*index):
"""Correlation of index with zero.
Turns a correlation function in terms of index distance
into one in terms of indices on a periodic domain.
Parameters
----------
index: tuple of int
Returns
-------
float[-1, 1]
The correlation of the given index with the origin.
See Also
--------
DistanceCorrelationFunction.correlation_from_index
"""
comp2_1 = square(index)
# Components of distance to shifted origin
comp2_2 = square(broadcastable_shape - index)
# use the smaller components to get the distance to the
# closest of the shifted origins
comp2 = fmin(comp2_1, comp2_2)
return corr_func(sqrt(array_sum(comp2, axis=0)))
def get_probabilities(histogram):
probabilites = []
count = array_sum(histogram[0])
for i in range(0, len(histogram[0])):
times = histogram[0][i]
tempo = histogram[1][i] + 0.5
if times != 0.0:
probabilites.append([tempo, times / count])
return probabilites
def assemble_stiffness_sum(num_dof, edm, elements, u):
"""
Assembles "summed" stiffness matrix (each row is summed, taking absolute value)
"""
K_sum = zeros(num_dof)
for elem_id, elem in enumerate(elements):
u_loc = empty(elem.num_dof)
gather_element_vector(edm[elem_id], u, u_loc)
K_elem = array_sum(abs(elem.calc_linear_stiffness([], u_loc)), axis=1)
for local_dof, dof in enumerate(edm[elem_id]):
K_sum[dof] += K_elem[local_dof]
return K_sum
def _sequence_updates_uni(sequence, last_model, label_dom, state_ids, \
beat_ids, d_ids, d_func):
"""Same as L{_sequence_updates}, modified for unigram models. """
num_beats = len(beat_ids)
num_ds = len(d_ids)
num_ktrans = 12
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
ems_local = zeros((num_beats, num_ds), float64)
# Compute the forward and backward probabilities
alpha, scale, seq_logprob = last_model.normal_forward_probabilities(
sequence)
beta, scale = last_model.normal_backward_probabilities(sequence)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence, alpha, beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, alpha, beta)
T = len(sequence)
for time in range(T):
for state in label_dom:
tonic, mode, chord = state
state_i = state_ids[state]
# We don't update the transition distribution here, because it's fixed
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc, beat in sequence[time]:
beat_i = beat_ids[beat]
d = d_func(pc, state)
d_i = d_ids[d]
ems_local[beat_i][d_i] += gamma[time][state_i]
# Calculate the denominators
ems_denom_local = array_sum(ems_local, axis=1)
# Wrap this all up in a tuple to return to the master
return (ems_local, ems_denom_local, seq_logprob)
def _sequence_updates_uni(sequence, last_model, label_dom, state_ids, \
beat_ids, d_ids, d_func):
"""Same as L{_sequence_updates}, modified for unigram models. """
num_beats = len(beat_ids)
num_ds = len(d_ids)
num_ktrans = 12
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
ems_local = zeros((num_beats,num_ds), float64)
# Compute the forward and backward probabilities
alpha,scale,seq_logprob = last_model.normal_forward_probabilities(sequence)
beta,scale = last_model.normal_backward_probabilities(sequence)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence, alpha, beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, alpha, beta)
T = len(sequence)
for time in range(T):
for state in label_dom:
tonic,mode,chord = state
state_i = state_ids[state]
# We don't update the transition distribution here, because it's fixed
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc,beat in sequence[time]:
beat_i = beat_ids[beat]
d = d_func(pc, state)
d_i = d_ids[d]
ems_local[beat_i][d_i] += gamma[time][state_i]
# Calculate the denominators
ems_denom_local = array_sum(ems_local, axis=1)
# Wrap this all up in a tuple to return to the master
return (ems_local, ems_denom_local, seq_logprob)
def test_accuracy_for_single_center(self):
"""Test that parameters obtained are correct for a single center."""
centers, batches = self.streamingKMeansDataGenerator(
batches=5, numPoints=5, k=1, d=5, r=0.1, seed=0)
stkm = StreamingKMeans(1)
stkm.setInitialCenters([[0., 0., 0., 0., 0.]], [0.])
input_stream = self.ssc.queueStream(
[self.sc.parallelize(batch, 1) for batch in batches])
stkm.trainOn(input_stream)
t = time()
self.ssc.start()
self._ssc_wait(t, 10.0, 0.01)
self.assertEquals(stkm.latestModel().clusterWeights, [25.0])
realCenters = array_sum(array(centers), axis=0)
for i in range(5):
modelCenters = stkm.latestModel().centers[0][i]
self.assertAlmostEqual(centers[0][i], modelCenters, 1)
self.assertAlmostEqual(realCenters[i], modelCenters, 1)
def _sequence_updates(sequence, last_model, label_dom, state_ids, mode_ids, \
chord_ids, beat_ids, d_ids, d_func):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
"""
num_chords = len(chord_ids)
num_beats = len(beat_ids)
num_modes = len(mode_ids)
num_ds = len(d_ids)
num_ktrans = 12
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
ctrans_local = zeros((num_chords, num_chords), float64)
ems_local = zeros((num_beats, num_ds), float64)
ktrans_local = zeros((num_modes, num_ktrans, num_modes), float64)
uni_chords_local = zeros(num_chords, float64)
# Compute the forward and backward probabilities
alpha, scale, seq_logprob = last_model.normal_forward_probabilities(
sequence)
beta, scale = last_model.normal_backward_probabilities(sequence)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence, alpha, beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, alpha, beta)
T = len(sequence)
for time in range(T):
for state in label_dom:
tonic, mode, chord = state
state_i = state_ids[state]
mode_i = mode_ids[mode]
if time < T - 1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
ntonic, nmode, nchord = next_state
state_j = state_ids[next_state]
mode_j = mode_ids[nmode]
## Key transition dist update ##
tonic_change = (ntonic - tonic) % 12
ktrans_local[mode_i][tonic_change][mode_j] += \
xi[time][state_i][state_j]
## Chord transition dist update ##
chord_i, chord_j = chord_ids[chord], chord_ids[nchord]
if tonic == ntonic and mode == nmode:
# Add to chord transition dist for this chord pair
ctrans_local[chord_i][chord_j] += xi[time][state_i][
state_j]
else:
uni_chords_local[chord_j] += xi[time][state_i][state_j]
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc, beat in sequence[time]:
beat_i = beat_ids[beat]
d = d_func(pc, state)
d_i = d_ids[d]
ems_local[beat_i][d_i] += gamma[time][state_i]
# Calculate the denominators
ctrans_denom_local = array_sum(ctrans_local, axis=1)
ems_denom_local = array_sum(ems_local, axis=1)
ktrans_denom_local = array_sum(array_sum(ktrans_local, axis=2), axis=1)
uni_chords_denom_local = array_sum(uni_chords_local)
# Wrap this all up in a tuple to return to the master
return (ktrans_local, ctrans_local, ems_local, \
uni_chords_local, \
ktrans_denom_local, ctrans_denom_local, \
ems_denom_local, uni_chords_denom_local, \
seq_logprob)
def sequence_updates(sequence,
last_model,
empty_arrays,
array_ids,
update_initial=True):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: usually you want to update all distributions,
including the initial state distribution. If update_initial=False,
the initial state distribution updates won't be made for this sequence.
We want this when the sequence is actually a non-initial fragment of
a longer sequence
"""
try:
trans, ems, trans_denom, ems_denom = empty_arrays
state_ids, em_ids = array_ids
# Compute the forwards with seq_prob=True
fwds, seq_logprob = last_model.normal_forward_probabilities(
sequence, seq_prob=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.gamma_probabilities(sequence, forward=fwds)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence)
label_dom = last_model.label_dom
T = len(sequence)
for time in range(T):
for state in label_dom:
state_i = state_ids[state]
if time < T - 1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
state_j = state_ids[next_state]
## Transition dist update ##
trans[state_i][state_j] += xi[time][state_i][state_j]
## Emission dist update ##
ems[state_ids[state]][em_ids[sequence[time]]] += \
gamma[time][state_i]
# Calculate the denominators by summing
trans_denom = array_sum(trans, axis=1)
ems_denom = array_sum(ems, axis=1)
# Wrap this all up in a tuple to return to the master
return (trans, ems, trans_denom, ems_denom, seq_logprob)
except KeyboardInterrupt:
return
def sequence_updates(sequence, last_model, empty_arrays, array_ids, update_initial=True):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: if update_initial=False,
the initial state distribution updates won't be made for this sequence
"""
try:
(
initial_keys,
initial_chords,
key_trans,
chord_trans,
ems,
initial_keys_denom,
initial_chords_denom,
key_trans_denom,
chord_trans_denom,
ems_denom,
) = empty_arrays
chord_ids, chord_type_ids = array_ids
# Compute the forwards with seq_prob=True
fwds, seq_logprob = last_model.normal_forward_probabilities(sequence, seq_prob=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.gamma_probabilities(sequence, forward=fwds)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence)
label_dom = last_model.label_dom
# Enumerate the label dom
state_ids = dict([(state, id) for (id, state) in enumerate(label_dom)])
T = len(sequence)
for time in range(T):
for state in label_dom:
keyi, rooti, labeli = state
state_i = state_ids[state]
chord_i = chord_ids[((rooti - keyi) % 12, labeli)]
if time == 0:
# Update initial distributions
initial_keys[keyi] += gamma[time][state_i]
initial_chords[chord_i] += gamma[time][state_i]
if time == T - 1:
# Last timestep
# Update the transition dists for transitions to final state
chord_trans[chord_i][-1] += gamma[time][state_i]
else:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
keyj, rootj, labelj = next_state
state_j = state_ids[next_state]
chord_j = chord_ids[((rootj - keyj) % 12, labelj)]
key_change = (keyj - keyi) % 12
## Transition dist updates ##
key_trans[key_change] += xi[time][state_i][state_j]
chord_trans[chord_i][chord_j] += xi[time][state_i][state_j]
## Emission dist update ##
for note in sequence[time]:
pc = (note - rooti) % 12
ems[chord_type_ids[labeli]][pc] += gamma[time][state_i]
# Calculate the denominators by summing
initial_keys_denom[0] = array_sum(initial_keys)
initial_chords_denom[0] = array_sum(initial_chords)
key_trans_denom[0] = array_sum(key_trans)
chord_trans_denom = array_sum(chord_trans, axis=1)
ems_denom = array_sum(ems, axis=1)
# Wrap this all up in a tuple to return to the master
return (
initial_keys,
initial_chords,
key_trans,
chord_trans,
ems,
initial_keys_denom,
initial_chords_denom,
key_trans_denom,
chord_trans_denom,
ems_denom,
seq_logprob,
)
except KeyboardInterrupt:
return
def normal_forward_probabilities(self, sequence, array=False):
"""If you want the normalized matrix of forward probabilities, it's
ok to use normal (non-log) probabilities and these can be computed
more quickly, since you don't need to sum logs (which is time
consuming).
Returns the matrix, and also the vector of values that each timestep
was divided by to normalize (i.e. total probability of each timestep
over all states).
Also returns the total log probability of the sequence.
@type array: bool
@param array: if True, returns a numpy 2d array instead of a list of
dicts.
@return: (matrix,normalizing vector,log prob)
"""
T = len(sequence)
N = len(self.label_dom)
alpha = numpy.zeros((T, N), numpy.float64)
scale = numpy.zeros(T, numpy.float64)
# Prepare the first column of the matrix: probs of all states in the
# first timestep
for i,state in enumerate(self.label_dom):
alpha[0,i] = self.transition_probability(state, None) * \
self.emission_probability(sequence[0], state)
# Normalize by dividing all values by the total probability
total = array_sum(alpha[0,:])
alpha[0,:] /= total
scale[0] = total
# Iterate over the other timesteps
for t in range(1, T):
for j,sj in enumerate(self.label_dom):
# Multiply each previous state's prob by the transition prob
# to this state and sum them all together
prob = sum(
(alpha[t-1, i] * self.transition_probability(sj, si) \
for i,si in enumerate(self.label_dom)), 0.0)
# Also multiply this by the emission probability
alpha[t, j] = prob * \
self.emission_probability(sequence[t], sj)
# Normalize by dividing all values by the total probability
total = array_sum(alpha[t,:])
alpha[t,:] /= total
scale[t] = total
# Multiply together the probability of each timestep to get the whole
# probability of the sequence
# This gets the same result as if we did:
# alpha = model.forward_log_probabilities(sequence, normalize=False, array=True)
# log_prob = sum_logs(alpha[T-1,:])
log_prob = sum((logprob(total) for total in scale), 0.0)
if not array:
# Convert this into a list of dicts
matrix = []
for t in range(T):
timestep = {}
for (i,label) in enumerate(self.label_dom):
timestep[label] = alpha[t,i]
matrix.append(timestep)
return matrix,scale,log_prob
else:
return alpha,scale,log_prob
def _sequence_updates(sequence, last_model, label_dom, state_ids, mode_ids, \
chord_ids, beat_ids, d_ids, d_func):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
"""
num_chords = len(chord_ids)
num_beats = len(beat_ids)
num_modes = len(mode_ids)
num_ds = len(d_ids)
num_ktrans = 12
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
ctrans_local = zeros((num_chords,num_chords), float64)
ems_local = zeros((num_beats,num_ds), float64)
ktrans_local = zeros((num_modes,num_ktrans,num_modes), float64)
uni_chords_local = zeros(num_chords, float64)
# Compute the forward and backward probabilities
alpha,scale,seq_logprob = last_model.normal_forward_probabilities(sequence)
beta,scale = last_model.normal_backward_probabilities(sequence)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence, alpha, beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, alpha, beta)
T = len(sequence)
for time in range(T):
for state in label_dom:
tonic,mode,chord = state
state_i = state_ids[state]
mode_i = mode_ids[mode]
if time < T-1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
ntonic,nmode,nchord = next_state
state_j = state_ids[next_state]
mode_j = mode_ids[nmode]
## Key transition dist update ##
tonic_change = (ntonic - tonic) % 12
ktrans_local[mode_i][tonic_change][mode_j] += \
xi[time][state_i][state_j]
## Chord transition dist update ##
chord_i, chord_j = chord_ids[chord], chord_ids[nchord]
if tonic == ntonic and mode == nmode:
# Add to chord transition dist for this chord pair
ctrans_local[chord_i][chord_j] += xi[time][state_i][state_j]
else:
uni_chords_local[chord_j] += xi[time][state_i][state_j]
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc,beat in sequence[time]:
beat_i = beat_ids[beat]
d = d_func(pc, state)
d_i = d_ids[d]
ems_local[beat_i][d_i] += gamma[time][state_i]
# Calculate the denominators
ctrans_denom_local = array_sum(ctrans_local, axis=1)
ems_denom_local = array_sum(ems_local, axis=1)
ktrans_denom_local = array_sum(array_sum(ktrans_local, axis=2), axis=1)
uni_chords_denom_local = array_sum(uni_chords_local)
# Wrap this all up in a tuple to return to the master
return (ktrans_local, ctrans_local, ems_local, \
uni_chords_local, \
ktrans_denom_local, ctrans_denom_local, \
ems_denom_local, uni_chords_denom_local, \
seq_logprob)
def compute_xi(self, sequence, forward=None, backward=None,
emission_matrix=None, transition_matrix=None,
use_logs=False):
"""
Computes the xi matrix used by Baum-Welch. It is the matrix of joint
probabilities of occupation of pairs of consecutive states:
P(i_t, j_{t+1} | O).
As with L{compute_gamma} forward and backward matrices can optionally
be passed in to avoid recomputing.
@type use_logs: bool
@param use_logs: by default, this function does not use logs in its
calculations. This can lead to underflow if your forward/backward
matrices have sufficiently low values. If C{use_logs=True}, logs
will be used internally (though the returned values are
exponentiated again). This makes the function an order of magnitude
slower.
"""
if forward is None:
forward = self.normal_forward_probabilities(sequence)
if backward is None:
backward = self.normal_backward_probabilities(sequence)
# T is the number of timesteps
# N is the number of states
T,N = forward.shape
# Create the empty array to fill
xi = numpy.zeros((T-1,N,N), numpy.float64)
# Precompute all the emission probabilities
if emission_matrix is None:
emission_matrix = self.get_emission_matrix(sequence)
# And transition probabilities: we'll need these many times over
if transition_matrix is None:
transition_matrix = self.get_transition_matrix()
if not use_logs:
# Do it without logs - much faster
for t in range(T-1):
total = 0.0
# Transpose the forward probabilities so that we multiply them
# along the vertical axis
fwd_trans = forward[t,:, numpy.newaxis]
# Compute the xi values by multiplying the arrays together
xi[t] = transition_matrix.T * fwd_trans * backward[t+1] * \
emission_matrix[t+1]
# Normalize all the probabilities
# Sum all the probs for the timestep and divide them all by total
total = array_sum(xi[t])
xi[t] /= total
else:
# Take logs of all the matrices we need
emission_matrix = numpy.log2(emission_matrix)
transition_matrix = numpy.log2(transition_matrix)
forward = numpy.log2(forward)
backward = numpy.log2(backward)
for t in range(T-1):
total = 0.0
fwd_trans = forward[t,:, numpy.newaxis]
xi[t] = transition_matrix.T + fwd_trans + backward[t+1] + \
emission_matrix[t+1]
# This takes a (relatively) long time
total = numpy.logaddexp2.reduce(xi[t])
xi[t] -= total
# Exponentiate all the probabilities again
# This also takes a while
xi = numpy.exp2(xi)
return xi
def sequence_updates(sequence,
last_model,
empty_arrays,
array_ids,
update_initial=True):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: if update_initial=False,
the initial state distribution updates won't be made for this sequence
"""
try:
(initial_keys, initial_chords, key_trans, chord_trans, ems,
initial_keys_denom, initial_chords_denom, key_trans_denom,
chord_trans_denom, ems_denom) = empty_arrays
chord_ids, chord_type_ids = array_ids
# Compute the forwards with seq_prob=True
fwds, seq_logprob = last_model.normal_forward_probabilities(
sequence, seq_prob=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.gamma_probabilities(sequence, forward=fwds)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence)
label_dom = last_model.label_dom
# Enumerate the label dom
state_ids = dict([(state, id) for (id, state) in enumerate(label_dom)])
T = len(sequence)
for time in range(T):
for state in label_dom:
keyi, rooti, labeli = state
state_i = state_ids[state]
chord_i = chord_ids[((rooti - keyi) % 12, labeli)]
if time == 0:
# Update initial distributions
initial_keys[keyi] += gamma[time][state_i]
initial_chords[chord_i] += gamma[time][state_i]
if time == T - 1:
# Last timestep
# Update the transition dists for transitions to final state
chord_trans[chord_i][-1] += gamma[time][state_i]
else:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
keyj, rootj, labelj = next_state
state_j = state_ids[next_state]
chord_j = chord_ids[((rootj - keyj) % 12, labelj)]
key_change = (keyj - keyi) % 12
## Transition dist updates ##
key_trans[key_change] += xi[time][state_i][state_j]
chord_trans[chord_i][chord_j] += xi[time][state_i][
state_j]
## Emission dist update ##
for note in sequence[time]:
pc = (note - rooti) % 12
ems[chord_type_ids[labeli]][pc] += gamma[time][state_i]
# Calculate the denominators by summing
initial_keys_denom[0] = array_sum(initial_keys)
initial_chords_denom[0] = array_sum(initial_chords)
key_trans_denom[0] = array_sum(key_trans)
chord_trans_denom = array_sum(chord_trans, axis=1)
ems_denom = array_sum(ems, axis=1)
# Wrap this all up in a tuple to return to the master
return (initial_keys, initial_chords, key_trans, chord_trans, ems,
initial_keys_denom, initial_chords_denom, key_trans_denom,
chord_trans_denom, ems_denom, seq_logprob)
except KeyboardInterrupt:
return
def _sequence_updates(sequence,
last_model,
label_dom,
schema_ids,
emission_cond_ids,
update_initial=True,
catch_interrupt=False):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: usually you want to update all distributions,
including the initial state distribution. If update_initial=False,
the initial state distribution updates won't be made for this sequence.
We want this when the sequence is actually a non-initial fragment of
a longer sequence
@type catch_interrupt: bool
@param catch_interrupt: catch KeyboardInterrupt exceptions and return
None. This is useful behaviour when calling this in a process pool,
since it allows the parent process to handle the interrupt, but should
be set to False (default) if calling directly.
"""
try:
# Get the sizes we'll need for the matrix
num_schemata = len(last_model.schemata)
num_root_changes = 12
num_chord_classes = len(last_model.chord_classes)
num_emission_conds = len(emission_cond_ids)
num_emissions = 12
T = len(sequence)
state_ids = dict([(state,id) for (id,state) in \
enumerate(last_model.label_dom)])
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
schema_trans = zeros((num_schemata, num_schemata + 1), float64)
root_trans = zeros((num_schemata, num_schemata, num_root_changes),
float64)
ems = zeros((num_emission_conds, num_emissions), float64)
sinit = zeros(num_schemata, float64)
# Compute the forward and backward probabilities
# These are normalized, but that makes no difference to the outcome of
# compute_gamma and compute_xi
alpha, scale, seq_logprob = last_model.normal_forward_probabilities(
sequence, array=True)
beta, scale = last_model.normal_backward_probabilities(sequence,
array=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence,
forward=alpha,
backward=beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, forward=alpha, backward=beta)
# Update the initial state distribution if requested
if update_initial:
for state in label_dom:
schema, root, chord_class = state
schema_i = schema_ids[schema]
# Add this contribution to the sum of the states with this schema
sinit[schema_i] += gamma[0][state_ids[state]]
for time in range(T):
for state in label_dom:
schema, root, chord_class = state
schema_i = schema_ids[schema]
state_i = state_ids[state]
if time < T - 1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
next_schema, next_root, next_chord_class = next_state
schema_j = schema_ids[next_schema]
state_j = state_ids[next_state]
## Transition dist update ##
root_change = (next_root - root) % 12
schema_trans[schema_i][schema_j] += \
xi[time][state_i][state_j]
root_trans[schema_i][schema_j][root_change] += \
xi[time][state_i][state_j]
else:
# Final state: update the probs of transitioning to end
schema_trans[schema_i][num_schemata] += gamma[T -
1][state_i]
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc, beat in sequence[time]:
# Take the pitch class relative to the root
rel_pc = (pc - root) % 12
ems[emission_cond_ids[(chord_class,beat)]][rel_pc] += \
gamma[time][state_i]
# Calculate the denominators
schema_trans_denom = array_sum(schema_trans, axis=1)
root_trans_denom = array_sum(root_trans, axis=2)
ems_denom = array_sum(ems, axis=1)
# This should come to 1.0
sinit_denom = array_sum(sinit)
# Wrap this all up in a tuple to return to the master
return (schema_trans, root_trans, ems, sinit, \
schema_trans_denom, root_trans_denom, ems_denom, sinit_denom, \
seq_logprob)
except KeyboardInterrupt:
if catch_interrupt:
return
else:
raise
def _sequence_updates(sequence, last_model, label_dom, schema_ids,
emission_cond_ids, update_initial=True, catch_interrupt=False):
"""
Evaluates the forward/backward probability matrices for a
single sequence under the model that came from the previous
iteration and returns matrices that contain the updates
to be made to the distributions during this iteration.
This is wrapped up in a function so it can be run in
parallel for each sequence. Once all sequences have been
evaluated, the results are combined and model updated.
@type update_initial: bool
@param update_initial: usually you want to update all distributions,
including the initial state distribution. If update_initial=False,
the initial state distribution updates won't be made for this sequence.
We want this when the sequence is actually a non-initial fragment of
a longer sequence
@type catch_interrupt: bool
@param catch_interrupt: catch KeyboardInterrupt exceptions and return
None. This is useful behaviour when calling this in a process pool,
since it allows the parent process to handle the interrupt, but should
be set to False (default) if calling directly.
"""
try:
# Get the sizes we'll need for the matrix
num_schemata = len(last_model.schemata)
num_root_changes = 12
num_chord_classes = len(last_model.chord_classes)
num_emission_conds = len(emission_cond_ids)
num_emissions = 12
T = len(sequence)
state_ids = dict([(state,id) for (id,state) in \
enumerate(last_model.label_dom)])
# Local versions of the matrices store the accumulated values
# for just this sequence (so we can normalize before adding
# to the global matrices)
# The numerators
schema_trans = zeros((num_schemata,num_schemata+1), float64)
root_trans = zeros((num_schemata,num_schemata,num_root_changes), float64)
ems = zeros((num_emission_conds,num_emissions), float64)
sinit = zeros(num_schemata, float64)
# Compute the forward and backward probabilities
# These are normalized, but that makes no difference to the outcome of
# compute_gamma and compute_xi
alpha,scale,seq_logprob = last_model.normal_forward_probabilities(sequence, array=True)
beta,scale = last_model.normal_backward_probabilities(sequence, array=True)
# gamma contains the state occupation probability for each state at each
# timestep
gamma = last_model.compute_gamma(sequence, forward=alpha, backward=beta)
# xi contains the probability of every state transition at every timestep
xi = last_model.compute_xi(sequence, forward=alpha, backward=beta)
# Update the initial state distribution if requested
if update_initial:
for state in label_dom:
schema, root, chord_class = state
schema_i = schema_ids[schema]
# Add this contribution to the sum of the states with this schema
sinit[schema_i] += gamma[0][state_ids[state]]
for time in range(T):
for state in label_dom:
schema, root, chord_class = state
schema_i = schema_ids[schema]
state_i = state_ids[state]
if time < T-1:
# Go through all possible pairs of states to update the
# transition distributions
for next_state in label_dom:
next_schema, next_root, next_chord_class = next_state
schema_j = schema_ids[next_schema]
state_j = state_ids[next_state]
## Transition dist update ##
root_change = (next_root - root) % 12
schema_trans[schema_i][schema_j] += \
xi[time][state_i][state_j]
root_trans[schema_i][schema_j][root_change] += \
xi[time][state_i][state_j]
else:
# Final state: update the probs of transitioning to end
schema_trans[schema_i][num_schemata] += gamma[T-1][state_i]
## Emission dist update ##
# Add the state occupation probability to the emission numerator
# for every note
for pc,beat in sequence[time]:
# Take the pitch class relative to the root
rel_pc = (pc - root) % 12
ems[emission_cond_ids[(chord_class,beat)]][rel_pc] += \
gamma[time][state_i]
# Calculate the denominators
schema_trans_denom = array_sum(schema_trans, axis=1)
root_trans_denom = array_sum(root_trans, axis=2)
ems_denom = array_sum(ems, axis=1)
# This should come to 1.0
sinit_denom = array_sum(sinit)
# Wrap this all up in a tuple to return to the master
return (schema_trans, root_trans, ems, sinit, \
schema_trans_denom, root_trans_denom, ems_denom, sinit_denom, \
seq_logprob)
except KeyboardInterrupt:
if catch_interrupt:
return
else:
raise
def compute_xi(self,
sequence,
forward=None,
backward=None,
emission_matrix=None,
transition_matrix=None,
use_logs=False):
"""
Computes the xi matrix used by Baum-Welch. It is the matrix of joint
probabilities of occupation of pairs of consecutive states:
P(i_t, j_{t+1} | O).
As with L{compute_gamma} forward and backward matrices can optionally
be passed in to avoid recomputing.
@type use_logs: bool
@param use_logs: by default, this function does not use logs in its
calculations. This can lead to underflow if your forward/backward
matrices have sufficiently low values. If C{use_logs=True}, logs
will be used internally (though the returned values are
exponentiated again). This makes the function an order of magnitude
slower.
"""
if forward is None:
forward = self.normal_forward_probabilities(sequence)
if backward is None:
backward = self.normal_backward_probabilities(sequence)
# T is the number of timesteps
# N is the number of states
T, N = forward.shape
# Create the empty array to fill
xi = numpy.zeros((T - 1, N, N), numpy.float64)
# Precompute all the emission probabilities
if emission_matrix is None:
emission_matrix = self.get_emission_matrix(sequence)
# And transition probabilities: we'll need these many times over
if transition_matrix is None:
transition_matrix = self.get_transition_matrix()
if not use_logs:
# Do it without logs - much faster
for t in range(T - 1):
total = 0.0
# Transpose the forward probabilities so that we multiply them
# along the vertical axis
fwd_trans = forward[t, :, numpy.newaxis]
# Compute the xi values by multiplying the arrays together
xi[t] = transition_matrix.T * fwd_trans * backward[t+1] * \
emission_matrix[t+1]
# Normalize all the probabilities
# Sum all the probs for the timestep and divide them all by total
total = array_sum(xi[t])
xi[t] /= total
else:
# Take logs of all the matrices we need
emission_matrix = numpy.log2(emission_matrix)
transition_matrix = numpy.log2(transition_matrix)
forward = numpy.log2(forward)
backward = numpy.log2(backward)
for t in range(T - 1):
total = 0.0
fwd_trans = forward[t, :, numpy.newaxis]
xi[t] = transition_matrix.T + fwd_trans + backward[t+1] + \
emission_matrix[t+1]
# This takes a (relatively) long time
total = numpy.logaddexp2.reduce(xi[t])
xi[t] -= total
# Exponentiate all the probabilities again
# This also takes a while
xi = numpy.exp2(xi)
return xi