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# Copyright 2020 Princeton University
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Event segmentation using a Hidden Markov Model
Given an ROI timeseries, this class uses an annealed fitting procedure to
segment the timeseries into events with stable activity patterns. After
learning the signature activity pattern of each event, the model can then be
applied to other datasets to identify a corresponding sequence of events.
Full details are available in:
Christopher Baldassano, Janice Chen, Asieh Zadbood,
Jonathan W Pillow, Uri Hasson, Kenneth A Norman
Discovering event structure in continuous narrative perception and memory
Neuron, Volume 95, Issue 3, 709 - 721.e5
https://doi.org/10.1016/j.neuron.2017.06.041
This class also extends the model described in the Neuron paper:
1) It allows transition matrices that are composed of multiple separate
chains of events rather than a single linear path. This allows a model to
contain patterns for multiple event sequences (e.g. narratives), and
fit probabilities along each of these chains on a new, unlabeled timeseries.
To use this option, pass in an event_chain vector labeling which events
belong to each chain, define event patterns using set_event_patterns(),
then fit to a new dataset with find_events.
2) To obtain better fits when the underlying event structure contains
events that vary substantially in length, the split_merge option allows
the fit() function to re-distribute events during fitting. The number of
merge/split proposals is controlled by split_merge_proposals, which
controls how thorough versus fast the fitting process is.
"""
# Authors: Chris Baldassano and Cătălin Iordan (Princeton University)
import numpy as np
from scipy import stats
import logging
import copy
from sklearn.base import BaseEstimator
from sklearn.utils.validation import check_is_fitted, check_array
from sklearn.exceptions import NotFittedError
import itertools
from . import _utils as utils # type: ignore
logger = logging.getLogger(__name__)
__all__ = [
"EventSegment",
]
class EventSegment(BaseEstimator):
"""Class for event segmentation of continuous fMRI data
Parameters
----------
n_events: int
Number of segments to learn
step_var: Callable[[int], float] : default 4 * (0.98 ** (step - 1))
The Gaussian variance to use during fitting, as a function of the
number of steps. Should decrease slowly over time.
n_iter: int, default: 500
Maximum number of steps to run during fitting
event_chains: ndarray with length = n_events
Array with unique value for each separate chain of events, each linked
in the order they appear in the array
split_merge: bool, default: False
Determines whether merge/split proposals are used during fitting with
fit(). This can improve fitting performance when events are highly
uneven in size, but requires additional time
split_merge_proposals: int, default: 1
Number of merges and splits to consider at each step. Computation time
scales as O(proposals^2) so this should usually be a small value
Attributes
----------
p_start, p_end: length n_events+1 ndarray
initial and final prior distributions over events
P: n_events+1 by n_events+1 ndarray
HMM transition matrix
ll_ : ndarray with length = number of training datasets
Log-likelihood for training datasets over the course of training
segments_: list of (time by event) ndarrays
Learned (soft) segmentation for training datasets
event_var_ : float
Gaussian variance at the end of learning
event_pat_ : voxel by event ndarray
Learned mean patterns for each event
"""
def _default_var_schedule(step):
return 4 * (0.98 ** (step - 1))
def __init__(self, n_events=2,
step_var=_default_var_schedule,
n_iter=500, event_chains=None,
split_merge=False, split_merge_proposals=1):
self.n_events = n_events
self.step_var = step_var
self.n_iter = n_iter
self.split_merge = split_merge
self.split_merge_proposals = split_merge_proposals
if event_chains is None:
self.event_chains = np.zeros(n_events)
else:
self.event_chains = event_chains
def _fit_validate(self, X):
"""Validate input to fit()
Validate data passed to fit(). Includes a transpose operation to
change the row/column order of X and z-scoring in time.
Parameters
----------
X: time by voxel ndarray, or a list of such ndarrays
fMRI data to be segmented
Returns
-------
X: list of voxel by time ndarrays
"""
if len(np.unique(self.event_chains)) > 1:
raise RuntimeError("Cannot fit chains, use set_event_patterns")
# Copy X into a list and transpose
X = copy.deepcopy(X)
if type(X) is not list:
X = [X]
for i in range(len(X)):
X[i] = check_array(X[i])
X[i] = X[i].T
# Check that number of voxels is consistent across datasets
n_dim = X[0].shape[0]
for i in range(len(X)):
assert (X[i].shape[0] == n_dim)
# Double-check that data is z-scored in time
for i in range(len(X)):
X[i] = stats.zscore(X[i], axis=1, ddof=1)
return X
def fit(self, X, y=None):
"""Learn a segmentation on training data
Fits event patterns and a segmentation to training data. After
running this function, the learned event patterns can be used to
segment other datasets using find_events
Parameters
----------
X: time by voxel ndarray, or a list of such ndarrays
fMRI data to be segmented. If a list is given, then all datasets
are segmented simultaneously with the same event patterns
y: not used (added to comply with BaseEstimator definition)
Returns
-------
self: the EventSegment object
"""
X = self._fit_validate(X)
n_train = len(X)
n_dim = X[0].shape[0]
self.classes_ = np.arange(self.n_events)
# Initialize variables for fitting
log_gamma = []
for i in range(n_train):
log_gamma.append(np.zeros((X[i].shape[1], self.n_events)))
step = 1
best_ll = float("-inf")
self.ll_ = np.empty((0, n_train))
while step <= self.n_iter:
iteration_var = self.step_var(step)
# Based on the current segmentation, compute the mean pattern
# for each event
seg_prob = [np.exp(lg) / np.sum(np.exp(lg), axis=0)
for lg in log_gamma]
mean_pat = np.empty((n_train, n_dim, self.n_events))
for i in range(n_train):
mean_pat[i, :, :] = X[i].dot(seg_prob[i])
mean_pat = np.mean(mean_pat, axis=0)
# Based on the current mean patterns, compute the event
# segmentation
self.ll_ = np.append(self.ll_, np.empty((1, n_train)), axis=0)
for i in range(n_train):
logprob = self._logprob_obs(X[i], mean_pat, iteration_var)
log_gamma[i], self.ll_[-1, i] = self._forward_backward(logprob)
if step > 1 and self.split_merge:
curr_ll = np.mean(self.ll_[-1, :])
self.ll_[-1, :], log_gamma, mean_pat = \
self._split_merge(X, log_gamma, iteration_var, curr_ll)
# If log-likelihood has started decreasing, undo last step and stop
if np.mean(self.ll_[-1, :]) < best_ll:
self.ll_ = self.ll_[:-1, :]
break
self.segments_ = [np.exp(lg) for lg in log_gamma]
self.event_var_ = iteration_var
self.event_pat_ = mean_pat
best_ll = np.mean(self.ll_[-1, :])
logger.debug("Fitting step %d, LL=%f", step, best_ll)
step += 1
return self
def _logprob_obs(self, data, mean_pat, var):
"""Log probability of observing each timepoint under each event model
Computes the log probability of each observed timepoint being
generated by the Gaussian distribution for each event pattern
Parameters
----------
data: voxel by time ndarray
fMRI data on which to compute log probabilities
mean_pat: voxel by event ndarray
Centers of the Gaussians for each event
var: float or 1D array of length equal to the number of events
Variance of the event Gaussians. If scalar, all events are
assumed to have the same variance
Returns
-------
logprob : time by event ndarray
Log probability of each timepoint under each event Gaussian
"""
n_vox = data.shape[0]
t = data.shape[1]
# z-score both data and mean patterns in space, so that Gaussians
# are measuring Pearson correlations and are insensitive to overall
# activity changes
data_z = stats.zscore(data, axis=0, ddof=1)
mean_pat_z = stats.zscore(mean_pat, axis=0, ddof=1)
logprob = np.empty((t, self.n_events))
if type(var) is not np.ndarray:
var = var * np.ones(self.n_events)
for k in range(self.n_events):
logprob[:, k] = -0.5 * n_vox * np.log(
2 * np.pi * var[k]) - 0.5 * np.sum(
(data_z.T - mean_pat_z[:, k]).T ** 2, axis=0) / var[k]
logprob /= n_vox
return logprob
def _forward_backward(self, logprob):
"""Runs forward-backward algorithm on observation log probs
Given the log probability of each timepoint being generated by
each event, run the HMM forward-backward algorithm to find the
probability that each timepoint belongs to each event (based on the
transition priors in p_start, p_end, and P)
See https://en.wikipedia.org/wiki/Forward-backward_algorithm for
mathematical details
Parameters
----------
logprob : time by event ndarray
Log probability of each timepoint under each event Gaussian
Returns
-------
log_gamma : time by event ndarray
Log probability of each timepoint belonging to each event
ll : float
Log-likelihood of fit
"""
logprob = copy.copy(logprob)
t = logprob.shape[0]
logprob = np.hstack((logprob, float("-inf") * np.ones((t, 1))))
# Initialize variables
log_scale = np.zeros(t)
log_alpha = np.zeros((t, self.n_events + 1))
log_beta = np.zeros((t, self.n_events + 1))
# Set up transition matrix, with final sink state
self.p_start = np.zeros(self.n_events + 1)
self.p_end = np.zeros(self.n_events + 1)
self.P = np.zeros((self.n_events + 1, self.n_events + 1))
label_ind = np.unique(self.event_chains, return_inverse=True)[1]
n_chains = np.max(label_ind) + 1
# For each chain of events, link them together and then to sink state
for c in range(n_chains):
chain_ind = np.nonzero(label_ind == c)[0]
self.p_start[chain_ind[0]] = 1 / n_chains
self.p_end[chain_ind[-1]] = 1 / n_chains
p_trans = (len(chain_ind) - 1) / t
if p_trans >= 1:
raise ValueError('Too few timepoints')
for i in range(len(chain_ind)):
self.P[chain_ind[i], chain_ind[i]] = 1 - p_trans
if i < len(chain_ind) - 1:
self.P[chain_ind[i], chain_ind[i+1]] = p_trans
else:
self.P[chain_ind[i], -1] = p_trans
self.P[-1, -1] = 1
# Forward pass
for i in range(t):
if i == 0:
log_alpha[0, :] = self._log(self.p_start) + logprob[0, :]
else:
log_alpha[i, :] = self._log(np.exp(log_alpha[i - 1, :])
.dot(self.P)) + logprob[i, :]
log_scale[i] = np.logaddexp.reduce(log_alpha[i, :])
log_alpha[i] -= log_scale[i]
# Backward pass
log_beta[-1, :] = self._log(self.p_end) - log_scale[-1]
for i in reversed(range(t - 1)):
obs_weighted = log_beta[i + 1, :] + logprob[i + 1, :]
offset = np.max(obs_weighted)
log_beta[i, :] = offset + self._log(
np.exp(obs_weighted - offset).dot(self.P.T)) - log_scale[i]
# Combine and normalize
log_gamma = log_alpha + log_beta
log_gamma -= np.logaddexp.reduce(log_gamma, axis=1, keepdims=True)
ll = np.sum(log_scale[:(t - 1)]) + np.logaddexp.reduce(
log_alpha[-1, :] + log_scale[-1] + self._log(self.p_end))
log_gamma = log_gamma[:, :-1]
return log_gamma, ll
def _log(self, x):
"""Modified version of np.log that manually sets values <=0 to -inf
Parameters
----------
x: ndarray of floats
Input to the log function
Returns
-------
log_ma: ndarray of floats
log of x, with x<=0 values replaced with -inf
"""
xshape = x.shape
_x = x.flatten()
y = utils.masked_log(_x)
return y.reshape(xshape)
def set_event_patterns(self, event_pat):
"""Set HMM event patterns manually
Rather than fitting the event patterns automatically using fit(), this
function allows them to be set explicitly. They can then be used to
find corresponding events in a new dataset, using find_events().
Parameters
----------
event_pat: voxel by event ndarray
"""
if event_pat.shape[1] != self.n_events:
raise ValueError(("Number of columns of event_pat must match "
"number of events"))
self.event_pat_ = event_pat.copy()
def find_events(self, testing_data, var=None, scramble=False):
"""Applies learned event segmentation to new testing dataset
After fitting an event segmentation using fit() or setting event
patterns directly using set_event_patterns(), this function finds the
same sequence of event patterns in a new testing dataset.
Parameters
----------
testing_data: timepoint by voxel ndarray
fMRI data to segment based on previously-learned event patterns
var: float or 1D ndarray of length equal to the number of events
default: uses variance that maximized training log-likelihood
Variance of the event Gaussians. If scalar, all events are
assumed to have the same variance. If fit() has not previously
been run, this must be specifed (cannot be None).
scramble: bool : default False
If true, the order of the learned events are shuffled before
fitting, to give a null distribution
Returns
-------
segments : time by event ndarray
The resulting soft segmentation. segments[t,e] = probability
that timepoint t is in event e
test_ll : float
Log-likelihood of model fit
"""
if var is None:
if not hasattr(self, 'event_var_'):
raise NotFittedError(("Event variance must be provided, if "
"not previously set by fit()"))
else:
var = self.event_var_
if not hasattr(self, 'event_pat_'):
raise NotFittedError(("The event patterns must first be set "
"by fit() or set_event_patterns()"))
if scramble:
mean_pat = self.event_pat_[:, np.random.permutation(self.n_events)]
else:
mean_pat = self.event_pat_
logprob = self._logprob_obs(testing_data.T, mean_pat, var)
lg, test_ll = self._forward_backward(logprob)
segments = np.exp(lg)
return segments, test_ll
def predict(self, X):
"""Applies learned event segmentation to new testing dataset
Alternative function for segmenting a new dataset after using
fit() to learn a sequence of events, to comply with the sklearn
Classifier interface
Parameters
----------
X: timepoint by voxel ndarray
fMRI data to segment based on previously-learned event patterns
Returns
-------
Event label for each timepoint
"""
check_is_fitted(self, ["event_pat_", "event_var_"])
X = check_array(X)
segments, test_ll = self.find_events(X)
return np.argmax(segments, axis=1)
def calc_weighted_event_var(self, D, weights, event_pat):
"""Computes normalized weighted variance around event pattern
Utility function for computing variance in a training set of weighted
event examples. For each event, the sum of squared differences for all
timepoints from the event pattern is computed, and then the weights
specify how much each of these differences contributes to the
variance (normalized by the number of voxels).
Parameters
----------
D : timepoint by voxel ndarray
fMRI data for which to compute event variances
weights : timepoint by event ndarray
specifies relative weights of timepoints for each event
event_pat : voxel by event ndarray
mean event patterns to compute variance around
Returns
-------
ev_var : ndarray of variances for each event
"""
Dz = stats.zscore(D, axis=1, ddof=1)
ev_var = np.empty(event_pat.shape[1])
for e in range(event_pat.shape[1]):
# Only compute variances for weights > 0.1% of max weight
nz = weights[:, e] > np.max(weights[:, e])/1000
sumsq = np.dot(weights[nz, e],
np.sum(np.square(Dz[nz, :] -
event_pat[:, e]), axis=1))
ev_var[e] = sumsq/(np.sum(weights[nz, e]) -
np.sum(np.square(weights[nz, e])) /
np.sum(weights[nz, e]))
ev_var = ev_var / D.shape[1]
return ev_var
def model_prior(self, t):
"""Returns the prior probability of the HMM
Runs forward-backward without any data, showing the prior distribution
of the model (for comparison with a posterior).
Parameters
----------
t: int
Number of timepoints
Returns
-------
segments : time by event ndarray
segments[t,e] = prior probability that timepoint t is in event e
test_ll : float
Log-likelihood of model (data-independent term)"""
lg, test_ll = self._forward_backward(np.zeros((t, self.n_events)))
segments = np.exp(lg)
return segments, test_ll
def _split_merge(self, X, log_gamma, iteration_var, curr_ll):
"""Attempt to improve log-likelihood with a merge/split
The simulated annealing used in fit() is susceptible to getting
stuck in a local minimum if there are some very short events. This
function attempts to find
a) pairs of neighboring events that are highly similar, to merge
b) events that can be split into two dissimilar events
It then tests to see whether simultaneously merging one of the
pairs from (a) and splitting one of the events from (b) can improve
the log-likelihood. The number of (a)/(b) pairs tested is determined
by the split_merge_proposals class attribute.
Parameters
----------
X: list of voxel by time ndarrays
fMRI datasets being fit
log_gamma : list of time by event ndarrays
Log probability of each timepoint belonging to each event,
for each dataset
iteration_var : float
Current variance in simulated annealing
curr_ll: float
Log-likelihood of current model
Returns
-------
return_ll : ndarray with length equal to length of X
Log-likelihood after merge/split (same as curr_ll if no
merge/split improved curr_ll)
return_lg : list of time by event ndarrays
Log probability of each timepoint belonging to each event,
for each dataset (same as log_gamma if no merge/split
improved curr_ll)
return_mp : voxel by event ndarray
Mean patterns of events (after possible merge/split)
"""
# Compute current probabilities and mean patterns
n_train = len(X)
n_dim = X[0].shape[0]
seg_prob = [np.exp(lg) / np.sum(np.exp(lg), axis=0)
for lg in log_gamma]
mean_pat = np.empty((n_train, n_dim, self.n_events))
for i in range(n_train):
mean_pat[i, :, :] = X[i].dot(seg_prob[i])
mean_pat = np.mean(mean_pat, axis=0)
# For each event, merge its probability distribution
# with the next event, and also split its probability
# distribution at its median into two separate events.
# Use these new event probability distributions to compute
# merged and split event patterns.
merge_pat = np.empty((n_train, n_dim, self.n_events))
split_pat = np.empty((n_train, n_dim, 2 * self.n_events))
for i, sp in enumerate(seg_prob): # Iterate over datasets
m_evprob = np.zeros((sp.shape[0], sp.shape[1]))
s_evprob = np.zeros((sp.shape[0], 2 * sp.shape[1]))
cs = np.cumsum(sp, axis=0)
for e in range(sp.shape[1]):
# Split distribution at midpoint and normalize each half
mid = np.where(cs[:, e] >= 0.5)[0][0]
cs_first = cs[mid, e] - sp[mid, e]
cs_second = 1 - cs_first
s_evprob[:mid, 2 * e] = sp[:mid, e] / cs_first
s_evprob[mid:, 2 * e + 1] = sp[mid:, e] / cs_second
# Merge distribution with next event distribution
m_evprob[:, e] = sp[:, e:(e + 2)].mean(1)
# Weight data by distribution to get event patterns
merge_pat[i, :, :] = X[i].dot(m_evprob)
split_pat[i, :, :] = X[i].dot(s_evprob)
# Average across datasets
merge_pat = np.mean(merge_pat, axis=0)
split_pat = np.mean(split_pat, axis=0)
# Correlate the current event patterns with the split and
# merged patterns
merge_corr = np.zeros(self.n_events)
split_corr = np.zeros(self.n_events)
for e in range(self.n_events):
split_corr[e] = np.corrcoef(mean_pat[:, e],
split_pat[:, (2 * e):(2 * e + 2)],
rowvar=False)[0, 1:3].max()
merge_corr[e] = np.corrcoef(merge_pat[:, e],
mean_pat[:, e:(e + 2)],
rowvar=False)[0, 1:3].min()
merge_corr = merge_corr[:-1]
# Find best merge/split candidates
# A high value of merge_corr indicates that a pair of events are
# very similar to their merged pattern, and are good candidates for
# being merged.
# A low value of split_corr indicates that an event's pattern is
# very dissimilar from the patterns in its first and second half,
# and is a good candidate for being split.
best_merge = np.flipud(np.argsort(merge_corr))
best_merge = best_merge[:self.split_merge_proposals]
best_split = np.argsort(split_corr)
best_split = best_split[:self.split_merge_proposals]
# For every pair of merge/split candidates, attempt the merge/split
# and measure the log-likelihood. If any are better than curr_ll,
# accept this best merge/split
mean_pat_last = mean_pat.copy()
return_ll = curr_ll
return_lg = copy.deepcopy(log_gamma)
return_mp = mean_pat.copy()
for m_e, s_e in itertools.product(best_merge, best_split):
if m_e == s_e or m_e+1 == s_e:
# Don't attempt to merge/split same event
continue
# Construct new set of patterns with merge/split
mean_pat_ms = np.delete(mean_pat_last, s_e, axis=1)
mean_pat_ms = np.insert(mean_pat_ms, [s_e, s_e],
split_pat[:, (2 * s_e):(2 * s_e + 2)],
axis=1)
mean_pat_ms = np.delete(mean_pat_ms,
[m_e + (s_e < m_e), m_e + (s_e < m_e) + 1],
axis=1)
mean_pat_ms = np.insert(mean_pat_ms, m_e + (s_e < m_e),
merge_pat[:, m_e], axis=1)
# Measure log-likelihood with these new patterns
ll_ms = np.zeros(n_train)
log_gamma_ms = list()
for i in range(n_train):
logprob = self._logprob_obs(X[i],
mean_pat_ms, iteration_var)
lg, ll_ms[i] = self._forward_backward(logprob)
log_gamma_ms.append(lg)
# If better than best ll so far, save to return to fit()
if ll_ms.mean() > return_ll:
return_mp = mean_pat_ms.copy()
return_ll = ll_ms
for i in range(n_train):
return_lg[i] = log_gamma_ms[i].copy()
logger.debug("Identified merge %d,%d and split %d",
m_e, m_e+1, s_e)
return return_ll, return_lg, return_mp