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# Copyright 2016 Intel Corporation
#
# 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.
"""Shared Response Model (SRM)
The implementations are based on the following publications:
.. [Chen2015] "A Reduced-Dimension fMRI Shared Response Model",
P.-H. Chen, J. Chen, Y. Yeshurun-Dishon, U. Hasson, J. Haxby, P. Ramadge
Advances in Neural Information Processing Systems (NIPS), 2015.
http://papers.nips.cc/paper/5855-a-reduced-dimension-fmri-shared-response-model
.. [Anderson2016] "Enabling Factor Analysis on Thousand-Subject Neuroimaging
Datasets",
Michael J. Anderson, Mihai Capotă, Javier S. Turek, Xia Zhu, Theodore L.
Willke, Yida Wang, Po-Hsuan Chen, Jeremy R. Manning, Peter J. Ramadge,
Kenneth A. Norman, arXiv preprint, 2016.
https://arxiv.org/abs/1608.04647
"""
# Authors: Po-Hsuan Chen (Princeton Neuroscience Institute) and Javier Turek
# (Intel Labs), 2015
import logging
import numpy as np
import scipy
from sklearn.base import BaseEstimator, TransformerMixin
from sklearn.utils import assert_all_finite
from sklearn.utils.validation import NotFittedError
__all__ = [
"SRM", "DetSRM"
]
logger = logging.getLogger(__name__)
def _init_w_transforms(data, features):
"""Initialize the mappings (Wi) for the SRM with random orthogonal matrices.
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
features : int
The number of features in the model.
Returns
-------
w : list of array, element i has shape=[voxels_i, features]
The initialized orthogonal transforms (mappings) :math:`W_i` for each
subject.
voxels : list of int
A list with the number of voxels per subject.
Note
----
This function assumes that the numpy random number generator was
initialized.
Not thread safe.
"""
w = []
subjects = len(data)
voxels = np.empty(subjects, dtype=int)
# Set Wi to a random orthogonal voxels by features matrix
for subject in range(subjects):
voxels[subject] = data[subject].shape[0]
rnd_matrix = np.random.random((voxels[subject], features))
q, r = np.linalg.qr(rnd_matrix)
w.append(q)
return w, voxels
class SRM(BaseEstimator, TransformerMixin):
"""Probabilistic Shared Response Model (SRM)
Given multi-subject data, factorize it as a shared response S among all
subjects and an orthogonal transform W per subject:
.. math:: X_i \\approx W_i S, \\forall i=1 \\dots N
Parameters
----------
n_iter : int, default: 10
Number of iterations to run the algorithm.
features : int, default: 50
Number of features to compute.
rand_seed : int, default: 0
Seed for initializing the random number generator.
Attributes
----------
w_ : list of array, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) for each subject.
s_ : array, shape=[features, samples]
The shared response.
sigma_s_ : array, shape=[features, features]
The covariance of the shared response Normal distribution.
mu_ : list of array, element i has shape=[voxels_i]
The voxel means over the samples for each subject.
rho2_ : array, shape=[subjects]
The estimated noise variance :math:`\\rho_i^2` for each subject
Note
----
The number of voxels may be different between subjects. However, the
number of samples must be the same across subjects.
The probabilistic Shared Response Model is approximated using the
Expectation Maximization (EM) algorithm proposed in [Chen2015]_. The
implementation follows the optimizations published in [Anderson2016]_.
This is a single node version.
The run-time complexity is :math:`O(I (V T K + V K^2 + K^3))` and the
memory complexity is :math:`O(V T)` with I - the number of iterations,
V - the sum of voxels from all subjects, T - the number of samples, and
K - the number of features (typically, :math:`V \\gg T \\gg K`).
"""
def __init__(self, n_iter=10, features=50, rand_seed=0):
self.n_iter = n_iter
self.features = features
self.rand_seed = rand_seed
return
def fit(self, X, y=None):
"""Compute the probabilistic Shared Response Model
Parameters
----------
X : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
y : not used
"""
logger.info('Starting Probabilistic SRM')
# Check the number of subjects
if len(X) <= 1:
raise ValueError("There are not enough subjects "
"({0:d}) to train the model.".format(len(X)))
# Check for input data sizes
if X[0].shape[1] < self.features:
raise ValueError(
"There are not enough samples to train the model with "
"{0:d} features.".format(self.features))
# Check if all subjects have same number of TRs
number_trs = X[0].shape[1]
number_subjects = len(X)
for subject in range(number_subjects):
assert_all_finite(X[subject])
if X[subject].shape[1] != number_trs:
raise ValueError("Different number of samples between subjects"
".")
# Run SRM
self.sigma_s_, self.w_, self.mu_, self.rho2_, self.s_ = self._srm(X)
return self
def transform(self, X, y=None):
"""Use the model to transform matrix to Shared Response space
Parameters
----------
X : list of 2D arrays, element i has shape=[voxels_i, samples_i]
Each element in the list contains the fMRI data of one subject
note that number of voxels and samples can vary across subjects
y : not used (as it is unsupervised learning)
Returns
-------
s : list of 2D arrays, element i has shape=[features_i, samples_i]
Shared responses from input data (X)
"""
# Check if the model exist
if hasattr(self, 'w_') is False:
raise NotFittedError("The model fit has not been run yet.")
# Check the number of subjects
if len(X) != len(self.w_):
raise ValueError("The number of subjects does not match the one"
" in the model.")
s = [None] * len(X)
for subject in range(len(X)):
s[subject] = self.w_[subject].T.dot(X[subject])
return s
def _init_structures(self, data, subjects):
"""Initializes data structures for SRM and preprocess the data.
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
subjects : int
The total number of subjects in `data`.
Returns
-------
x : list of array, element i has shape=[voxels_i, samples]
Demeaned data for each subject.
mu : list of array, element i has shape=[voxels_i]
Voxel means over samples, per subject.
rho2 : array, shape=[subjects]
Noise variance :math:`\\rho^2` per subject.
trace_xtx : array, shape=[subjects]
The squared Frobenius norm of the demeaned data in `x`.
"""
x = []
mu = []
rho2 = np.zeros(subjects)
trace_xtx = np.zeros(subjects)
for subject in range(subjects):
mu.append(np.mean(data[subject], 1))
rho2[subject] = 1
trace_xtx[subject] = np.sum(data[subject] ** 2)
x.append(data[subject] - mu[subject][:, np.newaxis])
return x, mu, rho2, trace_xtx
def _likelihood(self, chol_sigma_s_rhos, log_det_psi, chol_sigma_s,
trace_xt_invsigma2_x, inv_sigma_s_rhos, wt_invpsi_x,
samples):
"""Calculate the log-likelihood function
Parameters
----------
chol_sigma_s_rhos : array, shape=[features, features]
Cholesky factorization of the matrix (Sigma_S + sum_i(1/rho_i^2)
* I)
log_det_psi : float
Determinant of diagonal matrix Psi (containing the rho_i^2 value
voxels_i times).
chol_sigma_s : array, shape=[features, features]
Cholesky factorization of the matrix Sigma_S
trace_xt_invsigma2_x : float
Trace of :math:`\\sum_i (||X_i||_F^2/\\rho_i^2)`
inv_sigma_s_rhos : array, shape=[features, features]
Inverse of :math:`(\\Sigma_S + \\sum_i(1/\\rho_i^2) * I)`
wt_invpsi_x : array, shape=[features, samples]
samples : int
The total number of samples in the data.
Returns
-------
loglikehood : float
The log-likelihood value.
"""
log_det = (np.log(np.diag(chol_sigma_s_rhos) ** 2).sum() + log_det_psi
+ np.log(np.diag(chol_sigma_s) ** 2).sum())
loglikehood = -0.5 * samples * log_det - 0.5 * trace_xt_invsigma2_x
loglikehood += 0.5 * np.trace(
wt_invpsi_x.T.dot(inv_sigma_s_rhos).dot(wt_invpsi_x))
# + const --> -0.5*nTR*nvoxel*subjects*math.log(2*math.pi)
return loglikehood
def _srm(self, data):
"""Expectation-Maximization algorithm for fitting the probabilistic SRM.
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
Returns
-------
sigma_s : array, shape=[features, features]
The covariance :math:`\\Sigma_s` of the shared response Normal
distribution.
w : list of array, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) :math:`W_i` for each subject.
mu : list of array, element i has shape=[voxels_i]
The voxel means :math:`\\mu_i` over the samples for each subject.
rho2 : array, shape=[subjects]
The estimated noise variance :math:`\\rho_i^2` for each subject
s : array, shape=[features, samples]
The shared response.
"""
samples = data[0].shape[1]
subjects = len(data)
np.random.seed(self.rand_seed)
# Initialization step: initialize the outputs with initial values,
# voxels with the number of voxels in each subject, and trace_xtx with
# the ||X_i||_F^2 of each subject.
w, voxels = _init_w_transforms(data, self.features)
x, mu, rho2, trace_xtx = self._init_structures(data, subjects)
shared_response = np.zeros((self.features, samples))
sigma_s = np.identity(self.features)
# Main loop of the algorithm (run
for iteration in range(self.n_iter):
logger.info('Iteration %d' % (iteration + 1))
# E-step:
# Sum the inverted the rho2 elements for computing W^T * Psi^-1 * W
rho0 = (1 / rho2).sum()
# Invert Sigma_s using Cholesky factorization
(chol_sigma_s, lower_sigma_s) = scipy.linalg.cho_factor(
sigma_s, check_finite=False)
inv_sigma_s = scipy.linalg.cho_solve(
(chol_sigma_s, lower_sigma_s), np.identity(self.features),
check_finite=False)
# Invert (Sigma_s + rho_0 * I) using Cholesky factorization
sigma_s_rhos = inv_sigma_s + np.identity(self.features) * rho0
(chol_sigma_s_rhos, lower_sigma_s_rhos) = scipy.linalg.cho_factor(
sigma_s_rhos, check_finite=False)
inv_sigma_s_rhos = scipy.linalg.cho_solve(
(chol_sigma_s_rhos, lower_sigma_s_rhos),
np.identity(self.features), check_finite=False)
# Compute the sum of W_i^T * rho_i^-2 * X_i, and the sum of traces
# of X_i^T * rho_i^-2 * X_i
wt_invpsi_x = np.zeros((self.features, samples))
trace_xt_invsigma2_x = 0.0
for subject in range(subjects):
wt_invpsi_x += (w[subject].T.dot(x[subject])) / rho2[subject]
trace_xt_invsigma2_x += trace_xtx[subject] / rho2[subject]
log_det_psi = np.sum(np.log(rho2) * voxels)
# Update the shared response
shared_response = sigma_s.dot(
np.identity(self.features) - rho0 * inv_sigma_s_rhos).dot(
wt_invpsi_x)
# M-step
# Update Sigma_s and compute its trace
sigma_s = (inv_sigma_s_rhos
+ shared_response.dot(shared_response.T) / samples)
trace_sigma_s = samples * np.trace(sigma_s)
# Update each subject's mapping transform W_i and error variance
# rho_i^2
for subject in range(subjects):
a_subject = x[subject].dot(shared_response.T)
perturbation = np.zeros(a_subject.shape)
np.fill_diagonal(perturbation, 0.001)
u_subject, s_subject, v_subject = np.linalg.svd(
a_subject + perturbation, full_matrices=False)
w[subject] = u_subject.dot(v_subject)
rho2[subject] = trace_xtx[subject]
rho2[subject] += -2 * np.sum(w[subject] * a_subject).sum()
rho2[subject] += trace_sigma_s
rho2[subject] /= samples * voxels[subject]
if logger.isEnabledFor(logging.INFO):
# Calculate and log the current log-likelihood for checking
# convergence
loglike = self._likelihood(
chol_sigma_s_rhos, log_det_psi, chol_sigma_s,
trace_xt_invsigma2_x, inv_sigma_s_rhos, wt_invpsi_x,
samples)
logger.info('Objective function %f' % loglike)
return sigma_s, w, mu, rho2, shared_response
class DetSRM(BaseEstimator, TransformerMixin):
"""Deterministic Shared Response Model (DetSRM)
Given multi-subject data, factorize it as a shared response S among all
subjects and an orthogonal transform W per subject:
.. math:: X_i \\approx W_i S, \\forall i=1 \\dots N
Parameters
----------
n_iter : int, default: 10
Number of iterations to run the algorithm.
features : int, default: 50
Number of features to compute.
rand_seed : int, default: 0
Seed for initializing the random number generator.
Attributes
----------
w_ : list of array, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) for each subject.
s_ : array, shape=[features, samples]
The shared response.
Note
----
The number of voxels may be different between subjects. However, the
number of samples must be the same across subjects.
The Deterministic Shared Response Model is approximated using the
Block Coordinate Descent (BCD) algorithm proposed in [Chen2015]_.
This is a single node version.
The run-time complexity is :math:`O(I (V T K + V K^2))` and the memory
complexity is :math:`O(V T)` with I - the number of iterations, V - the
sum of voxels from all subjects, T - the number of samples, K - the
number of features (typically, :math:`V \\gg T \\gg K`), and N - the
number of subjects.
"""
def __init__(self, n_iter=10, features=50, rand_seed=0):
self.n_iter = n_iter
self.features = features
self.rand_seed = rand_seed
return
def fit(self, X, y=None):
"""Compute the Deterministic Shared Response Model
Parameters
----------
X : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
y : not used
"""
logger.info('Starting Deterministic SRM')
# Check the number of subjects
if len(X) <= 1:
raise ValueError("There are not enough subjects "
"({0:d}) to train the model.".format(len(X)))
# Check for input data sizes
if X[0].shape[1] < self.features:
raise ValueError(
"There are not enough samples to train the model with "
"{0:d} features.".format(self.features))
# Check if all subjects have same number of TRs
number_trs = X[0].shape[1]
number_subjects = len(X)
for subject in range(number_subjects):
assert_all_finite(X[subject])
if X[subject].shape[1] != number_trs:
raise ValueError("Different number of samples between subjects"
".")
# Run SRM
self.w_, self.s_ = self._srm(X)
return self
def transform(self, X, y=None):
"""Use the model to transform data to the Shared Response subspace
Parameters
----------
X : list of 2D arrays, element i has shape=[voxels_i, samples_i]
Each element in the list contains the fMRI data of one subject.
y : not used
Returns
-------
s : list of 2D arrays, element i has shape=[features_i, samples_i]
Shared responses from input data (X)
"""
# Check if the model exist
if hasattr(self, 'w_') is False:
raise NotFittedError("The model fit has not been run yet.")
# Check the number of subjects
if len(X) != len(self.w_):
raise ValueError("The number of subjects does not match the one"
" in the model.")
s = [None] * len(X)
for subject in range(len(X)):
s[subject] = self.w_[subject].T.dot(X[subject])
return s
def _objective_function(self, data, w, s):
"""Calculate the objective function
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
w : list of 2D arrays, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) :math:`W_i` for each subject.
s : array, shape=[features, samples]
The shared response
Returns
-------
objective : float
The objective function value.
"""
subjects = len(data)
objective = 0.0
for m in range(subjects):
objective += \
np.linalg.norm(data[m] - w[m].dot(s), 'fro')**2
return objective * 0.5 / data[0].shape[1]
def _compute_shared_response(self, data, w):
""" Compute the shared response S
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
w : list of 2D arrays, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) :math:`W_i` for each subject.
Returns
-------
s : array, shape=[features, samples]
The shared response for the subjects data with the mappings in w.
"""
s = np.zeros((w[0].shape[1], data[0].shape[1]))
for m in range(len(w)):
s = s + w[m].T.dot(data[m])
s /= len(w)
return s
def _srm(self, data):
"""Expectation-Maximization algorithm for fitting the probabilistic SRM.
Parameters
----------
data : list of 2D arrays, element i has shape=[voxels_i, samples]
Each element in the list contains the fMRI data of one subject.
Returns
-------
w : list of array, element i has shape=[voxels_i, features]
The orthogonal transforms (mappings) :math:`W_i` for each subject.
s : array, shape=[features, samples]
The shared response.
"""
subjects = len(data)
np.random.seed(self.rand_seed)
# Initialization step: initialize the outputs with initial values,
# voxels with the number of voxels in each subject.
w, _ = _init_w_transforms(data, self.features)
shared_response = self._compute_shared_response(data, w)
if logger.isEnabledFor(logging.INFO):
# Calculate the current objective function value
objective = self._objective_function(data, w, shared_response)
logger.info('Objective function %f' % objective)
# Main loop of the algorithm
for iteration in range(self.n_iter):
logger.info('Iteration %d' % (iteration + 1))
# Update each subject's mapping transform W_i:
for subject in range(subjects):
a_subject = data[subject].dot(shared_response.T)
perturbation = np.zeros(a_subject.shape)
np.fill_diagonal(perturbation, 0.001)
u_subject, _, v_subject = np.linalg.svd(
a_subject + perturbation, full_matrices=False)
w[subject] = u_subject.dot(v_subject)
# Update the shared response:
shared_response = self._compute_shared_response(data, w)
if logger.isEnabledFor(logging.INFO):
# Calculate the current objective function value
objective = self._objective_function(data, w, shared_response)
logger.info('Objective function %f' % objective)
return w, shared_response