Signal Processing - January 2016 - 24

1:d

Z (s)

v (s)

X (s)

)
∑(s
e

A(s)

s = 1, . . . , S
[FIG4] a graphical model representation of the group source
imaging algorithm, gMsP, in the presence of S subjects. X (s) and
Z (s) are the observed EEG data and unobserved source activity
for the sth subject. A (s) and R (es) are the associated lead-field
matrix and noise covariance matrix, respectively. y (s) is a subjectspecific scale hyperparameter, and c 1: d c are source-specific scale
hyperparameters that act as the group constraint. all unknown
latent variables and parameters are shown as unshaded nodes
and estimated from the data. see the section "VB for Group EEG/
MEG analysis" for details.

mean-field approximation is combined with the Laplace
approximation by assuming that the posterior distribution can
be factorized into Gaussian marginals (i.e., a combination of
VB-MF and VB-LA; see the section "SBL Methods"):
q ( Z, m) = q ( Z) q (m)
_ N (n zr , R zr ) N (n m, R m) .
This leads to variational free energies at both the individual sub-
ject and group levels. The gMSP algorithm then proceeds as fol-
lows: in the first stage, q (m 1: d ) is estimated by maximizing the
group-level variational free energy; in the second stage,
q (m (d + 1):(d + 2S + 1)) is estimated by maximizing the subject-level
variational free energy. With all the hyperparameters estimated,
q ( Z) can be obtained as the MAP estimates of the source activity.
Note that, to achieve sparsity on the source covariance compo-
nents, both ARD and a greedy search approach were developed to
optimize the source-specific hyperparameters. Refer to [23] and
[24] for algorithmic details. Only gMSP with ARD is illustrated in
this article (see the section "Group Electromagnetic Brain Imag-
ing Using gMSP"). This hierarchical model approach allows one to
place within- and between-subject constraints on the recon-
structed source activity in the context of group studies.
c

c

c

sBL MEtHods
Sparse learning, also known as compressed sensing in signal pro-
cessing [25], is referred to as a collection of learning methods
that seek a tradeoff between certain goodness-of-fit measure and
sparsity of the solution, the latter of which allows better inter-
pretability and enhanced generalization ability of the model.
Sparse learning is particularly suited for analyzing EEG/MEG sig-
nals with high dimensionality, small sample size, and low SNR.

See [26] for a recent review on the applications of sparse learning
to brain signal processing.
Compared with its most common counterpart, two advantages
of SBL [9], [27] are noteworthy:
■ SBL allows automatic model selection. This can be achieved
by both EB (see the "Introduction" section), and fully Bayesian
methods. In EB, maximizing p (X | c) provides a natural regu-
larizing mechanism that yields sparse solutions. In fully Bayes-
ian methods, as suggested by the VB methods presented in the
sections "VB for Learning Common EEG Components" and
"VB for Group EEG/MEG Analysis," automatic sparse learning
can be achieved by imposing noninformative priors on c,
which leads to sparsity since the hierarchical priors on the
parameters are typically sparse priors. For instance, by margin-
alizing the variance, the normal-inverse-Gamma prior for each
brain source in (16) amounts to a Student t-distribution:
p (z (mk), t) =
=

#

p (z (mk), t | t (mk)) p (t (mk)) dt (mk)

(k) 2
C (a + 1 )
2 b a [b + [z m, t] ] - (a + 12 ),
2
C (a) 2r

(32)

which is sharply peaked at zero in the noninformative case
(i.e., when a " 0, b " 0) .
■ SBL is more capable of finding sparse solutions than conven-
tional methods. In typical electromagnetic brain-imaging prob-
lem setups, many columns of the lead-field matrix are highly
correlated; in this case, the convex l 1-norm-based MAP (sparse)
solution performs poorly since the restricted isometry property
is violated. In spatiotemporal decomposition problems where A
is unknown, the MAP estimation may suffer from too many
local minima in the solution space due to the multiplicative
structure of the spatiotemporal model (2). Consider the multi-
ple covariance parameterization in (7). According to [28], if
C i = e i e 



Table of Contents for the Digital Edition of Signal Processing - January 2016

Signal Processing - January 2016 - Cover1

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