Signal Processing - May 2016 - 95
correlation between modulation profiles, allowing both highly
and weakly connected modulations as well as joint independent
components [47]. It is particularly suitable for jointly analyzing
data sets from multiple modalities in the same set of subjects.
For instance, it has been applied to fMRI and DTI data from
healthy controls and people with schizophrenia patients and
bipolar disorder [47].
Three-way pICA
Recall that pICA analyzes two data modalities concurrently
and allows the integration of data from different contexts
and feature spaces. This flexibility makes pICA particularly
suitable for data fusion (e.g., MRI and genetic data). Threeway pICA extends the concept of pICA to include three
modalities (M = 3), which is formulated as in Figure 2(b)
[49]. It also maximizes independence within each modality
using an entropy-based cost function, while identifying
intermodality correlations through adding three squared
correlation terms. Similar to pICA, an adaptive optimization
procedure is employed. These online adjustments guarantee the convergence of the three-way pICA. Compared to
MCCA for multimodal data fusion [23], [45], three-way pICA
exploits HOS for obtaining statistically independent components, which likely leads to a more accurate and meaningful
estimation. Compared to jICA and linked ICA, three-way
pICA relaxes the strong assumption of the same modulation profile. Also, in contrast to mCCA+jICA, three-way
pICA explicitly incorporates the information provided by all
modalities in one comprehensive data decomposition. It has
been applied to fMRI, sMRI, and SNP data for investigating
genetic effects on alcohol dependence and performed better
than pICA and separate ICA (sICA) in identifying pairwise
links between modalities and estimating independent components [49].
IVA
IVA is a generalization of ICA from one to multiple data
sets (M $ 2) , originally designed to address the permutation problem in the frequency domain for the separation of
acoustic sources [15]. IVA can be formulated within a general JBSS framework to ensure that the extracted source
components are independent within each data set and maximally correlated across multiple data sets [17]. Specifically,
the goal of IVA is to identify L independent SCVs s l from
M data sets X m . This can be achieved by minimizing the
mutual information among the estimated SCVs su l . It can be
proven that minimizing the IVA cost function is equivalent
to simultaneously minimizing the entropy of all components
su [l m] and maximizing the mutual information within each
estimated SCV su l . IVA can ultimately solve the problem of
permutation ambiguity when applying BSS techniques to
multiple data sets. IVA has been shown to achieve superior
performance than previous techniques in simulation studies [17]. Although IVA was formulated as in Figure 2(a) in
[17], it is quite straightforward to formulate IVA as in Figure
2(b), resembling the relationship between MCCA for multiset
data analysis and MCCA for multimodal data fusion. IVA
generalizes MCCA to the case where both SOS and HOS
are taken into account and where the demixing matrix is not
constrained to be orthogonal [16]. In a recent study, IVA was
used to fuse EEG, functional, and sMRI in the manner of
Figure 2(b) [71].
Different implementation algorithms of IVA involve
the assumption of specific SCV distributions. The most
widely used ones include IVA-L [15], which assumes that
each SCV follows a multivariate Laplace distribution that
is isotropic and possesses no second-order correlation,
and IVA-G [17], which assumes each SCV is multivariate
Gaussian distributed. In applications like speech recognition [15], the second-order information across data sets
may be minimal. However, in most neurophysiological
applications, a second-order dependence across data sets is
likely. IVA has been utilized in a number of applications,
such as group fMRI analysis [16] and concurrent multidimensional EEG and unidimensional KIN data analysis
[19]. Recently, IVA-GL, using the IVA-G solution to initialize the IVA-L algorithm, has been recommended for
fMRI applications [18]. The implementation first takes
into account full second-order dependence among entries
of an SCV by IVA-G. The estimates of the demixing matrices are then employed to initialize IVA-L, and HOS are
taken into account by assuming a Laplacian distribution
for each entry within an SCV. However, since it is a twostep method, IVA-GL may not work very well if the SCVs
are not Laplacian distributed. To summarize the different
methods, Table 1 provides a comprehensive summary of
all aforementioned JBSS methods in terms of different categories, motivations, optimization criteria, solutions, software, and related major works.
Numerical simulations
In this section, we provide numerical simulations to illustrate
the applicability and the performance of several fundamental
JBSS methods. Without loss of generality, we generate three
data sets (M = 3) and use the JBSS methods related to the first
formulation [shown in Figure 2(a)] for demonstration. The
studied representative methods include jICA, MCCA, JDIAGSOS, MCCA+jICA, IVA-G, IVA-GL, and sICA. In the following simulation, FastICA is employed as the ICA algorithm [61],
and the SSQCOR cost function is used to implement MCCA
due to its robustness [17], [22].
data generation
The following six sources were generated and analyzed as
in [48]:
IEEE Signal Processing Magazine
s1 = sin (0.015n) + cos (0.005n)
s 2 = 2 cos (0.08n) sin (0.006n)
s 3 = ECG, s 4 = EMG
s 5 = 1.5 cos (0.01n) sin (0.5n)
s 6 = 1.5sin [0.025 (n + 63)] sin (0.2n),
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