Signal Processing - May 2016 - 92
applied to the PLS-derived LVs. After these two steps, it is
ensured that the extracted components are maximally correlated across data sets while still explaining the variation
within each individual data set.
Biset/Bimodal using Hos
for data fusion problems especially in the medical realm. For
instance, it has been employed to link fMRI and genetic data
[41], and EEG and SNP data [42].
IC-PLS
IC-PLS is designed to combine the advantages of PLS and ICA
[28]. The goal of IC-PLS is to extract maximally independent
Both PLS and CCA exploit SOS and can only extract uncorresources from two data sets while keeping the correspondlated LVs. This lack of uniqueness may impede interpretations
ing sources correlated across the two data sets (M = 2).
of extracted LVs in real applications [39], which can sometimes
be circumvented by exploiting HOS. If the data are not strictly
Thus, the following conditions should be satisfied simultanemultivariate Gaussian, SOS (i.e., correlation and covariance)
ously: first, the covariance between the corresponding LVs
will be insufficient for obtaining a unique LV model. ICA
across the two data sets should be maximized; second, the
attempts to find mutually statistically independent sources
independence (e.g., non-Gaussianity) of the extracted LVs
that are linearly superimposed to create
within each data set should be maximized.
multivariate data. Since independence is a
This leads to a multi-objective optimizaICA has been empirically
much stronger condition than uncorrelatedtion problem, encapsulating three maxishown to be very useful
ness, algorithms for ICA, which typically
mization objectives. A solution using an
in many biomedical
employ HOS criteria related to information
approximate Newton iteration approach
theory and/or non-Gaussianity, can obtain
has been suggested [28]. Similar to pICA,
applications, suggesting
unique solutions. ICA has been empiricare
must be taken during weight adjusta use for incorporating
cally shown to be very useful in many bioment
to balance the three terms during
ICA into existing JBSS
medical applications [1], suggesting a use
cost function optimization. IC-PLS is formethods.
for incorporating ICA into existing JBSS
mulated as in Figure 2(a). It emphasizes
methods. The second category of JBSS
the role of the source components S [m] ,
methods, represented by the parallel ICA (pICA) [41] and ICbut pays little attention to the mixing matrices A [m] . It has
PLS [28], were designed for this purpose.
been applied to corticomuscular coupling analysis, extracting
maximally independent source pairs from concurrent EEG
and EMG data in order of relevance [28].
pICA
pICA was developed to identify maximally independent components from each of two modalities (M = 2) and connections
mUltiset/mUltimodal using sos
between them through enhancing intrinsic interrelationships
The previously described methods were designed for analysis
[40]. PICA largely relaxes the rigid assumption of sharing the
of biset or bimodal data. However, more than two data sets
same mixing matrix in the joint ICA (jICA) model (M $ 2)
from different modalities (i.e., multimodal) or from the same
modality (i.e., multiset) are frequently available. Thus a third
[26], which is also designed to fuse multimodal data (but will
category of JBSS methods exploiting SOS have been timely
be subsequently described in the section "Multiset/Multideveloped, including multiset CCA (MCCA) [20], joint multimodal Using HOS"). It maximizes the independence within
modal statistical framework (JMSF) [27], EEMD-MCCA [25],
each modality using an entropy-based cost function while
and joint diagonalization of second-order cumulant matrices
also identifying intermodality correlations through adding a
(JDIAG-SOS) [44]. These concepts are expanded upon below.
squared correlation term. In the pICA model, there are a total
of three terms that need to be optimized simultaneously-
two of them relate to maximizing the independence of sourcMULTISET CCA
es for the two modalities separately, while the third term is
MCCA ( M $ 2 ) extends the theory of CCA (M = 2) [60] to
the determination of the relationship between the two modalimore than two random vectors and identifies canonical varities. During optimization, adaptively adjusting the learning
ates that summarize the correlation structure among mulrates is critically important for balancing the three aspects in
tiple random vectors by linear transformations [20]. Unlike
the cost function [41].
CCA, where correlation between two canonical variates is
pICA is formulated as in Figure 2(b). The vertical dimenmaximized, MCCA aims to optimize an objective function
sions Pm of the data sets X [m] may represent, for example,
to make the canonical variates achieve the maximum overall
correlation [20]. Recently, it has been shown that MCCA can
the number of healthy subjects and patients with disease.
be used to achieve JBSS [22], allowing for jointly analyzing
The assumption is that the patterns of intersubject modulamultiset data. In such an approach, MCCA is implemented
tion across two modalities are similar or covarying. This
in multiple stages, such that one group of canonical varicould be reflected by the correlation between the correates is obtained at each stage through optimizing the objecsponding columns of modulation profiles A [m] . The associtive function with respect to a set of transformation weight
ated components or sources S [m] from different modalities
vectors [22]. Thus the lth group of canonical variates from
may provide the interpretation from a different view of brain
M data sets X [m] (m = 1, 2, f, M) is defined as w [l m] T X [m]
function or structure. Therefore, pICA is particularly suited
92
IEEE Signal Processing Magazine
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May 2016
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Table of Contents for the Digital Edition of Signal Processing - May 2016
Signal Processing - May 2016 - Cover1
Signal Processing - May 2016 - Cover2
Signal Processing - May 2016 - 1
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Signal Processing - May 2016 - 120
Signal Processing - May 2016 - Cover3
Signal Processing - May 2016 - Cover4
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