IEEE ENGINEERING IN MEDICINE AND BIOLOGY
MAGAZINE
MARCH/APRIL 2006
79
FUNCTIONAL MAGNETIC RESONANCE IMAGING
Unmixing fMRI
with Independent
Component Analysis
Using ICA to Characterize High-Dimensional
fMRI Data in a Concise Manner.
BY VINCE D. CALHOUN
AND TÜLAY ADALI
©DIGITAL STOCK
0739-5175/06/$20.00©2006IEEE
hat an antithetical mind!—tenderness, rough-
ness—delicacy, coarseness—sentiment, sensual-
ity—soaring and groveling, dirt and deity—all
mixed up in that one compound of inspired clay!
—Lord Byron
Independent component analysis (ICA) is a statistical method
used to discover hidden factors (sources or features) from a set
of measurements or observed data such that the sources are
maximally independent. Typically, it assumes a generative
model where observations are assumed to be linear mixtures
of independent sources, and unlike principal component analy-
sis (PCA), which uncorrelates the data, ICA works with high-
er-order statistics to achieve independence.
An intuitive example of ICA can be given by a scatter-plot of
two independent signals
s
1
and
s
2
. Figure 1(a) shows a plot of
the two independent signals (
s
1
,
s
2
) in a scatter plot. Figure 1(b)
and (c) shows the projections for PCA and ICA, respectively,
for a linear mixture of
s
1
and
s
2
. PCA finds the orthogonal
vectors
u
1
,
u
2
but does not find independent vectors. In con-
trast, ICA is able to find the independent vectors
a
1
,
a
2
of the
linear mixed signals (
s
1
,
s
2
) and is thus able to restore the orig-
inal sources.
A typical ICA model assumes that the source signals are not
observable, are statistically independent, and are non-
Gaussian, with an unknown but linear mixing process.
Consider an observed
M-
dimensional random vector denoted
by
x
=
(
x
1
,...
x
M
)
T
, which is generated by the ICA model:
x
=
As
,(
1
)
where
s
=
[
s
1
,
s
2
s
N
]
T
is an
N-
dimensional vector whose
elements are assumed independent sources and
A
M
×
N
is an
unknown mixing matrix. Typically
M
>
=
N
, so
A
is usually
of full rank. The goal of ICA is to estimate an unmixing
matrix
W
N
×
M
such that
y
[defined in (2)] is a good approxima-
tion to the true sources:
s
.
y
=
Wx
(
2
)
ICA is hence an approach to solving the blind source
separation problem, which traditionally addresses the solu-
tion of the cocktail party problem in which several people
are speaking simultaneously in the same room. The prob-
lem is to separate the voices of the different speakers by
using recordings of several microphones in the room [2].
The basic ICA model for blind source separation is shown
in Figure 2.
Popular approaches for performing ICA include maximiza-
tion of information transfer, which is equivalent to maximum
likelihood estimation, maximization of non-Gaussianity,
mutual information minimization, and tensorial methods. The
most commonly used ICA algorithms are Infomax [3],
FastICA [4], and joint approximate diagonalization of eigen-
matrices (JADE) [5]. The original Infomax algorithm for
blind separation by [3] is better suited to estimation of super-
Gaussian sources. To overcome this limitation, techniques