Mathematics in Independent Component Analysis

Chapter 19. LNCS 3195:977-984, 2004 253
978 Ingo R. Keck et al.
In this text we compare these analysis techniques in a study of an auditory
task. We show an example where traditional model based methods do not yield
reasonable results. Rather blind source separation techniques have to be used
to get meaningful and interesting results concerning the networks of activations
related to a combined word recognition and motor task.
1.1 Model Based Approach: General Linear Model
The general linear model as a kind of regression analysis has been the classic
way to analyze fMRI data in the past [3]. Basically it uses second order statistics
to find the voxels whose activations correlate best to given time courses. The
measured signal for each voxel in time y =(y(t1), ..., y(tn)) T is written as a
linear combination of independent variables y = Xb + e, with the vector b of
regression coefficients and the matrix X of the independent variables which in
case of an fMRI-analysis consist of the assumed time courses in the data and
additional filters to account for the serial correlation of fMRI data. The residual
error e ought to be minimized. The normal equation X T Xb = X T y of the
problem is solved by b =(X T X) −1 X T y and has a unique solution if XX T has
full rank. Finally a significance test using e is applied to estimate the statistical
significance of the found correlation.
As the model X must be known in advance to calculate b, this method is
called “model-based”. It can be used to test the accuracy of a given model, but
cannot by itself find a better suited model even if one exists.
1.2 Model Free Approach:
BSS Using Independent Component Analysis
In case of fMRI data blind source separation refers to the problem of separating
a given sensor signal, i.e. the fMRI data at the time t
x(t) =A [s(t)+snoise(t)] =
n�
aisi(t)+
i=1
n�
i=1
aisnoise,i(t)
into its underlying n source signals s with ai(t) being its contribution to the
sensor signal, hence its mixing coefficient. A and s are unique except for permutation
and scaling. The functional segregation of the brain [3] closely matches
the requirement of spatially independent sources as assumed in spatial ICA.
The term snoise(t) is the time dependent noise. Unfortunately, in fMRI the noise
level is of the same order of magnitude as the signal, so it has to be taken into
account. As the noise term will depend on time, it can be included as additional
components into the problem. This problem is called “under-determined”
or “over-complete” as the number of independent sources will always exceed the
number of measured sensor signals x(t).
Various algorithms utilizing higher order statistics have been proposed to solve
the BSS problem. In fMRI analysis, mostly the extended Infomax (based on
entropy maximisation [4, 5]) and FastICA (based on negentropy using fix-point