Mathematics in Independent Component Analysis

254 Chapter 19. LNCS 3195:977-984, 2004
3D Spatial Analysis of fMRI Data on a Word Perception Task 979
iteration [6]) algorithm have been used so far. While the extended Infomax algorithm
is expected to perform slightly better on real data due to its adaptive
nature, FastICA does not depend on educated guesses about the probability
density distribution of the unknown source signals. In this paper we choose to
utilize FastICA because of its low demands on computational power.
2 Results
First, we will present the implementation of the algorithm we used. Then we will
discuss an example of an event-designed experiment and its BSS based analysis
where we were able to identify a network of brain areas which could not be
detected using classic regression methods.
2.1 Method
To implement spatial ICA for fMRI data, every three-dimensional fMRI image
is considered as a single mixture of underlying independent components. The
rows of every image matrix have to be concatenated to a single row-vector and
with these image-vectors the mixture matrix X is constructed.
For FastICA the second order correlation in the data has to be eliminated by
a “whitening” preprocessing. This is done using a principal component analysis
(PCA) step prior to the FastICA algorithm. In this step a data reduction can be
applied by omitting principal components (PC) with a low variance in the signal
reconstruction process. However, this should be handled with care as valuable
high order statistical information can be contained in these low variance PCs.
The maximal variations in the timetrends of the supposed word-detection ICs
in our example account only for 0.7 % of the measured fMRI Signal.
The FastICA algorithm calculates the de-mixing matrix W = A −1 . Then the
underlying sources S can be reconstructed as well as the original mixing matrix
A. ThecolumnsofA represent the time-courses of the underlying sources which
are contained in the rows of S. To display the ICs the rows of S have to be
converted back to three-dimensional image matrixes.
As noted before because of the high noise present in fMRI data the ICA
problem will always be under-determined or over-complete. As FastICA cannot
separate more components than the number of mixtures available, the resulting
IC will always be composed of a noise part and the “real” IC superimposed on
that noise. This can be compensated by individually de-noising the IC. As a rule
of thumb we decided that to be considered a noise signal the value has to be
below 10 times the mean variance in the IC which corresponds to a standard
deviation of about 3.
2.2 Example: Analysis of an Event-Based Experiment
This experiment was part of a study to investigate the network involved in
the perception of speech and the decoding of auditory speech stimuli. Therefor