Mathematics in Independent Component Analysis

44 Chapter 1. Statistical machine learning of biomedical data
(a) general linear model analysis (b) one independent component
Figure 1.22: Comparison of model-based and model-free analysis of a word-perception fMRI
experiment. (a) illustrates the result of a regression-based analysis, which shows actitity mostly
in the auditive cortex. (b) is a single component extracted by ICA, which is corresponds to a
word-detection network.
with a resolution of 3 × 3 × 4 mm. A single 2d-slice is analyzed, which is oriented parallel to
the calcarine fissure. Photic stimulation was performed using an 8 Hz alternating checkerboard
stimulus with a central fixation point and a dark background.
At first, we show an example result using spatial ICA. We perform a dimension reduction using
PCA to n = 6 dimensions, which still contained 99.77% of the eigenvalues. Then, we applied
HessianICA with K = 100 Hessians evaluated at randomly chosen samples, see section 1.2.1 and
Theis (2004a). The resulting 6-dimensional sources are interpreted as the 6 component maps
that encode the data set. The columns of the mixing matrix contain the relative contribution of
each component map to the mixtures at the given time point, so they represent the components’
time courses. The maps together with the corresponding time courses are shown in figure 1.23.
A single highly task-related component (#4) is found, which after a shift of 4s has a high crosscorrelation
with the block-based stimulus (cc = 0.89). Other component maps encode artifacts,
e.g. in the interstitial brain region and other background activity.
We then tested the usefulness of taking into account additional information contained in
the data set such as the spatiotemporal dependencies. For this, we analyzed the data using
spatiotemporal BSS as described in section 1.3.2 and Theis et al. (2005b), Theis et al. (2007b),
see chapter 9. In order to make things more challenging, only 4 components were to be extracted
from the data, with preprocessing either by PCA only or by the slightly more general singular
value decomposition, a necessary preprocessing for spatiotemporal BSS. We based the algorithms
on joint diagonalization, for which K = 10 autocorrelation matrices were used, both for spatial
and temporal decorrelation, weighted equally (α = 0.5). Although the data was reduced to only
4 components, stSOBI was able to extract the stimulus component very well, with a equally
high crosscorrelation of cc = 0.89. We compared this result with some established algorithms
for blind fMRI analysis by discussing the single component that is maximally autocorrelated