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For a threeway data array being a sample of size (number of voxels of the brain image) of
a random matrix of dimensions (T= number of Time points , S=number of Subjects),
different twoways ICA can be investigated. It is possible to look for an Independent Components
in the spatial dimension associated to
unmixing matrices. We
call it SingleICA by analogy with single subject ICA. Note this is done with classical ICA on a
data matrix, the unmixing matrices are in fact the unmixing vectors of length the product .

(9) 
Intuitively one supposes the unmixing vector as ``different" from subject to subject but the
sources should be similar i.e. we hope the subcomponents (unmixing for each subjects) to be
similar. This should be considered as a random subject model but here is considered only as a
fixed subject effect as in fact we may want to look at the variation a posteriori. At this
stage in a similar way of nested PCA one could look at the SVD decomposition of the unmixing
vector as bilinear forms (
) and probably retain the best low rank
approximation of the unmixing matrix.
Figure 3:
SingleICA of the detrended time
series and subject scaled of a 12 subjects fMRI data: unmixing matrix split by subject with best
mean timecourse lagged correlation to paradigm; the mean timecourse is in black.

One gets very similar results as with PTAk for the correlation with the paradigm and
spatial representation. For the subject representation on figure (4) is slightly
different even if the interpretation of it remains the same. The variances within group seem
similar not like for PTAk where the Female group had a smaller variance.
Figure 4:
SingleICA of the detrended time
series and subject scaled of a 12 subjects fMRI data: brain component associated to the best
correlated average timecourse to paradigm; is the Jones
& Sibson approximation of the Negentropy for this component.

Figure 5:
SingleICA smooth SVD [4] of the unmixing matrix (fig.4) of ICA6; the
first component well correlated with the paradigm is associated with no apparent difference for
sex; on the scree plot is for cumulated and an estimation of a risk in choice dimension.

It is also possible to look for combined subjectbrain Independent Components associated to
the same timecourse mixing vector:

(10) 
It is called MultipleICA as the ``traditional" spatial (brain) component is multiple. One would
hope the component to reflect a ``narrow" distribution over subjects of this traditional spatial
component. This last model correspond more to the intuitive consideration that the subjects
responses to an fMRI experiment are independent identically distributed and we are looking for a
random variation of activation pattern hopefully i.e. the random variation between subjects is
accessible at each voxel. Nonetheless this last method is biased towards nonGaussian subject
samples, i.e. as well as looking for spatial distribution the least Gaussian the method will try
to separate the subjects and with small samples of subjects this may destroy any hope of getting
similar spatial distributions.
Figure 6:
MultipleICA standardised mean over subjects of ICA20 and unmixing timecourse.

Figure 7:
MultipleICA ICA20; the scale is different for each subject.

Next: Towards rank1ICA
Up: tr01dl1
Previous: Negentropy criterion
Didier Leibovici
20010906