Single and Multiple ICA

For a three-way data array $Y$ being a sample of size $N$ (number of voxels of the brain image) of a random matrix $Y'$ of dimensions $T \times S$ (T= number of Time points , S=number of Subjects), different two-ways ICA can be investigated. It is possible to look for an Independent Components in the spatial dimension associated to $time-course \times subject$ unmixing matrices. We call it Single-ICA 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 $T.S$.

$$Y=\sum_i^r B_i \otimes T_{S_i}+ E$$ (9)

Intuitively one supposes the unmixing vector as ``different" from subject to subject but the sources should be similar i.e. we hope the sub-components (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 ( $time-course \times subject$) and probably retain the best low rank approximation of the unmixing matrix.

Figure 3: Single-ICA of the detrended time series and subject scaled of a 12 subjects fMRI data: unmixing matrix split by subject with best mean time-course lagged correlation to paradigm; the mean time-course 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: Single-ICA of the detrended time series and subject scaled of a 12 subjects fMRI data: brain component associated to the best correlated average time-course to paradigm; $2.121$ is the Jones & Sibson approximation of the Negentropy for this component.

Figure 5: Single-ICA 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 $c$ is for cumulated and $r$ an estimation of a risk in choice dimension.

It is also possible to look for combined subject-brain Independent Components associated to the same time-course mixing vector:

$$Y=\sum_i^r B_{S_i} \otimes T_i+ E$$ (10)

It is called Multiple-ICA 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 non-Gaussian 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: Multiple-ICA standardised mean over subjects of ICA20 and unmixing time-course.

Figure 7: Multiple-ICA ICA20; the scale is different for each subject.