What does the number of independent components produced by ICA depend on? I'm a student working on my bachelor thesis performing
independent component analysis (ICA) on some fMRI data using MELODIC FSL.
I would like to ask some questions regarding the results of ICA. Usually I
obtain about 50 components per subject but this number varies across
subjects and after changing some preprocessing steps as well.
For instance, I obtained a larger number of components (ten more than
before) after applying intensity normalization and slice timing correction
to the data. My question is what does the number of ICs depend on? Why do
some patients have more or less components? If they moved less during the
scan, does this mean they will have less components or more?
I would really appreciate if someone could help me figure it out. It would
be very useful in order to understand the differences between obtaining a
larger or smaller number of components from ICA.
 A: MELODIC FSL dimensionality selection uses a formula derived in the context of Bayesian PCA to estimate the number of components (Minka 2000). The formula (loosely) returns the index of the eigenvalue of the data covariance matrix after which all the remaining eigenvalues are roughly equivalent to each other (and thus understood to represent 'noise').
Understanding the more/less components question requires you to think about the geometry of brain images as vectors. PCA looks for the vectors along which the data varies most. If someone moves their head that's a massive changes versus having a slightly higher activation in some brain region. This would lead to some enormous eigenvalue for movement in that direction, ditto other artifacts. This may have the effect of making other directions look unimportant (relatively small eigenvalues). This may fool the formula into choosing a smaller number of pcs.  
You'll notice there are several 'may's in the above. It's devilishly hard to imagine the effects of different corrections on the eigenspectrum of the data.
