# 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.

## 1 Answer

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.

• Do I then understand correctly that first PCA is performed and a certain number of "significant" components extracted following Minka's approach, the rest is discarded, and then ICA is used to rotate the retained components? May 17, 2016 at 10:14
• Yeah, that's right. May 17, 2016 at 10:14
• I see. So the question is more about the number of components in PCA than about the number of components in ICA. Because ICA produces as many components as it receives as an input. (+1 and congratulations with passing 2k.) May 17, 2016 at 10:16
• It's a common pattern with ICA software, although there are some other methods. PCA usually comes first though. May 17, 2016 at 10:21
• Oh, and cheers btw. May 17, 2016 at 10:23