Is ICA appropriate for separating mixed signals when all source signals are NOT detectable at all sensors?

Question

A generic implementation of ICA for the separation of a mixture of $N$ signals into their $M$ constituent components requires that the signals be assumed to be a linear instantaneous mixture of the sources. Every description of ICA that I have come across seems to take for granted the fact that all $M$ sources are present to some extent in all $N$ signal mixtures.

My question is, what if the $M$ sources are only present in some but not all of the signal mixtures?

Does this scenario violate the fundamental assumptions necessary for ICA to be able to separate these signals? (Assume, for the sake of argument, that we are dealing with an overcomplete or complete system ($N>M$ or $N=M$), and that each of the $M$ source signals are in fact statistically independent from each other).

The implementation I am considering using ICA for, in which this situation arises, is the following: I have data from 4 different types of sensors, each with a different number of channels. Specifically, I have 24 channels of EEG data, 3 channels of electrooculographic (EOG) data, 4 channels of EMG data, and 1 channel of ECG data. All data is recorded simultaneously.

I would like to identify the contributions of the ECG, EMG, and EOG signals within the EEG data so that I can remove them. The expectation is that EMG+ECG+EOG signals will be picked up by the EEG sensors, but not vice versa. Also, EOG and EMG will likely contaminate each other and be contaminated by ECG, but ECG will probably be pretty isolated from all the other signals. Also, I'm assuming that where mixing does occur, it is linear and instantaneous.

My intuition tells me that, hypothetically, ICA should be smart enough to return mixing filters with very small (close to 0) coefficients to account for the lack of contribution of sources to a mixed signal. But I am worried that something about the way that ICA demixes the signals inherently enforces the expectation that all sources will be present in all mixtures. The implementation I am using is FastICA, which is a projection pursuit-based approach.

Answer 1

You should be fine, zeros in the mixing matrix are not a problem....and theoretically it should converge even quicker than if all sources existed in all sensors.

Answer 2

"My question is, what if the M sources are only present in some but not all of the signal mixtures?"

This is the same as saying that in your mixing matrix you will have some zeros. When M=N, I don't think it matters if you just make sure that the mixing matrix is non-singular. I'm not 100% sure though. But you could make a simple 3-by-3 toy experiment with one or more zeros in the mixing matrix to get some hands-on. If you read up on FastICA I bet you will find in the requirements placed on the mixing matrix that it has to be non-singular.

Answer 3

Your intuition is fine.

But you may want to give a shot to a novel tecnique (GUSSS, Guided Underdetermined Source Signal Separation), described in this paper. The intuition is to extract an already known signal from another one. Let's say we have your EEG as $x$ and ECG as $s$: we can see $x$ as a mixture of the signal $s$ plus another mixture $\tilde{s}$ so that $$x=c_ss+\tilde{s}$$ where $c_s$ is the weight of $s$ into $x$.

If we sum $$x_s = w_xx+w_ss = w_x(c_ss+\tilde{s})+w_ss = w_x\tilde{s}+ks$$ where $k=(w_xc_s+w_s)$, we can extract how $s$ is into $x$ (as $c_s$) since feeding $\left[\begin{array}{c} x\\ x_s \end{array}\right]$ into ICA gives you

$$ A=\left[\begin{array}{cc} 1 & c_s\\ w_x & k \end{array}\right], \; S = \left[\begin{array}{c} \tilde{s} \\ s \end{array}\right] $$

When you have $c_p$ you can subtract the signal from the EEG, repeating the process for every sensor.