I understand (mostly) how independent component analysis (ICA) works on a set of signals from one population, but I am failing to make it work if my observations (X matrix) includes signals from two different populations (having different means) and I am wondering if it is an inherent limitation of ICA or if I can resolve this. My signals are different than the common type being analyzed in that my source vectors are very short (e.g. 3 values long), but I have many (e.g. 1000's) of observations. Specifically, I am measuring fluorescence in 3 colors where the broad fluorescence signals can "spillover" into other detectors. I have 3 detectors and using 3 different fluorophores on particles. One could think of this as a very poor resolution spectroscopy. Any fluorescent particle could have an arbitrary amount of any of the 3 different fluorophors. However, I have a mixed set of particles which tend to have quite distinct concentrations of fluorophores. For example, one set may generally have lots of fluorophore #1 and little fluorophore #2, while the other set has little of #1 and lots of #2.
Basically, I want to deconvolve the spillover effect to estimate the actual amount of each fluorophore on each particle, rather than having a fraction of signal from one fluorophore add to the signal of another. It seemed like this would be possible for ICA, but after some significant failures (the matrix transform seems to prioritize separating the populations rather than rotating to optimize signal independence), I am wondering if ICA isn't the right solution or if I need to pre-process my data in some other manner to address this.
The graphs show my synthetic data used to demonstrate the problem. Starting with "true" sources (panel A) consisting of a mixture of 2 populations, I created a "true" mixing (A) matrix and calculated the observation (X) matrix (panel B). FastICA estimates the S matrix (shown in panel C) and instead of finding my true sources, it seems to me that it rotates the data to minimize the covariance between the 2 populations.
Looking for any suggestions or insight.