When two distributions overlap, how to separate one distribution from the mixture distribution if I the other distribution is known? I have a data which can be classified into two groups. As you see, Figure(a) shows that they are easily classified into group A and B. However, sometimes they are overlapped and it is impossible set a line between group A and B. Figure(b) shows that two groups are overlapped. Fortunately, I have a group A's single distribution and Figure(c) shows it.

And Figure(d) shows Group A's distribution along X and y axises.
Even though Group A and B are overlapped, I think there got to be a way to know the distribution of Group B, because we know the group A's distribution and it does not change well.
Whould you plase tell me how to solve this problem? I don't expect exact solution, just give me a idea or a proper mathmatical approach or something I have to study.
I hope many genius people help me.
Thank you:)


In UV-VIS spectroscopy, a 1-dimensional variant of this is common. Absorbance peaks often overlap. The spectra are “deconvoluted” by peak fitting. You have to be a bit careful about the lingo; many spectroscopists use this term without really knowing why it’s called this way. After all the inherent (routine, tacit) transforms to get to an absorbance spectrum it turns out that in spectroscopy the absorbance peaks are additive. So, it’s a matter of finding models to describe the peaks and then performing a fit routine to find the relative contributions.
Something similar may work here. You say you know A so you can put that distribution as constant. You probably have a sense of how to describe B. Then it is a matter of fitting the parameters for B so that the observed data comes out.
The 2-dimensionality makes it a bit more difficult, there are some additional things to consider (such as covariance). I have never had to do it that way so I can’t help you there. If you can modify the problem into “2x 1-D”, that would definitely be easier. But you require independence and I can’t tell if your data/problem allows that.
