I have two time series, $M_1(t)$ and $M_2(t)$, which can be seen as measurements of two different physical sources, $s_1(t)$ and $s_2(t)$. $M_1$ only depends on $s_1$, whereas $M_2$ depends on both. The dependence is known up to a certain parameter, like this:
$M_1(t) = f_1(s_1(t), \alpha) \\ M_2(t) = f_2(s_1(t), \alpha) + g(s_2(t), \gamma)$
My aim is to remove the $s_1$ dependence from $M_2$ as far as possible, by adjusting the parameter $\alpha$. My problem, I believe is to use the correct metric to measure the association between the time series. Using the correlation coefficient does not work, since at some point, the periods when the signals are correlated and anti-correlated will cancel. This is not the point I am looking for.
Note that $s_2$ and $s_1$ are also correlated, so it will not be possible to remove the correlation between the time series entirely, only minimize it. Also, the sampling intervals are highly irregular.