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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.

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Have you tried resampling to a regularly-spaced grid and then looking at a time-frequency decomposition of the signals? I say this because your description implies that the spectral components are time-dependent. However, not knowing anything more about $\alpha$ or the individual functions makes it impossible to know if changing $\alpha$ will allow you to manipulate the time-frequency characteristics of the signals and affect the correlation. –  AnonSubmitter85 Jul 23 '13 at 19:18
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