Compare two signals, find times when they are affected by the same processes
You can model your system as a hypothesis testing problem. For example, let $M_1$ and $M_2$ denote the two systems which are correlated. Let the observed signal in $M_1$ and $M_2$ be given as
\begin{align}
y_1(t) = s_1(t) + n_1(t) \\
y_2(t) = s_2(t) + n_2(t),
\end{align}
respectively, where $n_i(t) \sim \mathcal{N}(0, \sigma_i^2)$, for $i = \{1,2\}$, and $s_i(t)$, for $i = \{1,2\}$, is the original signal. Since the two systems are correlated, you can model them as
\begin{align}
\begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix} = \begin{bmatrix} 1 &amp; \rho \\ \rho &amp; 1 \end{bmatrix} \begin{bmatrix} s_1(t) \\ s_2(t) \end{bmatrix} + \begin{bmatrix} n_1(t) \\ n_2(t) \end{bmatrix},
\end{align}
where $-1 &lt; \rho &lt; 1$ denotes the correlation constant between the two systems. Now, we can model the system as a hypothesis testing problem as follows:
\begin{align}
\mathcal{H}_0: \rho = 0, &amp;\quad \mbox{uncorrelated model} \\
\mathcal{H}_1: \rho \ne 0, &amp;\quad \mbox{correlated model}.
\end{align}
If you assume that $s_1(t)$ and $s_2(t)$ are known, then find the maximum likelihood estimate (MLE) of $\rho$ under $\mathcal{H}_1$. If you assume $s_1(t)$ and $s_2(t)$ are unknown, then use generalized likelihood ratio test, i.e., find the MLE of $s_i(t)$ under each hypothesis and replace $s_i(t)$ with its MLE.