*Bounty: 50*

*Bounty: 50*

Assume I have two time series, A(t) and B(t). I **do know** the exact signals in A(t). I want to find out when a known signal in A(t) is causing a signal in B(t).

- In case of pure noise the data in both is distributed around the mean (not necessarily Gaussian, as there is systematic noise even after pre-whitening/detrending).
- Any signal will increase in a time-dependent manner, peak, and decrease.
- A(t) can cause a signal in B(t), but will not always cause one.
- If it causes a signal, the impact is immediate (at lag 0).
- Both have independent (systematic) noise properties.
- If there is no signal in A(t) there is no signal in B(t) expected

(however there might be systematic noise that looks similar).

My current analyses:

- First, I look at the data from the time series point-of-view, and perform auto-correlation, cross-correlation and rolling correlation analyses. I use the signal-to-noise ratio in e.g. the cross-correlation as a handle on ‘significance’ (not in a statistical sense).
- Second, I try to explore other statistical options. Assume I do know when in time the signal in A(t) starts and ends. I extract the time points in B(t) at these times as a seperate sample, B*.
- I perform a T-test to see whether B* data is distributed around the mean of B(t).
- I perform a binominal test to see whether B* data is randomly distributed around the mean, or biased in any direction.

I.e. if B* data contains a signal, both tests will reject the Null Hypotheses. However, this somewhat decreases my signal-to-noise ratio, as it includes the ramping parts of the signal, which are not that far off the mean.

My questions are:

- are any of these tests redundant?
- how do I retrieve values of ‘statistical significance’ from the cross-correlation etc?
- are there any other possibilities to analyse this data that I am not thinking of (either as time series or as seperate sample B*)?