# Correctly determine alpha for repeating statistical test in non-independent (interpolated) timeseries data

Problem

I have timeseries of measurements of which I also have determined the standard deviation of the noise. My goal is to determine whether or not the measurement is significantly different from 0 at every measured point with a net alpha of a.

Explanation

Now the timeseries are always of the same duration, but are not always sampled at the same interval. I have measurements that are sampled K=n_i . k (n_i = 1 to n_max ) times during the period of the measurement. Because of our setup however I always get N interpolated timepoints. N > n_max . k, so these N timepoints are never independent measurements.

If they were N independent measurements and I wanted them to be statistically different from 0 at every measured point with a net alpha of a, I would adjust the alpha to a_adjusted = a^(1/N). So that testing N times leads to a_adjusted^(N) = a.

But how do I adjust a when I do N tests but there are really only K independent measurements? I still perform the test N times but since the outcome of the tests are not independent I think I should apply a correction for that. Should I just make a_adjusted = a^(1/K) or some ratio of N and k?

Example

I measure the output flow velocity of a cyclic pump over one pump cycle. Because of the way it works, the measuring apparatus I use can not output independent measurements. From the settings I can determine how many independent measurements are in the data, say 4.5 measurements. I also know how many timepoints are in the output of the apparatus, say 10 calculated time points. I also know the standard deviation of noise that is in one independent measurement, say 2. I then calculate the confidence interval for each timepoint. But I that is the 'incorrect' 95% confidence interval because it assumes the timepoints to be independent.

Example measurement:

These settings are not always the same for various reasons, but I would still like to test with the same 'net' alpha value for any setting. That way I can compare results between outputs with different settings. That is why I am trying to correct this alpha.

Similar questions found on Cross Validated:

This question has a similar problem, except this person introduces the dependency himself and clusters the timeseries, seems more like averaging.

This question is very similar, but more broad. It asks 'is this correctable' instead of how to correct it in this specific situation.

This question also deals with dependent data, but in this case the different output points have a different number of measurements from which they are reconstructed. In my case this is equal for all points.

This question is related, but focuses on correction for family wise error in multiple testing of dependent data.

This question is very similar too, except it wants to compare two timeseries which are interpolated / dependent. It was asked two years ago and has not received a comment or answer. Maybe Cross Validated is not the place to get answers about this.