# mean of a fluctuating time series - what is the uncertainy?

#### ata5d

##### New Member
Hello. I very much appreciate your attention to my problem.

I have lists of data generated by my computer simulations collected over simulation time. These data are essentially time series showing stationary fluctuations. These fluctuations aren't noise, but rather the consequence of the physical nature of the problem.

I can easily compute the mean of these time series. I hope that by averaging over a suitable length of time, I can average over the natural fluctuations exhibited by the data to obtain an accurate estimation of the series' central tendency. I also would like to quantify the uncertainty in this calculation of the mean. The uncertainty in my mean calculation is not due to experimental error, or a small sample set, but due to the fact that I can only average over a few periods of the oscillations.

I'm stumped because I'm certain that neither the standard deviation nor the confidence interval will suit my purposes. Neither the standard deviation nor the confidence interval "understands" the data's fluctuation period relative to the length of time over which I average. Please take a moment to think about this last sentence because it is the crux of my problem.

What calculation should I instead perform to quantify the uncertainty of the mean? Thank you.

#### JohnM

##### TS Contributor
Does the fluctuation have a reliable repeating pattern or cycle length?

If so, could you identify points throughout the series, but based on their relative position within the cycle, and use the variation among the points at that same relative location to get a confidence interval?

#### ata5d

##### New Member
JohnM said:
Does the fluctuation have a reliable repeating pattern or cycle length?

If so, could you identify points throughout the series, but based on their relative position within the cycle, and use the variation among the points at that same relative location to get a confidence interval?
Hi John - I understand your suggestion. Unfortunately the fluctuations are very random and there is no similarity I could identify among different instants of time. Good idea though.

#### JohnM

##### TS Contributor
If it's as random as you suggest, then I would histogram it to see if it follows a normal distribution, or at least try to determine which distribution "fits" the best, then use that as the basis for confidence intervals.

#### ata5d

##### New Member
JohnM said:
If it's as random as you suggest, then I would histogram it to see if it follows a normal distribution, or at least try to determine which distribution "fits" the best, then use that as the basis for confidence intervals.
Yes, I've drawn the histogram and the distribution appears Gaussian. I used this to calculate a confidence interval, and I achieve a very tight confidence interval because I have 10,000 data points tracing out about 2 periods of the strongest fluctuation. However I know that I have more than enough data points (samples) and the real problem is this problem of the fluctuation time scales versus the time window over which I average. Therefore the confidence interval isn't giving me the confidence assessment I seek.