I need an estimate of the average value $\bar{X}$ of the data within the shaded area of this graph, and its corresponding confidence interval.

ideal case

In the ideal case, the data would reflect some minor oscillations. I'm tempted to ignore this, and use the $CLT$ so the standard error would be $SE=\frac{S}{\sqrt{N}}$, where $S$ is the sample standard deviation, and $N$ is the total number of data points in the range. But I'm uncomfortable with this approach because I know this is not completely correct.

But the main problem is that sometimes the data is not stationary for the whole range, and can show multiple stationary segments, as shown below.

worst case

These issues invalidate any assumption of normality. I'm struggling to identify a proper method to calculate a single value for the mean and confidence interval for the whole data range. Any ideas?


1 Answer 1


You may consider a nonstationary time series as consisting of a number of time segments that are stationary. You can do it either manually or using methods related to the algorithm discussed here. Once segments are identified, you can estimate the mean, variance and the autocovariance structure of each stationary segment.

The fact that time series is non-stationary does not automatically make the assumption of normality inappropriate. Also note that for a non-stationary time serires the long-run mean is undefined. For example, how would you define the mean of a random walk? The local means of stationary segements we refer to as levels of the process. To estimate levels at any given point of (discrete) time, we use methods appropriate for non-stationary time series, such as those of exponential smoothing.

  • $\begingroup$ Thanks @Hibernating, for pointing me to the article. Will take a look at it. This data corresponds to flow, and we are trying to measure the average and the confidence interval by selecting a time-frame were oscillations have dissipated enough and external processes are not affecting the measurement. Sometimes we don't know the exact effect of these external processes (like in the second graph). Also, I can't see how the oscillation component can be considered piece-wise stationary. I see what you are saying about random walks, but I still can't reconcile the answer with my needs. $\endgroup$
    – JAponte
    Jan 10, 2014 at 20:20

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