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I have a time series including one hundred thousand observations. In order to make the structure more visible I tried to apply a Moving Average for smoothing.

However there are two issues. First the time series is non-stationary due to a decreasing variance over time. Second the points in time are not equally spaced. Because of this there are a few observations at the beginning of the displayed time interval and more crowded areas at the middle plus end of it.

My question is the following. Can these issues cause any problems while applying the MA? Because I got the impression that the result is very sensitive to a change of the subset size.

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Can these issues cause any problems while applying the MA?

Yes, if you want to communicate the changing variance, the moving average usually will cause that feature to be lost. Just think about what the moving average of $\frac{1}{t} \sin(t)$ would look like if the window for averaging is larger than $2π$. Or consider the moving average of noise with a changing variance.

The non-equidistant sampling is also a problem as naïvely applying the moving average to it will cause the effective averaging window to be larger where the sampling is coarser. I suggest to take a look at kernel smoothing.

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