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Is there a statistical measure for how much a variable fluctuates over time? For example a noise signal fluctuates a lot. However, if you would sort all values of the signal in time, you would have a signal that does not fluctuate at all. My first idea was to use the variance of the derivative, but I’m not sure whether this is the best measure. Another idea might be to use information entropy, but I’m not sure how to approach it that way.

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Variance of the first derivative would mean looking for variations in derivative of your variable. Rather, I would recommend taking derivative of your variable with respect to time and see the results. This result would be rate of change of your variable with respect to time.

rate_of_change = d(variable)/d(time)

Note : This rate_of_change should be calculated by taking difference between consecutive values of the variable and then dividing this number by difference between time of those two samples(take both the differences in same order). This way the rate_of_change will reflect instantaneous fluctuations in the variable over time. Further you can explore taking population standard deviation of the rate of change and then taking a moving standard deviation as well.

Additionally one can take moving average of the variable across certain time duration and then do the same rate of change analysis as mentioned earlier. This new rate_of_change values would represent fluctuations in the variable of interest over predefined duration of time. Larger the duration for moving average, more global would be the rate of change in variable.

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Sunil, can you differentiate between global rate of change and recent/instantaneous rate of change? The latter will tend to stabilize in stationary processes, but to grow with integrated or nearly-integrated processes. – Alexis Jul 18 '14 at 15:53
@Alexis The answer is intended to show instantaneous rate of change, since the OP is interested in finding fluctuations in a variable over time. Added some clarification in the answer. – Sunil Jul 21 '14 at 7:15

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