I have a time-series and I'd like to find a way to measure how 'shaky' and basically have a statistic (or couple of statistics) that could describe that.
The standard-deviation isn't suitable because in the case of a very shaky time series but small variations, the standard deviation could still turn out to be very low.
Any suggestions on approaches that would be suitable here ?
I interpret „being shaky“ as: The differences between nearby samples are constantly swapping their sign.
This leads me to the suggestion: Model it using ARMA (autoregressive moving averages) models. These are models which check for differential or integrative relationships in the data. You can also apply these models on the absolute value of some kind of numerical derivative of the time series.
These work best when you already substracted trend and periodicity, so that the expectation of the trend is zero and there is no periodicity involved. Removing the trend can be done by fitting a suitable (its up to you to define that word) function to the time series. Periodicities can be found as peaks in the autocorrelation function or some kind of Fourier or wavelet transforms.
Another way could be just to normalize the variance of time series, or a combination of different approaches. The question is always: Can you interpret your mathematical tools in terms of the language of your data.