# Measures of Variability and/or Polarity Changes for Time Series

I am trying to measure the variability or "choppiness" of a time series but I am aware that standard measures of standard deviation do not apply due to auto-correlations between the observations.

I am specifically trying to differentiate between series that "flip" sign more than other time series, e.g. consider the two time series

1,1,1,1,-1,-1,-1,-1

1,-1,1,-1,1,-1,1,-1

In both, the sum is the same, the average is the same, the max/min are the same, the standard deviation is the same, yet one sequence switches sign only once while the other switches 7 times. Is there a way to measure this? (note I am looking at data this highly non-stationary)

The only idea I can come up with to measure this "choppiness" is to draw a linear line connecting the first number in the series and the last and measure the absolute area between the time series and this straight line. This at least will measure the spread around an ideal path from beginning to end with a smaller choppier series having a smaller value than a larger choppier series, and a perfectly smooth series being close to zero.

• I see no problem in calculating a SD for time series. but its use and interpretation will not be the same as for independent data. Looking at SD of changes ($\Delta y = y_t - y_{t-\tau}$, say) to me would be less ad hoc and more flexible than your proposal. In fact this is just a bridge to the autocovariance function once you vary $\tau \ge 1$. What is generally true is that it is often difficult to separate "trend" from other variability and the chicken and egg question is that adequate description often requires an adequate model of a time series, rather than precedes such a model. – Nick Cox Oct 27 '18 at 17:23
• @NickCox I see your point, but I have also measure SD of the first difference and there was little to no information I could extract from that. However,I do agree with your chicken/egg description of the problem. I am basically trying to determine the variability around a given trend. – guy Oct 27 '18 at 17:44
• Your series are very short, which makes the (sample) uncertainty on anything measuring the rate of flips rather high. For "long" series (e.g., 100 or more samples), have you thought of calculating covariance, correlation (coefficient), or spectral density --and if so, why have you discarded that route? If nonstationary, the result will depend on the pivot (reference) sample point, therefore a series of coefficients arises rather than a single measure. – Lucozade Oct 27 '18 at 20:43
• @Lucozade Thanks for the advice. In reality, my series are much longer (each series is hundreds of points long). I don't know how auto-covariance compares between different time series. I am not familiar with spectral density though i don't think that is applicable in my case as my time series cannot be represented in the frequency domain. I am not sure what you mean but the last sentence. – guy Oct 27 '18 at 21:16