# Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify the extent of periodicity between of a time series (for comparison with other time series), considering the time series is almost periodic. By almost periodic I mean: if I were to take one time series and segment (consider it is possible and allowed to do this) its periods, there would be a fairly strong cross-correlation (say >8) between any two of these segmented periods within one time series.

The simplest metric I can think of is average of cross-correlations between pairs of segmented periods within the time series. I would expect the time series with a higher average cross-correlation to be more periodic than a time series with a lower average cross-correlation measure. This seems intuitively fair, because if we were to take the average cross-correlation measure for a perfectly periodic time series say a sinusoid, I would expect the cross-correlation between any two of this periodic segments to be one (as they are the same) and hence find the average cross-correlation among a set of such segmented periods to be 1 (which is the maximum value in this metric).

I would also like to know if there is a way to do it using fractal analyses, considering only one time-scale resolution (self similarity i.e. of the periods in one time scale and not in multiple time scale resolutions)? I understand that the point of use of fractal analysis is that it considers multiple time-scales, but I want to understand its use and where it stands from the perspective of single time-scale resolution. I also want to see if there is a generalization of the the measurement of self similarity when time scales are varied differently (single time scale being an extreme case where it is not varied at all).

I am currently using fractal dimension measures as heuristics to quantify the smoothness/roughness of a time series. If I could find a metric using fractal analysis to also quantify extents of periodicity of time series, it would complement the other time series analyses I am performing.

Here's a link to a csv with two vectors that I would like to compare the extent of periodicity of, for example.

• Can you please address the following to make your question answerable? 1) Do you want to compare two time series or evaluate one time series? 2) The cross correlation is always between −1 and 1, unless you do not normalise, in which case the value itself is meaningless. 3) The whole point of fractal analysis is that it considers multiple scales. If you consider a single scale, you can use simpler methods. Why are you so keen on fractal analysis anyway? 4) Can you roughly say, why you want to do this? Sep 2 '15 at 19:37
• @Wrzlprmft I have improved the problem description as you suggested.
– np20
Sep 3 '15 at 6:54

Consider the following methods:

• The autocorrelation of a time series for a given time delay $τ$ is nothing but the cross-correlation of a time series with a respectively delayed version of itself. The autocorrelation is 1 for $τ=0$ and also for $τ=θ$, if the time series is perfectly periodic with a period length $θ$.

To quantify the periodicness of a time series, consider the height of the highest non-trivial maximum of the autocorrelation function. (Non-trivial means $τ≠0$.)

• Periodograms such as the discrete Fourier transform yield, roughly speaking, to which extent certain frequencies (in sinusoidal form) are present in the time series.

To quantify the periodicness of a time series, you can measure the area under the most prominent peak in the periodogram and the corresponding subharmonics. (Without the latter your measure would be biased, yielding higher periodicness for sinusoidal time series.)

Periodogram-based approaches are more robust to certain effects like frequency drifts (which only broaden the peaks in the periodogram but may destroy the autocorrelation). Whether that’s a good thing for your application is something you have to decide.

Note that the above methods come with all sorts of caveats and open parameters and thus I would strongly advise against blindly applying them and for looking into the literature.