I would start by looking at their cross correlation. Best, David Booth

The cross correlations should be high enough to show closeness of the two series.

I agre with the other answers. Cross-correlation would be the simplest (and probably the first thing) one should try out. Its easily computed either in Pandas (between the dataframes) or in MATLAB, as you would prefer.

Crosscorrelation would have been my first approach, too.

By distance you mean difference? If so, the measure you propose may or may not reflect the nature of the dissimilarity you are interested in. If e.g. one of the signals is consistently larger or smaller than the other for extended periods of time, you will have very few zero-line crossings of their differenc, falsely suggesting a high degree of similarity (if I get your idea correctly). By contrast, identical series which differ only by tiny amounts of white noise would give you a large count, indicating strong dissimilarity.

Depending on what exactly you're after, you may want to consider cross-entropy measures, for example cross-approximate entropy (if you're interested in Matlab code, let me know).

Refs: Kreuzer, M., Hentschke, H., Antkowiak, B., Schwarz, C., Kochs, E.F., & Schneider, G. (2010) Cross-approximate entropy of cortical local field potentials quantifies effects of anesthesia - a pilot study in rats. BMC Neurosci., 11, 122. Skoric, T., Sarenac, O., Milovanovic, B., Japundzic-Zigon, N., & Bajic, D. (2017) On Consistency of Cross-Approximate Entropy in Cardiovascular and Artificial Environments. Complexity, 2017, 1–15.

https://cran.r-project.org/web/packages/dtwclust/vignettes/dtwclust.pdf

https://wiki.smu.edu.sg/18191isss608g1/img_auth.php/f/fd/Time_Series_Clustering_A_Decade_Review.pdf

As previously said you can compare the cross correlation between two time series. May be you can use PCA (SVD) for these two series, which will tell similarity between the disturbances affecting the time series as well.

Emilija B Nikolic-Djoric pretty interesting papers, thanks for sharing!

Another approach based on Euclidean distances that may be useful and which I just remembered (but have never used): the so-called matrix profile. https://towardsdatascience.com/introduction-to-matrix-profiles-5568f3375d90

The measure of similarity needs to be adapted to the time series patterns you need to match.

If you use R, I recommend the following packages where you have a collection of time series metrics you can try on your data:

https://cran.r-project.org/web/packages/TSdist/index.html

https://cran.r-project.org/web/packages/TSclust/index.html

covariance and correlation best measure if you are using R tool then the life become too easy

In addition to the degree of correlation you may wish to check if the time series are cointegrated (if they are nonstationary) to see if they generally move together or if they diverge without bound (no cointegration). The Engle and Granger, Johansen or ARDL bounds testing procedure would be useful in this regard.

I agree with David Eugene Booth and the others about cross-correlations. Don't forget, however, that sample cross-correlations are severely biased and correlated when the time series series are autocorrelated, in particular when there is a unit root (a case where the population counterpart has no meaning). I remember Box & Jenkins recommended removing the unit root by differencing for one series and working with the residuals of a good model for the other series. That way, the biases and correlations of the estimates are reduced.

In the case of the nonstationary time series detrended cross-correlation coefficient may be applied:

https://arxiv.org/pdf/1310.3984.pdf

https://cran.r-project.org/web/packages/DCCA/index.html

If you can divide your time series into different phases, I would suggest measuring probability density function-based distance or cumulative distribution function-based distance as a sign of similarity. I have attached some of the approaches that are well-known in this area.

I would suggest also to check the following paper that has a nice discussion regarding the similarity measure of time series.

Article A comparison of time series similarity measures for classifi...

Johannes Burdack the previous answers gave a good overview of papers in the topic. Let me just add that in R it is available a package called TSDist that integrates several distance measurements on time series - https://cran.r-project.org/web/packages/TSdist/TSdist.pdf.

If you're looking to measure similarity in temporal domain (i.e. phase between time series), you can consider using the Time Alignment Measurement (TAM). The paper is available at Article Time Alignment Measurement for Time Series

Cross correlation.

Hi Johannes,

My suggestion is to put both series on the same footing, that is, estimate an AR model for each series and then calculate the distance between the corresponding estimated AR parameters.

This is in line with the method proposed by Piccolo (1990), "A distance measure for classifying ARIMA models." Journal of Time Series Analysis, Vol. 11, 153-164.

I hope this helps.

Regards

technology has given the flexibility , but the same depends on algorithms, many people have work on this , what matter is the scope of how your predictions evolve

You can extract time series features using tsfeature R package, see the intro: https://cran.r-project.org/web/packages/tsfeatures/vignettes/tsfeatures.html.

Then you can apply similarity matrix you want.

many thanks for the very helpful answers!

I suggest that you check out Matrix Profile [a research contribution by Prof. Eeamon Koeghh & Team of University of California Riverside] and STUMPY [A Python library for Matrix Profile]. You can compare time series subsequences easily using Matrix Profile.