To answer this question, you need to formalize dissimilarity. A simple dissimilarity measure could be a geometrical distance, e.g., the Euclidean distance between two time-series. This dissimilarity has certain properties that can be advantageous or disadvantageous depending on your research question. For example, if your dissimilarity is supposed to be invariant to scaling, Euclidean distance is typically a poor choice (unless you normalize but this imposes additional assumptions). Among other approaches, dissimilarity can be formalized in terms of shape, based on parameters of a hypothesized generating model, or complexity of the time series. The R package PDC provides complexity-based dissimilarity calculation and clustering, and also provides p values for a null hypothesis of identical underlying generating permutation distributions. The R package TSclust was recently updated and provides (among PDC) a number of approaches to time series dissimilarities.

References:

Andreas M. Brandmaier (2014). pdc: Permutation Distribution

Clustering. R package version 0.5.

Pablo Montero Manso and José Antonio Vilar (2014). TSclust: Time

series clustering utilities. R package version 1.2.

http://CRAN.R-project.org/package=TSclust

To answer this question, you need to formalize dissimilarity. A simple dissimilarity measure could be a geometrical distance, e.g., the Euclidean distance between two time-series. This dissimilarity has certain properties that can be advantageous or disadvantageous depending on your research question. For example, if your dissimilarity is supposed to be invariant to scaling, Euclidean distance is typically a poor choice (unless you normalize but this imposes additional assumptions). Among other approaches, dissimilarity can be formalized in terms of shape, based on parameters of a hypothesized generating model, or complexity of the time series. The R package PDC provides complexity-based dissimilarity calculation and clustering, and also provides p values for a null hypothesis of identical underlying generating permutation distributions. The R package TSclust was recently updated and provides (among PDC) a number of approaches to time series dissimilarities.

References:

Andreas M. Brandmaier (2014). pdc: Permutation Distribution

Clustering. R package version 0.5.

Pablo Montero Manso and José Antonio Vilar (2014). TSclust: Time

series clustering utilities. R package version 1.2.

http://CRAN.R-project.org/package=TSclust

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before running head first into the "null ritual" and the "hunt for p-values", please have a look at the article below ("Mindless statistics" by G. Gigerenzer ; deep and deliciously written)

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There exist different method for measuring the similarity between time series. In the attached file I report some ideas that could be applied to compare signals and test for their difference.

Let’s focus on the method based on the phase-locking detection. The original series, after some opportune filtering, can be Hilbert-transformed into analytic-signal. Hilbert transformed series has the same amplitude and frequency content as the original real data and includes phase information that depends on the phase of the original data. The instantaneous components (phase and frequency) of the signal are therefore obtained.

Phase difference between the instantaneous phases of the two Hilbert-transformed data are easily computable. Some index of this phase difference can be defined. For example, one can consider information-based measures, such as “sample-entropy”, or spectrum-based measures, such as phase coherence, or even other measures…

The application of this method within running windows would face, locally, the issue of non-stationarity.

Surrogate data can be used to test the null hypothesis of independent pairs of signals. Only values that depart significantly from what would be expected for independent oscillators can be considered as revealing the presence of phase locking. We can obtain surrogate data which preserve the power spectrum of the instantaneous frequency, by means of the iterative Amplitude Adjusted Fourier Transform method (iAAFT) and take the surrogate data as representing the null hypothesis.

Under certain conditions, the Weibull distribution would fit for the vector of the phase difference.

I have gone through the same situation. But in my case, I have time series with a change point. My interest is to identified if it is the same (in temporal terms).

There are different approaches to this issue. Thinking in linear terms cross-correlation and coherence (Fourier) coefficient. Nonlinear approaches would be phase synch...by means of Hilbert Transf. or other synch. parameters...

I agree with Andreas Brandmaier's comment. First, you should to define what do you meant by dissimilarity measure. Second, you should use an appropriate procedure for evaluating the significance of the obtained value. For instance, the following two papers propose a subsampling procedure to compare the autocorrelation structure and the extremal behaviour for two (non)-independent time series, respectively.

Alonso, A.M. and Maharaj, E.A. (2006) Comparison of time series using subsampling, Computational Statistics and Data Analysis, 50 (10), 2589-2599.

Alonso, A.M., de Zea Bermudez, P. and Scotto, M. (2014) Comparing Generalized Pareto models fitted to extreme observations: An application to the largest temperatures in Spain, Stochastic Environmental Research and Risk Assessment, In press.

Additionally, for the first paper, you can obtain the routines at

http://halweb.uc3m.es/esp/Personal/personas/amalonso/esp/addenda.html

Not exactly an answer to your question, but let me bring to your attention my patent for characterizing system behavior from time-series (see attached).

Another approach to compute the differences between two time series is moving window root-mean-square. RMS can be run for both series separately. This way, you can compare the similarities in energy (gain) level of time series. You may vary the window length for best resolution.