Statistical distances for time series of distributions I am interested in clustering $N$ time series of $T$ 'values' each. These values are distributions (which can be represented by their cumulative distribution functions (cdf), or their probability density functions (pdf), or more convenient forms such as square-root pdfs yielding a simple spheric geometry).

For comparing given distributions, there is an extensive literature on statistical distances (KL, Hellinger, Wasserstein, and so on), but for comparing given time series of distributions, I am not sure whether there is any literature at all?

Such distances should somehow take into account dynamics information besides the distribution proximity at time t. Ideally, I wish I could have a kind of information factorization similar to this result.

I am wondering if such distances already exist and whether this kind of problem has already been formulated in the literature?

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edit for further precisions and answer to comments:

Thanks for your answer, but dynamic time warping does not suit to my need. This dp technique only captures a rough similarity of shapes by allowing non-linear time distortion. But, it does not amount for the whole information in these time series, e.g. what about the distribution of distortions? Do the distributions of a given time series vary smoothly through time or violently? DTW is not always the solution, for instance, when working with random walks, it does not make sense to use a DTW since there are no time patterns! In this case, the only information is "correlation" and "distribution" (cf. Sklar's theorem in Copula Theory), and the paper cited above.

-- edit 2 Here are the papers that are somehow related to my question:


Predicting the Future Behavior of a Time-Varying Probability Distribution
Clustering on the unit hypersphere using von Mises-Fisher distributions
Unsupervised clustering of multidimensional distributions using earth mover distance
Hilbert space embeddings of conditional distributions with applications to dynamical systems
This is similar to the fundamental information theory problem that Shannon explored.  In that domain, it is framed this way: given two rvs, X and Y, what information does X convey about Y?

An example would be that I create sequence of numbers/bit/letters from a known PDF (X) and you receive a distorted version of those values for which you also know the PDF (Y).  The mutual information is the number of bits that Y communicates about X which can be thought of as a type of correlation.

In the case of time varying PDFs (ie stochastic processes), information theory would just treat the ensemble of rvs as one joint rv and then calculate the mutual information of the joint PDFs.  If the PDFs are iid, then significant simplifications are possible.  Joint Gaussian PDFs also make things easier.

Another information theory concept that might be useful is the entropy rate of a process.  ER is a quantification of the amount of information contained in a process.  Depending on your problem, you might be able to compute the ER for each process and then use those values as realizations of a feature that can be grouped with a clustering algorithm.