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?
-- 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