The clustering behavior of sliding windows

Here's the abstract of this earlier paper:

> Given the recent explosion of interest in streaming data and online algorithms, clustering of time series subsequences, extracted via a sliding window, has received much attention. In this work we make a surprising claim. Clustering of time series subsequences is meaningless. More concretely, clusters extracted from these time series are forced to obey a certain constraint that is pathologically unlikely to be satisfied by any dataset, and because of this, the clusters extracted by any clustering algorithm are essentially random. While this constraint can be intuitively demonstrated with a simple illustration and is simple to prove, it has never appeared in the literature. We can justify calling our claim surprising, since it invalidates the contribution of dozens of previously published papers. We will justify our claim with a theorem, illustrative examples, and a comprehensive set of experiments on reimplementations of previous work. Although the primary contribution of our work is to draw attention to the fact that an apparent solution to an important problem is incorrect and should no longer be used, we also introduce a novel method which, based on the concept of time series motifs, is able to meaningfully cluster subsequences on some time series datasets.

Several commenters here seem to ask "okay, so then what's the right way to cluster windows of timeseries??" Perhaps the final sentence of this abstract suggests a solution in that direction?

Absolutely mindblowing. A strong claim, but they present very strong arguments. Thanks for sharing.

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Imagine you know that there are some repeating patterns in your dataset, e.g. heartbeats in an EKG. If you take two windows that happen to contain a heartbeat at the same position, the corresponding vectors will be close, and if the positions are different, the vectors should become more dissimilar the more the alignment is off.

Then if you create a number of clusters equal to the window size, you might expect that each cluster will correspond to one of the possible positions of the heartbeat within the window. But somehow that's not what happens...

My understanding is that people would mostly try this approach when the samples are periodic/regular.

> Unless the samples are nicely periodic

This is assumed from context.

... if I get it right the whole idea of clustering sliding windows is wrong, the question of "what you should do instead?" is an interesting one.

I'd imagine two answers are: (1) for time series which are somewhat periodic you might cut out individual days or weeks and try to cluster them for each other, (2) for time series which are intermittent you might create some definition of an "event" (an earthquake, or a particle passing through a detector) and then cluster events and maybe (3) for something episodic such as "heart rate during a workout" you would cluster episodes.