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I have some time series data I'm working with, where each at each step I have a sequence. For example I could have a bunch of graphs looking like the left figure, followed by an anomalous piece of data looking like the right figure.

enter image description here

I want to construct an online method where I'm able to say the anomalous graph is clearly from a different distribution or similar approach which only uses the recent data, say the previous 20 graphs. Is there a nice way to do this or some nice literature for this type of problem? I've tried some optimal transport methods to compare distributions and simple stats, but I want something that is fairly robust. Any comments welcome!

The anomalies can be anything - for instance the right-hand image is showing jagged behaviour rather than smooth, but generally the anomalies are not visually subtle, the distributions will look a lot different.

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    $\begingroup$ What kind of anomalies do you have in mind (it's not clear for me from the image)? $\endgroup$
    – Tim
    Apr 30 '18 at 12:11
  • $\begingroup$ Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. $\endgroup$ Apr 30 '18 at 13:00
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There is no general solution to the anomaly problem, but you can address specific cases. For example.

  • Are the "previous 20" all trusted cases (i.e. not anomalies)? If they are short pieces of time series all of equal length, you could do a functional PCA, hopefully work with one or two principal components, and then assess if your new curve departs significantly from the others on the basis of the first and/or second PC scores.
  • Are you primarily distinguishing smooth functions from jagged ones? I recently used double exponential smoothing in a case like this. With smooth data, exponential smoothing will optimize with a beta value close to 1 and an alpha close to 0 (it basically tries to grab first differences of the data as it attempts to estimate the derivative of the function). When you apply those parameter values to a jagged portion of the series, your forecast values will depart markedly from your actual values -- so the sequence of anomalies you get will be an indication that you have departed smooth territory.
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