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I am trying to find an appropriate distance measure that reflects the differences of the vectors seen in the image below: The green vector is compared with the blue one and the orange one.

Most of the distance measures (like the Euclidean for example) would yield the same value despite the "phase shift". Therefore the straight line would look "the same" as the blue and the orange line!

The features are ordered but I am also interested in the case they were unordered.

I was also thinking to "split" the vectors in the middle and treat them as bags of two subvectors (like in a multi-instance setting) but then I would not know how to combine the two resulting distances in one.

So I guess my question is two-fold:

  • Is there a distance measure that highlights these type of value symmetries in a single instance setting?
  • Is there a way to combine multiple distances (in a multi-instance setting) in a way that symmetry (and therefore order) is highlighted in the final distance result?

Thank you

enter image description here

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Is this an issue of features? From the explanation, it seems that the feature you are considering is distance. So the results are as expected (from a "bag of single feature distances").

You mention that order is important, so that can added as a feature. Or since you mentioned "phase shift", time (and/or frequency) -- giving you a time series. In both cases, you then can bring in all kinds of tools. You could start with regression and go from there to more complex methods (simple sometimes is best).

That would allow you to yield similarity metrics between single or multiple vectors.

Vectors contained in a sliding time window vs. population, is something you maybe can also consider, we did a solution like that once.

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