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I am looking for some distance (or similarity) measure between two sequences, possibly of different lengths. Conceptually, I would like a measure with a property that $[3,1,5]$ is similar to $[3,1,4,5]$ (just one insertion), and $d([1,2,3], [1,2,4]) < d([1,2,3], [1,2,99])$ because 3 is closer to 4 than to 99. I do not any more specific properties in my mind.

I thought of the Levenstein distance first, but its property that substitution between any two values is constant is not what I desire. I also thought of the Manhattan distance but it does not allow, as far as I know, the sequences have different lengths.

I admit this is a crude description of what I intend, but I do not have much more specific requirement for now. It will be great if you could point out some functions to measure the distance between sequences of different lengths. Just a link to a paper or websites would also be helpful.

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As Stephan Kolassa commented, the dynamic time warping is promising in this case.

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