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Can time series having multiple lengths be clustered using the k-medoids algorithm. I am essentially looking for a way to find a representative pattern from a set of time series using the k-medoids centroid.

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Not a big deal, but DTW is not a metric, it is only a measure.

Paper [a] does exactly what you want.

However, if you have one long time series, as opposed to many short time series, they you should look at time series snippets [b].

[a] https://www.cs.ucr.edu/~eamonn/ICDM_2014_DTW_average.pdf [b] https://www.cs.ucr.edu/~eamonn/Time_Series_Snippets_10pages.pdf

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You can adapt it via a suitable distance. Unlike k-means, k-medoid centers are chosen directly from data. So, you don't have to implement addition operation between data samples. It just remains to use a well-defined distance, i.e. $d(x_i,x_j)$. Several distances for time series with different lengths exist, e.g. Dynamic Time Warping. This library in R has many of them and also implements K-medoids algorithm with a lot of distance options.

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