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.
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].
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.