clusteringdistance-functions
I clustered a data set using PAM with a euclidean distance metric and a pearson correlation distance metric. The average silhouette value of the correlation clusters is higher at most points than the euclidean one, but their maximas are at the same number of clusters, and peaks occur at similar cluster numbers.
Is there any way to compare the two on the same scale?
I'm not sure why you are doing things this way. It seems to me that the first algorithm found initially 2 clusters (from where you chose one to be clustered by the 2nd algorithm). Why don't you cluster the whole data set with both algorithms and compare the clusterings?
Its very difficult to compare one cluster vs 3 clusters on the same data set. To have one clusters makes little sense, and one should reasonably expect the output of the K-Means criterion to be much smaller when you have 3 clusters (so you can't use this for comparison).
Now what's the silhouette of the 2nd algorithm generating 3 clusters? is it in average closer >=0.5? if not the meaning of this clustering is debatable.
Answering question 1:
You critic of DTW is met by introducing global constraints to the warping path. This effectively restrains both computational effort (since warping paths which are not allowed do not have to be computed) and prevents pathological warping.
Therefore the answer is: DTW with global constraints
There are several variants of such constraints such as the Sakoe-Chiba band and the Itakura Parallelogram as you can see in the following image. The image originates from a presentation, which is available online in a presentation done by Chotirat Ratanamahatana and Eamonn Keogh.
Another possibly relevant time series distance measure is:
LCSS - Longest Common Subsequence - has been originally developed to analyse string similarity but can also be used for numerical time series.
Best Answer
I don't think Silhouette scores with different metrics can be made comparable.
Some metrics have range $[0;\infty]$. Others only $[0;1]$ - most likely, the resulting silhouette values will be different.
Even Euclidean and squared Euclidean - which agree on what is "most similar" - will yield a different silhouette.