Q:

Evaluating time series clustering through EM/GMM

I perform clustering of time series into k=[2,N] number of clusters by using either DTW+kmedoids or DTW+single linkage+hierarchical clustering (HC), as advised in a previous post:  Dynamic Time Warping Clustering
Regarding the evaluation of optimum number of clusters, I want to use expectation–maximization (EM) Gaussian Mixture Models (GMM) and determine the k that maximizes the log-likelihood for each approach.
My questions are: 

What should be the input dataset in EM/GMM? The DTW similarity distance
or the raw dataset? 
Can the initializations in EM/GMM be the clusters
centers of either the kmedoids or randomly selected seeds from the HC
clusters?
Should I run k-fold cross-validation of the input dataset with the EM/GMM and return the average log-likelihood value? 

A:

Gaussian Mixture Modeling assumes ypur input data are coordinates in $R^d$ and contain Gaussian-shaped clusters. I don't think you can use this on time series.
Also, GMM is a clustering approach on its own, and I don't see how you could use that to evaluate other clusterings.

