Evaluating clusters of first-order Markov chains I clustered my dataset of several thousand first-order Markov chains into about 10 clusters.
Is there some recommended way how I can evaluate these clusters and find out what the items in the clusters share and how they differ from other clusters? So I can make statement like  "Processes in cluster A tend to stay in state Y once they get there, which is not true for processes in other clusters."
The transition matrices of those Markov chains are too large to just "look and see". They are relatively sparse, if that can help.
My idea was to take all the transition matrices in a cluster, sum them and plot it as intensity in a picture (in a scale from 0 to 255). Is there something more "professional" I should try out?
 A: To make a statement about the steady state behaviour of each cluster you could compute the steady state distributions of each transition matrix by eigenvectors, then compare box-plots by cluster. You're likely to run into issues in the computation of steady state without applying some sort of smoothing first.
How are you clustering the transition matrices? If it were me, I'd apply additive smoothing to each row then take the centered log-ratio transform of each row then flatten the matrices.
If you're clustering with K-means or a variant, you could analyze the normalized cluster centers. Or just pick a few observations from each cluster and analyze them.
A: First, to get an idea, are your matrices of dimension 105 x 105, corresponding with the applications that you mention? When you say 'stay in state Y' does that mean stick around application Y?
Then, I would assume that outcomes such as "Processes in cluster A tend to stay in state Y once they get there, which is not true for processes in other clusters" are a bit too fine-grained with just 10 clusters. Have you tried a clustering of the application domain -- if I understand correctly you could cluster the 105 applications based on user behaviour. Next, have you looked at simple presence of users rather than transition, i.e. look at profiles of users across the 105 applications? It sounds as if you could use Pearson coefficient between user profiles; either on clusters of applications, or on the applications themselves. This could perhaps be extended towards transitions between applications, but currently I feel there is a huge mismatch between the number of clusters and the type of outcome you are interested in.
