# unified estimation of discrete Markov Model

Background

I have a multivariate dataset, say M x N, where M is the number of variables and N is the number of samples. Now, the pattern of dependencies between the M variables changes across the N samples i.e. the pattern of dependencies is non-stationary.

Problem

I want to use a 'discrete' Markov Model to characterize this change in pattern of dependencies across samples. One way of doing this is by estimating the pattern of dependencies for each sample and using for e.g. k-means clustering to group these patterns into a small number of symbols. Then, I could use Markov Modelling to estimate transition probabilities between symbols.

My question is: can I do all the above in a single unified model i.e. a model which combines the Markov Modelling with estimating patterns of dependencies and clustering them into a small number of symbols. If so, wouldn't this be preferable to the sequential approach outlined above?

Any suggestions/thoughts/ideas welcome!

• I don't understand the statement that the pattern between dependencies is non-stationary. I take it that the N samples are ordered, and that you're talking about transitioning from one M-dimensional state to another? – jlimahaverford Sep 22 '15 at 1:39
• yes, that's right. however, the 'state' is actually defined not by the M values at a given sample, but by the M x M matrix of dependencies at that sample (where 'dependencies' can be considered as a measure of interaction). – Nitin Sep 22 '15 at 2:02
• to give an example, suppose I record activity from M brain regions. now, I want to model how the 'pattern of interaction' between brain regions changes w.r.t. to time, so I estimate an M x M matrix of dependencies at each of the N samples. hope that makes sense. – Nitin Sep 22 '15 at 2:06
• Sorry going to ask for more clarification. Is this matrix supposed to model transitioning from one vector to another (By multiplication). If not then it what sense are they dependencies. You said that the matrix is the state, so at every point you expect a new matrix? Are these observed? – jlimahaverford Sep 22 '15 at 2:14
• I might not be using the right term with 'dependencies'. I just mean a 'measure of interaction', for e.g. a correlation value. so, if there were M brain regions, the 'state' at a given sample would be the M x M symmetric matrix of correlations of each region to each other region. as for whether a new matrix is expected at every sample, yes they are. – Nitin Sep 22 '15 at 2:21