Hello. The usual formulation of a HMM is (pi0,A,B) where pi0 is the initial probability distribution, A is the transition matrix and B is teh observation matrix (for quantised observations). All three quantities relate to a finite state space of disctinct discrete states. It is not clear to me how you can combine different HMMs with the same state space (I assume this is what you mean) since they will be inconsistent.

If you mean that they have the same state space AND the same transition probability matrices, then how is it possible to have different probability densities for the observations? This is not consistent with the HMM formalism.

It is not clear what you mean by "objective". This term does not seem to relate to the HMM but maybe to the subsequent optimisation problem that you might wish to solve, e.g. the estimation of the states given some noisy observations.

Threre are some methods for reducing the number of states in a HMM (state aggregation) but I don't think this is what you are after. Nonetheless, here are some references that you might be able to use. The second one of these deals with aggregation of a process formed by combining two different Markov chains.

Article A Survey of Hidden Markov Model Tracking with Emphasis on OTHR

Article Evaluation and Estimation of Various Markov Models with Appl...

Dear Dr. Pulford

I'm so appreciative of your interest.

I analysis two different kind of drug doses with HMM and I found three matrices individually for these drugs in two HMM. Now, I want to describe influence of both of them simultaneously by combining them. Is it possible?

Best regards

You can certainly combine them.  There are a variety of ways to do this.  You need to clearly define what you hope to achieve by combining them.

One way to do this is to construct a larger HMM so that when each drug is present in isolation, the model can reduce to the original HMM (i.e. when you set the amount of all drugs but one to zero for each drug, you can get each of the original models).  There are any number of larger models containing the structures of your original models, so you have to think about how you want to do this, and how big the larger structure should be.

If you have two Markov chains with state spaces S1 and S2 and transition probability matrices P1 and P2, then the "combined" HMM with state space S1 x S2 (cartesian product) has TPM P1 (x) P2 (kronecker product of P1 and P2). This approach is theoretically OK but will get impractical quickly since you are "multiplying up" or "vectorising" the state spaces at each stage. In some situations you can then aggregate the Markov chain by looking at combined states that are, for example, sums of several vector states. In addition to the link I provided before, you should refer to textbooks on markov chain theory, like Kemeny and Snell, or Isaacson and Madsen.

I have a similar problem "How can we combine two semi-Markov chains with similar states with different transition matrix into one to obtain collective results". Later I have to show the theatrical and simulations results.