Recently I come up with a problem the observe variables contain 4 continuous variables and a discrete variable. I want to model it with HMM, but I don't know implement it. Do you have know any related papers about this?
Getting here a bit late, but maybe for future reference.
I have worked on this topic and here is a related paper (Disclamer: I am the author of the paper): Hybrid hidden Markov model for mixed continuous/continuous and discrete/continuous data modeling, E. Epaillard, N. Bouguila, MMSP'15
It learns the HMM parameters in a hybrid way, via an EM-algorithm. It basically computes some of the intermediate parameters of the EM-algorithm based only on the discrete part of the data, some other only from the continuous part of the data. Then it combines these parameters together to get the update of the hybrid-HMM.
I found some similarities in this problem to the problem of multi-stream HMMs. Here is the main reference about these special types of HMMs: O. Missaoui, H. Frigui, P. Gader, "Multi-stream continuous hidden Markov models with application to landmine detection", EURASIP J. Adv. Sig. Proc., 2013.
The theory behind multi-stream HMMs is not straightforward and I found myself getting good results with a much simpler method. On synthetic data, we could even get good results with different types of continuous emissions mixed with some discrete variables.