# Hidden Markov Model and volatile Matrix A

Suppose that we have a Hidden Markov Model (HMM) in which my Transition Distribution (also known as Matrix A) between hidden states dramatically change during the time; However, my Emission Distribution (also known as Matrix B) still the same during all times.

How can I possibly handle the volatile behavior of Matrix A during time?

By the way, I have fully observed the hidden states. Hence, I compute Matrix A, by counting the transitions between different states.

I would be very thankful if someone could possibly introduce me a similar paper work or give me a hint on how to handle this intermittence of matrix A during time?

Thank you very much :)

• how do you have fully observed hidden states? – Taylor Aug 9 '16 at 14:39
• How many training sequences do you observe? Just one, or do you have a panel? If you only observe one training sequence, I don't see how you'd get a good estimate of a time-varying transition matrix. – Adrian Aug 9 '16 at 17:34
• @Taylor As a matter of fact, It is just assumed that I have seen the hidden states. Imagine someone has just given the labels of hidden states to me! I want to use this model (which I learn through count of the jumps between states) to solve some other questions such as Decoding and Evaluation problems! – user126608 Aug 9 '16 at 18:58
• @Adrian Well, my sequence is comprised of four states which depict different time epochs. There also exists four different labels which can be assigned to hidden states. Let's say the labels come from the set of L={a,b,c,d} and one sample sequence can be a => b => b => a. Somehow I have the history of jumps between the hidden states through time which help me build my Transition Distributions (Matrix A). But the Matrix A itself changes over my jumps. For example imagine if in the first jump P(b|a)=0.45, however, in the second jump P(b|a) may be 0.34. – user126608 Aug 9 '16 at 19:19
• Suppose your training sequence has length $T$, meaning there are $T - 1$ transition matrices to estimate. If you observe only one training sequence, you won't be able to do much. Having multiple iid sequences would help. – Adrian Aug 9 '16 at 19:49

In sum, when we talk about a model, we may need to think how much free parameters in there and do we have sufficient data to reliably estimate these parameters. A matrix changes over time would cause too many free parameters and may not be useful in real world. (suppose we have $\Omega$ hidden stages, the transition matrix has $\Omega \times (\Omega-1)$ free parameters, prob need to sum to $1.0$, in addition, if sequence length is $N$ then there are $\Omega \times (\Omega-1) \times N$ free parameters), and we are not counting the free parameters in emission matrix. Such level of complexity is not practical / parameters cannot reliably estimated in real world with limited data.