Algorithm - finding the order of HMM from observations

Question

I am given a data that consists of N sequences of variable lengths of hidden variables and their corresponding observed variables (i.e., I have both the hidden variables and the observed variables for each sequence).

Is there a way to find the order K of the "best" HMM model for this data, without exhaustive search? (justified heuristics are also legitimate).

Answer 1

I think there may be a confusion about the word "order": A first-order HMM is an HMM which transition matrix depends only on the previous state. A 2nd-order HMM is an HMM which transition matrix depends only on the 2 previous states, and so on. As the order increases, the theory gets "thicker" (i.e., the equations) and very few implementations of such complex models are implemented in mainstream libraries. A search on your favorite browser with the keywords "second-order HMM" will bring you to meaningful readings about these models.

If by order you mean the number of states, and with the assumptions that you use single distributions assigned to each state (i.e., you do not use HMMs with mixtures of distributions) then, indeed the only hyperparameter you need to tune is the number of states.

You can estimate the optimal number of states using criteria such as the Bayesian Information Criterion, the Akaike Information Criterion, or the Minimum Message Length Criterion which are based on model's likelihood computations. Usually, the use of these criteria necessitates training multiple models in order to be able to compute some meaningful likelihood results to compare.

If you just want to get a blur idea of a good K value that may not be optimal, a k-means clustering combined with the percentage of variance explained can do the trick: if X clusters explain more than, let say, 90% of the variance of the observations in your training set then, going with an X-state HMM is a good start. The 3 first criteria are interesting because they include a penalty term that goes with the number of parameters of the model and can therefore prevent some overfitting.

These criteria can also be applied when one uses mixture-based HMMs, in which case there are more hyperparameters to tune (i.e., the number of states and the number of component of the mixture models).