Siddiqi et al 2009 Reduced-Rank Hidden Markov Models

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Summary

This paper introduces Reduced-Rank Hidden Markov Models (RR-HMMs). RR-HMMs are similar to standard HMMs, except the rank of the transition matrix is less than the number of hidden states. Thus the dynamics evolve in a subspace of the hidden state probability space.

Method

Sample singles, doubles, and triples from the from the observed output of the RR-HMM. Then let:

The learning algorithm uses a singular value decomposition (SVD) of the correlation matrix between past and future observations. The algorithm is borrowed from Hsu et al 2009, with no change for the reduced-rank case.

${\hat {b}}_{1}$ is the initial state distribution, ${\hat {b}}_{\infty }$ is the final state distribution, and ${\hat {B}}_{x}$ is the transition matrix when x is observed. m is the rank of the reduced state space. Note that $X^{+}$ denotes the Moore-Penrose pseudo-inverse of the matrix $X$.

Inference can be performed using the model parameters:

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