Hidden Semi-Markov Models

Abstract

As an extension to the popular hidden Markov model (HMM), a hidden semi-Markov model (HSMM) allows the underlying stochastic process to be a semi-Markov chain. Each state has variable duration and a number of observations being produced while in the state. This makes it suitable for use in a wider range of applications. Its forwardbackward algorithms can be used to estimate/update the model parameters, determine the predicted, filtered and smoothed probabilities, evaluate goodness of an observation sequence fitting to the model, and find the best state sequence of the underlying stochastic process. Since the HSMM was initially introduced in 1980 for machine recognition of speech, it has been applied in thirty scientific and engineering areas, such as speech recognition/synthesis, human activity recognition/prediction, handwriting recognition, functional MRI brain mapping, and network anomaly detection. There are about three hundred papers published in the literature. An overview of HSMMs is presented in this paper, including modelling, inference, estimation, implementation and applications. It first provides a unified description of various HSMMs and discusses the general issues behind them. The boundary conditions of HSMM are extended. Then the conventional models, including the explicit duration, variable transition, and residential time of HSMM, are discussed. Various duration distributions and observation models are presented. Finally, the paper draws an outline of the applications.

The discrete logic of the Brain - Explicit modelling of Brain State durations in EEG and MEG

“…The underlying process describes the hidden sequence of discrete states with its own dynamical evolution. This influences the continuous process that generates the data or emissions (Rabiner, 1989;Yu, 2015). interpreted as a hidden standard Markov chain on a set of "super-state" nodes ( ) =1 .…”

“…The partial correlation was computed using a general linear model analysis (Friston et al, 1996) with the BS signals as predictors and the full-rank envelop data (prior to dimension reduction and whitening) as response variables. Each BS signal was a binary vector indicating whether the BS is on or off at each point of the most probable BS path, which was estimated using the Viterbi algorithm for HSMM (Yu, 2015). Given the estimated Viterbi path ̂1 : , each element of the [ × ] design matrix is…”

“…That is, given a previously trained model HSMM (or HMM), with estimated parameters ̂, and a new observation sequence 1: (test dataset), what is the most probable BS path that produced the observations? Such a task is solved using the well-known Viterbi algorithm (Yu, 2015). In this simulation experiment, the test dataset was noisier than the training dataset.…”