Mixed Hidden Markov Models

Hidden Markov models (HMMs) are a useful tool for capturing the behavior of overdispersed, autocorrelated data. These models have been applied to many different problems, including speech recognition, precipitation modeling, and gene finding and profiling. Typically, HMMs are applied to individual stochastic processes; HMMs for simultaneously modeling multiple processes—as in the longitudinal data setting—have not been widely studied. In this article I present a new class of models, mixed HMMs (MHMMs), where I use both covariates and random effects to capture differences among processes. I define the models using the framework of generalized linear mixed models and discuss their interpretation. I then provide algorithms for parameter estimation and illustrate the properties of the estimators via a simulation study. Finally, to demonstrate the practical uses of MHMMs, I provide an application to data on lesion counts in multiple sclerosis patients. I show that my model, while parsimonious, can describe the heterogeneity among such patients.

[1]  J. Nash.  Compact Numerical Methods for Computers , 2018 .

[2]  L. R. Rabiner,et al.  An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition , 1983, The Bell System Technical Journal.

[3]  James P. Hughes,et al.  A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena , 1994 .

[4]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[5]  J A Frank,et al.  Time series for modelling counts from a relapsing-remitting disease: application to modelling disease activity in multiple sclerosis. , 1994, Statistics in medicine.

[6]  J. Nelder,et al.  Hierarchical Generalized Linear Models , 1996 .

[7]  A. Kuk,et al.  MAXIMUM LIKELIHOOD ESTIMATION FOR PROBIT-LINEAR MIXED MODELS WITH CORRELATED RANDOM EFFECTS , 1997 .

[8]  C. McCulloch.  Maximum Likelihood Algorithms for Generalized Linear Mixed Models , 1997 .

[9]  Anders Krogh,et al.  Chapter 4 - An introduction to hidden Markov models for biological sequences , 1998 .

[10]  J. Rost,et al.  Applications of Latent Trait and Latent Class Models in the Social Sciences , 1998 .

[11]  G. Ebers,et al.  Randomised double-blind placebo-controlled study of interferon β-1a in relapsing/remitting multiple sclerosis , 1998, The Lancet.

[12]  Murray A. Cameron,et al.  Hidden Markov chains in generalized linear models , 1998 .

[13]  Keith Humphreys,et al.  The Latent Markov Chain with Multivariate Random Effects , 1998 .

[14]  Anthony Y. C. Kuk,et al.  Pointwise and functional approximations in Monte Carlo maximum likelihood estimation , 1999, Stat. Comput..

[15]  D. Li,et al.  Magnetic resonance imaging results of the PRISMS trial: a randomized, double-blind, placebo-controlled study of interferon-beta1a in relapsing-remitting multiple sclerosis. Prevention of Relapses and Disability by Interferon-beta1a Subcutaneously in Multiple Sclerosis. , 1999, Annals of neurology.

[16]  Martin L. Puterman,et al.  Markov Poisson regression models for discrete time series. Part 1: Methodology , 1999 .

[17]  D. Paty,et al.  Magnetic resonance imaging results of the PRISMS trial: A randomized, double‐blind, placebo‐controlled study of interferon‐β1a in relapsing‐remitting multiple sclerosis , 1999 .

[18]  A. Norheim,et al.  A randomised , double-blind, placebo-controlled study , 2001 .

[19]  S. R. Searle,et al.  Generalized, Linear, and Mixed Models , 2005 .

[20]  Richard A. C. Hughes,et al.  PRISMS-4: Long-term efficacy of interferon-&bgr;-1a in relapsing MS , 2001, Neurology.

[21]  Rachel J. Mackay,et al.  Estimating the order of a hidden markov model , 2002 .

[22]  M. Filippi,et al.  MRI metrics as surrogate markers for clinical relapse rate in relapsing-remitting MS patients , 2002, Neurology.

[23]  Howard Seltman,et al.  Hidden Markov Models for Analysis of Biological Rhythm Data , 2002 .

[24]  R. J. MacKay,et al.  HIDDEN MARKOV MODELS: MULTIPLE PROCESSES AND MODEL SELECTION , 2003 .

[25]  A. Petkau,et al.  Application of hidden Markov models to multiple sclerosis lesion count data , 2005, Statistics in medicine.

[26]  B. Silverman,et al.  Maximum Penalized Likelihood Estimation , 2006 .

[27]  Rachel MacKay Altman.  A variance component test for mixed hidden Markov models , 2008 .

[28]  Jörn Dannemann,et al.  The likelihood ratio test for hidden Markov models in two-sample problems , 2008, Comput. Stat. Data Anal..