Testing lumpability for marginal discrete hidden Markov models

Colombi, Roberto; Giordano, Sabrina

doi:10.1007/s10182-011-0160-7

Cited by 3 publications

(5 citation statements)

References 16 publications

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“…This theorem is a special case of the results presented in Colombi and Giordano (2011) where the statement (1) is proved to ensure that the marginal process E T of the latent component E U is still a Markov chain. Condition (2) states that observable variables F R (t) at time t depend only on the latent variables E T (t).…”

Section: Multiple Hidden Markov Modelsmentioning

confidence: 89%

“…In this section, we fit different MHMMs on two data sets. The EM algorithm used for estimating the models is described in Colombi and Giordano, 2011, and implemented in the R-package hmmm by The data set of a soft-drink company (Ching et al, 2002, available also in the Rpackage hmmm) consists of a one-year time series of daily sales of soft-drinks: lemon tea, orange juice and apple juice, all with categories: low, medium, high level. Changes in sale outcomes over time can depend on time-varying unobservable factors and we consider an MHMM with two dichotomous latent variables to model these data.…”

Section: Examplesmentioning

confidence: 99%

“…In particular, condition c) implies that E U is a first order Markov chain (Colombi and Giordano, 2011). The marginal process…”

Section: Multiple Hidden Markov Modelsmentioning

confidence: 99%

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Multiple hidden Markov models for categorical time series

Colombi¹,

Giordano²

2015

Journal of Multivariate Analysis

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We introduce multiple hidden Markov models (MHMMs) where an observed multivariate categorical time series depends on an unobservable multivariate Markov chain. MHMMs provide an elegant framework for specifying various independence relationships between multiple discrete time processes. These independencies are interpreted as Markov properties of a mixed graph and a chain graph associated to the latent and observable components of the MHMM, respectively. These Markov properties are also translated into zero restrictions on the parameters of marginal models for the transition probabilities and the distributions of the observable variables given the latent states.

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Section: Multiple Hidden Markov Modelsmentioning

confidence: 89%

Section: Examplesmentioning

confidence: 99%

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Multiple hidden Markov models for categorical time series

Colombi¹,

Giordano²

2015

Journal of Multivariate Analysis

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“…In addition to this, the package can fit hidden Markov models where the conditional distribution of several observable variables and the transition probabilities of the latent chain can be specified by HMM models, see Colombi and Giordano (2011). The hidden.emfit() function computes the ML estimates of the parameters via an EM algorithm, but the current version of the package does not provide standard errors.…”

Section: Discussionmentioning

confidence: 99%

hmmm: AnRPackage for Hierarchical Multinomial Marginal Models

Colombi¹,

Giordano²,

Cazzaro³

2014

J. Stat. Soft.

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In this paper we show how complete hierarchical multinomial marginal (HMM) models for categorical variables can be defined, estimated and tested using the R package hmmm. Models involving equality and inequality constraints on marginal parameters are needed to define hypotheses of conditional independence, stochastic dominance or notions of positive dependence, or when the parameters are allowed to depend on covariates. The hmmm package also serves the need of estimating and testing HMM models under equality and inequality constraints on marginal interactions.

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“…Such systems are classified as lumpable [1]. The lumpability criterion (strong, weak or nearly) of Markov chains is an active field of research [2][3][4][5]. If a system is found to be lumpable and the states are joined together following this criterion, then the course-grained system continues to have the same measurable properties as the original system.…”

Section: Introductionmentioning

confidence: 99%

Mapping absorption processes onto a Markov chain, conserving the mean first passage time

Biswas¹

2013

J. Phys. A: Math. Theor.

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The dynamics of a multidimensional system is projected onto a discrete state master equation using the transition rates W(k → k′; t, t + dt) between a set of states {k} represented by the regions {ζk} in phase or discrete state space. Depending on the dynamics Γi(t) of the original process and the choice of ζk, the discretized process can be Markovian or non-Markovian. For absorption processes, it is shown that irrespective of these properties of the projection, a master equation with time-independent transition rates can be obtained, which conserves the total occupation time of the partitions of the phase or discrete state space of the original process. An expression for the transition probabilities is derived based on either time-discrete measurements {ti} with variable time stepping Δ(i + 1)i = ti + 1 − ti or the theoretical knowledge at continuous times t. This allows computational methods of absorbing Markov chains to be used to obtain the mean first passage time (MFPT) of the system. To illustrate this approach, the procedure is applied to obtain the MFPT for the overdamped Brownian motion of particles subject to a system with dichotomous noise and the escape from an entropic barrier. The high accuracy of the simulation results confirms with the theory.

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Testing lumpability for marginal discrete hidden Markov models

Abstract: Markov chains, Multinomial hidden Markov models, Conditional independence, Identifiability, EM algorithm,

Cited by 3 publications

References 16 publications

Multiple hidden Markov models for categorical time series

Multiple hidden Markov models for categorical time series

hmmm: AnRPackage for Hierarchical Multinomial Marginal Models

Mapping absorption processes onto a Markov chain, conserving the mean first passage time

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