Number Of Latent States Research Articles

Covariate-modulated rectangular latent Markov models with an unknown number of regime profiles

We derive a multivariate latent Markov model with number of latent states that can possibly change at each time point. We model both the manifest and latent distributions conditionally on explanatory variables. Bayesian inference is based on a transdimensional Markov Chain Monte Carlo approach, where Reversible Jump is separately performed for each time occasion. In a simulation study, we show how our approach can recover the true underlying sequence of latent states with high probability, and that it has lower bias than competitors. We conclude with an analysis of the well-being of 100 nations, as expressed by the dimensions of the Human Development Index, for six-time points spanning a period of 22 years. R code with an implementation is available as supplementary material, together with files for reproducing the data analysis.

Assessing Measurement Invariance for Longitudinal Data through Latent Markov Models

ABSTRACT We propose a general approach to detect measurement non-invariance in latent Markov models for longitudinal data. We define different notions of differential item functioning in the context of panel data. We then present a model selection approach based on the Bayesian information criterion (BIC) to choose both the number of latent states and the measurement structure. We show the practical relevance by means of an extensive simulation study, and illustrate its use on two real–data examples from the social sciences. Our results indicate that BIC is able to select the correct measurement equivalence structure more than 95% of times.

Rectangular Latent Markov Models for Time-Specific Clustering, with An Analysis of the Wellbeing of Nations

SummaryA latent Markov model admitting variation in the number of latent states at each time period is introduced. The model facilitates subjects switching latent states at each time period according to an inhomogeneous first-order Markov process, wherein transition matrices are generally rectangular. As a consequence, latent groups can merge, split or be rearranged. An application analysing the progress of wellbeing of nations, as measured by the three dimensions of the human development index over the last 25 years, illustrates the approach.

Ranking Nursing Homes’ Performances Through a Latent Markov Model with Fixed and Random Effects

In this paper, we aim at ranking a set of nursing homes based on their ability in maintaining their residents’ physical conditions as good as possible. In this respect, we propose a nursing home performance indicator, which is essentially a probability to avoid resident health status worsening. Specifically, latent Markov models with covariates and normally distributed continuous random effects are fitted to produce standardised 180-day ahead transition matrices, upon which the aforementioned index is based. Nursing home effects on these transition matrices are modelled through fixed as well as random effects. The performance index is used to build two distinct rankings, one of which also accounts for the variability induced by the estimation process. In this framework, several rankings can be obtained by combining the model specification (fixed vs. random effects), the kind of ranking and the number of latent states, which is the typical sensitivity parameter of latent Markov models. Our methodological approach is applied to a dataset which was gathered from a health protocol implemented in Umbria (Italy). Results for this data show a rather high degree of robustness, in the sense that the obtained rankings are almost the same.

Network Identifiability from Intrinsic Noise

This paper considers the problem of inferring an unknown network of dynamical systems driven by unknown, intrinsic , noise inputs. Equivalently we seek to identify direct causal dependencies among manifest variables only from observations of these variables. For linear, time-invariant systems of minimal order, we characterise under what conditions this problem is well posed. We first show that if the transfer matrix from the inputs to manifest states is minimum phase, this problem has a unique solution irrespective of the network topology. This is equivalent to there being only one valid spectral factor (up to a choice of signs of the inputs) of the output spectral density. If the assumption of phase-minimality is relaxed, we show that the problem is characterised by a single Algebraic Riccati Equation (ARE), of dimension determined by the number of latent states. The number of solutions to this ARE is an upper bound on the number of solutions for the network. We give necessary and sufficient conditions for any two dynamical networks to have equal output spectral density, which can be used to construct all equivalent networks. Extensive simulations quantify the number of solutions for a range of problem sizes. For a slightly simpler case, we also provide an algorithm to construct all equivalent networks from the output spectral density.

GP-Select: Accelerating EM Using Adaptive Subspace Preselection.

We propose a nonparametric procedure to achieve fast inference in generative graphical models when the number of latent states is very large. The approach is based on iterative latent variable preselection, where we alternate between learning a selection function to reveal the relevant latent variables and using this to obtain a compact approximation of the posterior distribution for EM. This can make inference possible where the number of possible latent states is, for example, exponential in the number of latent variables, whereas an exact approach would be computationally infeasible. We learn the selection function entirely from the observed data and current expectation-maximization state via gaussian process regression. This is in contrast to earlier approaches, where selection functions were manually designed for each problem setting. We show that our approach performs as well as these bespoke selection functions on a wide variety of inference problems. In particular, for the challenging case of a hierarchical model for object localization with occlusion, we achieve results that match a customized state-of-the-art selection method at a far lower computational cost.

Power Analysis for the Likelihood-Ratio Test in Latent Markov Models: Shortcutting the Bootstrap p-Value-Based Method

ABSTRACTThe latent Markov (LM) model is a popular method for identifying distinct unobserved states and transitions between these states over time in longitudinally observed responses. The bootstrap likelihood-ratio (BLR) test yields the most rigorous test for determining the number of latent states, yet little is known about power analysis for this test. Power could be computed as the proportion of the bootstrap p values (PBP) for which the null hypothesis is rejected. This requires performing the full bootstrap procedure for a large number of samples generated from the model under the alternative hypothesis, which is computationally infeasible in most situations. This article presents a computationally feasible shortcut method for power computation for the BLR test. The shortcut method involves the following simple steps: (1) obtaining the parameters of the model under the null hypothesis, (2) constructing the empirical distributions of the likelihood ratio under the null and alternative hypotheses via Monte Carlo simulations, and (3) using these empirical distributions to compute the power. We evaluate the performance of the shortcut method by comparing it to the PBP method and, moreover, show how the shortcut method can be used for sample-size determination.

Hidden Gibbs random fields model selection using Block Likelihood Information Criterion

Performing model selection between Gibbs random fields is a very challenging task. Indeed, because of the Markovian dependence structure, the normalizing constant of the fields cannot be computed using standard analytical or numerical methods. Furthermore, such unobserved fields cannot be integrated out, and the likelihood evaluation is a doubly intractable problem. This forms a central issue to pick the model that best fits an observed data. We introduce a new approximate version of the Bayesian Information Criterion. We partition the lattice into contiguous rectangular blocks, and we approximate the probability measure of the hidden Gibbs field by the product of some Gibbs distributions over the blocks. On that basis, we estimate the likelihood and derive the Block Likelihood Information Criterion (BLIC) that answers model choice questions such as the selection of the dependence structure or the number of latent states. We study the performances of BLIC for those questions. In addition, we present a comparison with ABC algorithms to point out that the novel criterion offers a better trade‐off between time efficiency and reliable results. Copyright © 2016 John Wiley & Sons, Ltd.