Latent Markov Research Articles

Extreme Overdispersion and Persistence in Time-Series of Counts

Time series of counts are often characterized by high overdispersion and persistence. These extreme features challenge the existing models. We approach this problem by combining the framework of INAR with a latent Markov structure. We call it HMM-INAR since it belongs to the class of hidden Markov models. We study the probabilistic properties of HMM-INAR model and illustrate conditions for the existence of an ergodic and stationary solution. We show that the HMM-INAR model is identifiable and can be estimated by maximum likelihood via an efficient expectation-maximization (EM) algorithm with steps available in closed form. The HMM-INAR well predicts the distributional and dynamic features of the time series of counts under investigation, i.e. the number of monthly bankruptcies in South Korea, and the number of trades and volume of several NYSE stocks observed at high frequency. Finally, the model proves empirically superior to other INAR specifications.

A latent discrete Markov random field approach to identifying and classifying historical forest communities based on spatial multivariate tree species counts

The Wisconsin Public Land Survey database describes historical forest composition at high spatial resolution and is of interest in ecological studies of forest composition in Wisconsin just prior to significant Euro-American settlement. For such studies it is useful to identify recurring subpopulations of tree species known as communities, but standard clustering approaches for subpopulation identification do not account for dependence between spatially nearby observations. Here, we develop and fit a latent discrete Markov random field model for the purpose of identifying and classifying historical forest communities based on spatially referenced multivariate tree species counts across Wisconsin. We show empirically for the actual dataset and through simulation that our latent Markov random field modeling approach improves prediction and parameter estimation performance. For model fitting we introduce a new stochastic approximation algorithm which enables computationally efficient estimation and classification of large amounts of spatial multivariate count data.

Multiple imputation of longitudinal categorical data through bayesian mixture latent Markov models

ABSTRACT Standard latent class modeling has recently been shown to provide a flexible tool for the multiple imputation (MI) of missing categorical covariates in cross-sectional studies. This article introduces an analogous tool for longitudinal studies: MI using Bayesian mixture Latent Markov (BMLM) models. Besides retaining the benefits of latent class models, i.e. respecting the (categorical) measurement scale of the variables and preserving possibly complex relationships between variables within a measurement occasion, the Markov dependence structure of the proposed BMLM model allows capturing lagged dependencies between adjacent time points, while the time-constant mixture structure allows capturing dependencies across all time points, as well as retrieving associations between time-varying and time-constant variables. The performance of the BMLM model for MI is evaluated by means of a simulation study and an empirical experiment, in which it is compared with complete case analysis and MICE. Results show good performance of the proposed method in retrieving the parameters of the analysis model. In contrast, competing methods could provide correct estimates only for some aspects of the data.

Bubble regime identification in an attention-based model for Bitcoin and Ethereum price dynamics

In this paper we extend the model in Cretarola, Figà-Talamanca, “Detecting bubbles in Bitcoin price dynamics via market exuberance”, Annals of Operations Research (2019), by allowing for a state-dependent correlation parameter between asset returns and market attention. We assume that the change of state is described by a continuous time latent Markov chain and we propose an estimation procedure based on the conditional maximum likelihood and on the Hamilton filter. Finally, model parameters, as well as Markov chain transition probabilities, are estimated on Bitcoin and Ethereum returns in case market attention is measured via the Google Search Volume Index for the keywords “bitcoin” and “ethereum”, respectively; up to four regimes are considered in the empirical application. The empirical outcomes show that the model is not only capable of identifying bubble and non-bubble regimes but also enables the interpretation of the correlation between cryptocurrencies and their market attention as a tuning to define the speed at which a bubble boosts.

Modeling material stress using integrated Gaussian Markov random fields

The equations of a physical constitutive model for material stress within tantalum grains were solved numerically using a tetrahedrally meshed volume. The resulting output included a scalar vonMises stress for each of the more than 94,000 tetrahedra within the finite element discretization. In this paper, we define an intricate statistical model for the spatial field of vonMises stress which uses the given grain geometry in a fundamental way. Our model relates the three-dimensional field to integrals of latent stochastic processes defined on the vertices of the one- and two-dimensional grain boundaries. An intuitive neighborhood structure of the said boundary nodes suggested the use of a latent Gaussian Markov random field (GMRF). However, despite the potential for computational gains afforded by GMRFs, the integral nature of our model and the sheer number of data points pose substantial challenges for a full Bayesian analysis. To overcome these problems and encourage efficient exploration of the posterior distribution, a number of techniques are now combined: parallel computing, sparse matrix methods, and a modification of a block update strategy within the sampling routine. In addition, we use an auxiliary variables approach to accommodate the presence of outliers in the data.

Continuous-Time Latent Markov Factor Analysis for Exploring Measurement Model Changes Across Time

Abstract. Drawing valid inferences about daily or long-term dynamics of psychological constructs (e.g., depression) requires the measurement model (indicating which constructs are measured by which items) to be invariant within persons over time. However, it might be affected by time- or situation-specific artifacts (e.g., response styles) or substantive changes in item interpretation. To efficiently evaluate longitudinal measurement invariance, and violations thereof, we proposed Latent Markov factor analysis (LMFA), which clusters observations based on their measurement model into separate states, indicating which measures are validly comparable. LMFA is, however, tailored to “discrete-time” data, where measurement intervals are equal, which is often not the case in longitudinal data. In this paper, we extend LMFA to accommodate unequally spaced intervals. The so-called “continuous-time” (CT) approach considers the measurements as snapshots of continuously evolving processes. A simulation study compares CT-LMFA parameter estimation to its discrete-time counterpart and a depression data application shows the advantages of CT-LMFA.

Uncontrolled diabetes and health care utilisation: A bivariate latent Markov model approach.

Although uncontrolled diabetes (UD) or poor glycaemic control is a widespread condition with potentially life-threatening consequences, there is sparse evidence of its effects on health care utilisation. We jointly model the propensities to consume health care and UD by employing an innovative bivariate latent Markov model that allows for dynamic unobserved heterogeneity, movements between latent states and the endogeneity of UD. We estimate the effects of UD on primary and secondary health care consumption using a panel dataset of rich administrative records from Spain and measure UD using a biomarker. We find that, conditional on time-varying unobservables, UD does not have a statistically significant direct effect on health care use. Furthermore, individuals appear to move across latent classes and increase their propensities to poor glycaemic control and health care use over time. Our results suggest that by ignoring time-varying unobserved heterogeneity and the endogeneity of UD, the effects of UD on health care utilisation might be overestimated and this could lead to biased findings. Our approach reveals heterogeneity in behaviour beyond standard groupings of frequent versus infrequent users of health care services. We argue that this dynamic latent Markov approach could be used more widely to model the determinants of health care use.

A Dynamic Framework for Modelling Set-Shifting Performances.

Higher-order cognitive functions can be seen as a class of cognitive processes which are crucial in situations requiring a flexible adjustment of behaviour in response to changing demands of the environment. The cognitive assessment of these functions often relies on tasks which admit a dynamic, or longitudinal, component requiring participants to flexibly adapt their behaviour during the unfolding of the task. An intriguing feature of such experimental protocols is that they allow the performance of an individual to change as the task unfolds. In this work, we propose a Latent Markov Model approach to capture some dynamic aspects of observed response patterns of both healthy and substance dependent individuals in a set-shifting task. In particular, data from a Wisconsin Card Sorting Test were analysed in order to represent performance trends in terms of latent cognitive states dynamics. The results highlighted how a dynamic modelling approach can considerably improve the amount of information a researcher, or a clinician, can obtain from the analysis of a set-shifting task.

Modelling the combined impact of interventions in averting deaths during a synthetic-opioid overdose epidemic.

The province of British Columbia (BC) Canada has experienced a rapid increase in illicit drug overdoses and deaths during the last 4years, with a provincial emergency declared in April 2016. These deaths have been driven primarily by the introduction of synthetic opioids into the illicit opioid supply. This study aimed to measure the combined impact of large-scale opioid overdose interventions implemented in BC between April 2016 and December 2017 on the number of deaths averted. We expanded on the mathematical modelling methodology of our previous study to construct a Bayesian hierarchical latent Markov process model to estimate monthly overdose and overdose-death risk, along with the impact of interventions. Overdose events and overdose-related deaths in BC from January 2012 to December 2017. The interventions considered were take-home naloxone kits, overdose prevention/supervised consumption sites and opioid agonist therapy MEASUREMENTS: Counterfactual simulations were performed with the fitted model to estimate the number of death events averted for each intervention and in combination. Between April 2016 and December 2017, BC observed 2177 overdose deaths (77% fentanyl-detected). During the same period, an estimated 3030 (2900-3240) death events were averted by all interventions combined. In isolation, 1580 (1480-1740) were averted by take-home naloxone, 230 (160-350) by overdose prevention services and 590 (510-720) were averted by opioid agonist therapy. A combined intervention approach has been effective in averting overdose deaths during British Columbia's opioid overdose crisis in the period since declaration of a public health emergency (April 2016-December 2017). However, the absolute numbers of overdose deaths have not changed.

Panel Data Analysis via Mechanistic Models

Panel data, also known as longitudinal data, consist of a collection of time series. Each time series, which could itself be multivariate, comprises a sequence of measurements taken on a distinct unit. Mechanistic modeling involves writing down scientifically motivated equations describing the collection of dynamic systems giving rise to the observations on each unit. A defining characteristic of panel systems is that the dynamic interaction between units should be negligible. Panel models therefore consist of a collection of independent stochastic processes, generally linked through shared parameters while also having unit-specific parameters. To give the scientist flexibility in model specification, we are motivated to develop a framework for inference on panel data permitting the consideration of arbitrary nonlinear, partially observed panel models. We build on iterated filtering techniques that provide likelihood-based inference on nonlinear partially observed Markov process models for time series data. Our methodology depends on the latent Markov process only through simulation; this plug-and-play property ensures applicability to a large class of models. We demonstrate our methodology on a toy example and two epidemiological case studies. We address inferential and computational issues arising due to the combination of model complexity and dataset size. Supplementary materials for this article are available online.

A Bayesian structural model for predicting algal blooms

AbstractA Bayesian structural model with two components is proposed to forecast the occurrence of algal blooms, multivariate mean‐reverting diffusion process (MMRD), and a binary probit model with latent Markov regime‐switching process (BPMRS). The model has three features: (a) forecast of the occurrence probability of algal bloom is directly based on oceanographic parameters, not the forecasting of special indicators in traditional approaches, such as phytoplankton or chlorophyll‐a; (b) augmentation of daily oceanographic parameters from the data collected every 2 weeks is based on MMRD. The proposed method solves the problem of unavailability of daily oceanographic parameters in practice; (c) BPMRS captures the unobservable factors which affect algal bloom occurrence and therefore improve forecast accuracy. We use panel data collected in Tolo Harbour, Hong Kong, to validate the model. The model demonstrates good forecasting for out‐of‐sample rolling forecasts, especially for algal bloom appearing for a longer period, which severely damages fisheries and the marine environment.

The unobserved pattern of material hardship and health among older Americans

This paper investigates the relationship between self-reported health and material hardship among older Americans. Differently from income-based measures, material hardship provides a more specific description of the concrete adversities faced by the elderly. We have used the last six waves of the Health and Retirement Study to explore the relative contributions of state dependence, unobserved heterogeneity and time-specific shocks on reporting poor health, experiencing food insecurity and medication cutbacks. We have used a Latent Markov model to estimate a multivariate non-linear system of equations for panel data allowing time-varying unobserved heterogeneity. Our results reveal a high state dependence of both health and material hardship conditions. Estimated trajectories reveal that pathways of material hardship are associated differently with health. Material hardship is also spread across its dimensions.

Latent Markov Factor Analysis for Exploring Measurement Model Changes in Time-Intensive Longitudinal Studies

When time-intensive longitudinal data are used to study daily-life dynamics of psychological constructs (e.g., well-being) within persons over time (e.g., by means of experience sampling methodology), the measurement model (MM)—indicating which constructs are measured by which items—can be affected by time- or situation-specific artifacts (e.g., response styles and altered item interpretation). If not captured, these changes might lead to invalid inferences about the constructs. Existing methodology can only test for a priori hypotheses on MM changes, which are often absent or incomplete. Therefore, we present the exploratory method “latent Markov factor analysis” (LMFA), wherein a latent Markov chain captures MM changes by clustering observations per subject into a few states. Specifically, each state gathers validly comparable observations, and state-specific factor analyses reveal what the MMs look like. LMFA performs well in recovering parameters under a wide range of simulated conditions, and its empirical value is illustrated with an example.

Bayesian analysis of latent Markov models with non-ignorable missing data

Latent Markov models (LMMs) are widely used in the analysis of heterogeneous longitudinal data. However, most existing LMMs are developed in fully observed data without missing entries. The main objective of this study is to develop a Bayesian approach for analyzing the LMMs with non-ignorable missing data. Bayesian methods for estimation and model comparison are discussed. The empirical performance of the proposed methodology is evaluated through simulation studies. An application to a data set derived from National Longitudinal Survey of Youth 1997 is presented.

A Nonlinear Dynamic Latent Class Structural Equation Model

In this article, we propose a nonlinear dynamic latent class structural equation modeling (NDLC-SEM). It can be used to examine intra-individual processes of observed or latent variables. These processes are decomposed into parts which include individual- and time-specific components. Unobserved heterogeneity of the intra-individual processes are modeled via a latent Markov process that can be predicted by individual- and time-specific variables as random effects. We discuss examples of sub-models which are special cases of the more general NDLC-SEM framework. Furthermore, we provide empirical examples and illustrate how to estimate this model in a Bayesian framework. Finally, we discuss essential properties of the proposed framework, give recommendations for applications, and highlight some general problems in the estimation of parameters in comprehensive frameworks for intensive longitudinal data.

Exploring the randomness of mentally generated head–tail sequences

It is well known that people deviate from randomness as they attempt to mentally generate head–tail sequences as randomly as possible. This deviation from randomness is quantified by an excess of repetitions or alternations between successive responses more than would be expected by chance. We conducted an experiment in which a sample of students was asked to mentally simulate a sequence as if it is produced by a fair coin. We propose several models based on Markov chains for analysing the dynamic of head–tail outcomes in these sequences. First, we explore observed Markov chains and suggest some practical solutions to reduce the number of parameters. However, there is a need for more sophisticated models, and in this case, we propose latent Markov models and mixture of Markov chains to analyse these head–tail sequences. A generalization of the so-called mixture transition distribution (MTD) model is also considered.

Marketing innovations to old-age consumers: A dynamic Bass model for different life stages

To identify context-dependent opportunities to market innovations to the elderly, this study empirically analyzes the most prevalent pathways through advanced age, demonstrating which circumstances in the old-age life course provide the strongest potential for specific targeting strategies. First, using a latent Markov model and longitudinal survey data spanning 15 years, we produce a dynamic life course model with transitions over time. Second, we link a modified Bass diffusion model — using both static and dynamic parameters — to our model, augmenting it with a second cross-sectional consumer behavior data set. The results show comparatively strong consumption spending, high media interaction, but diminishing social inclusion in old age, though all factors exhibit heterogeneity among old-age clusters. Employing dynamic diffusion models, we find that a static view of the elderly market that ignores life course transitions generally overestimates their spending power. Forecasts of cluster-specific adoption dynamics draw a differentiated picture of individual clusters' attractiveness. Our analysis underscores the influence of life events on individual behavior and shows that a dynamic view of elderly markets yields a more nuanced and accurate assessment of their potential and attractiveness. It also confirms that social status and income strongly affect consumer behavior and spending, though we identify several exceptions.

Automated detection of many-particle solvation states for accurate characterizations of diffusion kinetics.

Discrete-space kinetic models, i.e., Markov state models, have emerged as powerful tools for reducing the complexity of trajectories generated from molecular dynamics simulations. These models require configuration-space representations that accurately characterize the relevant dynamics. Well-established, low-dimensional order parameters for constructing this representation have led to widespread application of Markov state models to study conformational dynamics in biomolecular systems. On the contrary, applications to characterize single-molecule diffusion processes have been scarce and typically employ system-specific, higher-dimensional order parameters to characterize the local solvation state of the molecule. In this work, we propose an automated method for generating a coarse configuration-space representation, using generic features of the solvation structure-the coordination numbers about each particle. To overcome the inherent noisy behavior of these low-dimensional observables, we treat the features as indicators of an underlying, latent Markov process. The resulting hidden Markov models filter the trajectories of each feature into the most likely latent solvation state at each time step. The filtered trajectories are then used to construct a configuration-space discretization, which accurately describes the diffusion kinetics. The method is validated on a standard model for glassy liquids, where particle jumps between local cages determine the diffusion properties of the system. Not only do the resulting models provide quantitatively accurate characterizations of the diffusion constant, but they also reveal a mechanistic description of diffusive jumps, quantifying the heterogeneity of local diffusion.

Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R

Sequence analysis is being more and more widely used for the analysis of social sequences and other multivariate categorical time series data. However, it is often complex to describe, visualize, and compare large sequence data, especially when there are multiple parallel sequences per subject. Hidden (latent) Markov models (HMMs) are able to detect underlying latent structures and they can be used in various longitudinal settings: to account for measurement error, to detect unobservable states, or to compress information across several types of observations. Extending to mixture hidden Markov models (MHMMs) allows clustering data into homogeneous subsets, with or without external covariates. The seqHMM package in R is designed for the efficient modeling of sequences and other categorical time series data containing one or multiple subjects with one or multiple interdependent sequences using HMMs and MHMMs. Also other restricted variants of the MHMM can be fitted, e.g., latent class models, Markov models, mixture Markov models, or even ordinary multinomial regression models with suitable parameterization of the HMM. Good graphical presentations of data and models are useful during the whole analysis process from the first glimpse at the data to model fitting and presentation of results. The package provides easy options for plotting parallel sequence data, and proposes visualizing HMMs as directed graphs.

A Bayesian multiple changepoint model for marked poisson processes with applications to deep earthquakes

A multiple changepoint model for marked Poisson process is formulated as a continuous time hidden Markov model, which is an extension of Chib’s multiple changepoint models (J Econ 86:221–241, 1998). The inference on the locations of changepoints and other model parameters is based on a two-block Gibbs sampling scheme. We suggest a continuous time version of forward-filtering backward-sampling algorithm for simulating the full trajectories of the latent Markov chain without utilizing the uniformization method. To retrieve the optimal posterior path of the latent Markov chain, i.e. the maximum a posteriori estimation of changepoint locations, a continuous-time version of Viterbi algorithm (CT-Viterbi) is proposed. The set of changepoint locations is obtainable either from the CT-Viterbi algorithm or the posterior samples of the latent Markov chain. The number of changepoints is determined according to a modified BIC criterion tailored particularly for the multiple changepoint problems of a marked Poisson process. We then perform a simulation study to demonstrate the methods. The methods are applied to investigate the temporal variabilities of seismicity rates and the magnitude-frequency distributions of medium size deep earthquakes in New Zealand.