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We propose a multivariate regime switching model based on a Student- (Formula presented.) copula function with parameters controlling the strength of correlation between variables and that are governed by a latent Markov process. To estimate model parameters by maximum likelihood, we consider a two-step procedure carried out through the Expectation–Maximisation algorithm. To address the main computational burden related to the estimation of the matrix of dependence parameters and the number of degrees of freedom of the Student- (Formula presented.) copula, we show a novel use of the Lagrange multipliers, which simplifies the estimation process. The simulation study shows that the estimators have good finite sample properties and the estimation procedure is computationally efficient. An application concerning log-returns of five cryptocurrencies shows that the model permits identifying bull and bear market periods based on the intensity of the correlations between crypto assets.

We examine the effect of streaming based on ability levels on individuals’ civic participation throughout their adult life. The hypothesis we test is that ability grouping influences individuals’ general self-concept and, consequently, their civic participation choices across the life course. We employ data from the British National Child Development Study, which follows all UK citizens born during a certain week in 1958. Six binary variables observed at 33, 42, and 51 years of age are considered to measure civic participation. Our approach defines causal estimands with multiple treatments referring to the evolution of civic engagement over time in terms of potential versions of a sequence of latent variables assumed to follow a Markov chain with initial and transition probabilities depending on posttreatment time-varying covariates. The model also addresses partially or entirely missing data on one or more indicators at a given time occasion and missing posttreatment covariate values using dummy indicators. The model is estimated by maximizing a weighted log-likelihood function with weights corresponding to the inverse probability of the received treatment obtained from a multinomial logit model based on pretreatment covariates. Our results show that ability grouping affects the civic participation of high-ability individuals when they are 33 years old with respect to participation in general elections.

We propose an evolutionary optimization method for maximum likelihood and approximate maximum likelihood estimation of discrete latent variable models. The proposal is based on modified versions of the expectation–maximization (EM) and variational EM (VEM) algorithms, which are based on the genetic approach and allow us to accurately explore the parameter space, reducing the chance to be trapped into one of the multiple local maxima of the log-likelihood function. Their performance is examined through an extensive Monte Carlo simulation study where they are employed to estimate latent class, hidden Markov, and stochastic block models and compared with the standard EM and VEM algorithms. We observe a significant increase in the chance to reach global maximum of the target function and a high accuracy of the estimated parameters for each model. Applications focused on the analysis of cross-sectional, longitudinal, and network data are proposed to illustrate and compare the algorithms.

We propose a variable selection method for multivariate hidden Markov models with continuous responses that are partially or completely missing at a given time occasion. Through this procedure, we achieve a dimensionality reduction by selecting the subset of the most informative responses for clustering individuals and simultaneously choosing the optimal number of these clusters corresponding to latent states. The approach is based on comparing different model specifications in terms of the subset of responses assumed to be dependent on the latent states, and it relies on a greedy search algorithm based on the Bayesian information criterion seen as an approximation of the Bayes factor. A suitable expectation-maximization algorithm is employed to obtain maximum likelihood estimates of the model parameters under the missing-at-random assumption. The proposal is illustrated via Monte Carlo simulation and an application where development indicators collected over eighteen years are selected, and countries are clustered into groups to evaluate their growth over time.

In longitudinal data analysis, hidden Markov (HM) models are fundamental tools, especially when the analysis is focused on transitions or the need to cluster individuals dynamically. When individual covariates are available in the dataset, a typical problem is how to parametrize the transition probabilities based on these covariates in a parsimonious way. In fact, standard multinomial parametrizations of these probabilities lead to models with many parameters, which are also difficult to interpret and, consequently, to unstable parameter estimates. To overcome the above problems, different parametrizations of the transition probabilities of HM models with covariates are introduced based on multinomial logit models formulated by two different choices of the reference state of each logit. These parametrizations rely on constraints having a straightforward interpretation, making the model much more parsimonious. Estimation based on the maximum likelihood (ML) approach is developed under different constraints based on the Expectation-Maximization algorithm. Steps of Newton-Raphson type are also included to improve the algorithm’s convergence speed

We propose assessing the causal effects of a dynamic treatment in a longitudinal observational study, given observed confounders under suitable assumptions. The causal hidden Markov model is based on potential versions of discrete latent variables, and it accounts for the estimated propensity to be assigned to each treatment level over time using inverse probability weighting. Estimation of the model parameters is carried out through a weighted maximum log-likelihood approach. Standard errors for the parameter estimates are provided by nonparametric bootstrap. The proposal is validated through a simulation study aimed at comparing different model specifications. As an illustrative example, we consider a marketing campaign conducted by a large European bank over time on its customers. Findings provide straightforward managerial implications.

In the context of longitudinal data, we show that a general class of hidden Markov (HM, [1]) models may be equivalent to a class of finite mixture (FM, [3]) models based on an augmented set of components and suitable constraints on the conditional response probabilities, given these components. We formulate a misspecification test for the latent structure of an HM model comparing maximum likelihood values of the two models for the same data, and when the number of possible latent state sequences is excessive, we propose a multiple version of this test including the Bonferroni correction. The procedure is simple since it is based on the output of the Expectation-Maximization estimation algorithm [2]. The properties of this testing procedure are evaluated through a simulation study. An empirical application illustrates it through data from the National Longitudinal Survey of Youth, in which we jointly consider wages and years of experience after labour force entry. We show that the proposed testing procedure may also be used as an alternative model selection criterion for the number of latent states of an HM model to those usually employed.

Young people’s place in the labor market has been a topic of interest to the European Union and national governments for many years. This study analyzes young people who are Not in Employment nor in Education or Training (NEET) and Youth Unemployment (YU) in the European Union member states, through data collected over a period of sixteen years, considering the influence of some macroeconomic factors through an hidden Markov model. This approach is based on maximum likelihood estimation of the model parameters, and provides a dynamic classification of the countries into clusters representing different levels of the phenomena. We discover three clusters of countries, and we show that whereas Italy was the worst performing country in terms of both NEETs and YU, the Czech Republic was the best performing country in reducing NEETs, and Poland and Slovakia were the best performing in reducing YU.

A hidden Markov model is proposed for the analysis of time-series of daily log-returns of the last 4 years of Bitcoin, Ethereum, Ripple, Litecoin, and Bitcoin Cash. These log-returns are assumed to have a multivariate Gaussian distribution conditionally on a latent Markov process having a finite number of regimes or states. The hidden regimes represent different market phases identified through distinct vectors of expected values and variance–covariance matrices of the log-returns, so that they also differ in terms of volatility. Maximum-likelihood estimation of the model parameters is carried out by the expectation–maximisation algorithm, and regimes are singularly predicted for every time occasion according to the maximum-a-posteriori rule. Results show three positive and three negative phases of the market. In the most recent period, an increasing tendency towards positive regimes is also predicted. A rather heterogeneous correlation structure is estimated, and evidence of structural medium term trend in the correlation of Bitcoin with the other cryptocurrencies is detected.

Flexible statistical models have an important role in explaining the joint distribution of financial returns. In these analyses, it is necessary to consider abrupt switches in the market conditions, especially if the focus is on cryptoassets, the market of which is characterized by high instabilities. Regime switching (RS) copula models represent a powerful tool to formulate the joint distribution of time-series accurately: they are based on a copula distribution with parameters governed by a hidden Markov process of first-order so as to account for the correlation patterns between series. The hidden states represent different market regimes, each described by a state-specific vector of copula parameters. We propose RS copula models as a valuable instrument for describing the joint behavior of log- returns. We choose a Student-t copula function to consider extreme dependent values appropriately as they are often observed in financial returns. We split the modeling process into two steps: the first one consists in fitting the marginal distribution of each univariate time-series, while the second one deals with the estimation of the joint distribution of the log-returns described by a RS copula model. Maximum likelihood estimation of the model parameters is carried out by the expectation-maximization (EM) algorithm, which alternates two steps until convergence: at the E-step, we compute the expectation of the log-likelihood evaluated using the current values for the parameters and, at the M-step, parameters estimates are updated by maximizing the expected complete-data log-likelihood computed at the previous step. The main computational burdens deal with estimating the correlation matrix (R) and the number of degrees of freedom (v) of the Student t-copula. At this aim, we propose performing the M-step by computing R given v using a closed form solution obtained from a constrained optimization of the log-likelihood using Lagrange multipliers. Then, we numerically maxim

Methods: We used open-source tools to develop a public European COVID-19 Forecast Hub. We invited groups globally to contribute weekly forecasts for COVID-19 cases and deaths reported by a standardised source for 32 countries over the next 1–4 weeks. Teams submitted forecasts from March 2021 using standardised quantiles of the predictive distribution. Each week we created an ensemble forecast, where each predictive quantile was calculated as the equally-weighted average (initially the mean and then from 26th July the median) of all individual models’ predictive quantiles. We measured the performance of each model using the relative Weighted Interval Score (WIS), comparing models’ forecast accuracy relative to all other models. We retrospectively explored alternative methods for ensemble forecasts, including weighted averages based on models’ past predictive performance. Results: Over 52 weeks, we collected forecasts from 48 unique models. We evaluated 29 models’ forecast scores in comparison to the ensemble model. We found a weekly ensemble had a consistently strong performance across countries over time. Across all horizons and locations, the ensemble performed better on relative WIS than 83% of participating models’ forecasts of incident cases (with a total N=886 predictions from 23 unique models), and 91% of participating models’ forecasts of deaths (N=763 predictions from 20 models). Across a 1–4 week time horizon, ensemble performance declined with longer forecast periods when forecasting cases, but remained stable over 4 weeks for incident death forecasts. In every forecast across 32 countries, the ensemble outperformed most contributing models when forecasting either cases or deaths, frequently outperforming all of its individual component models. Among several choices of ensemble methods we found that the most influential and best choice was to use a median average of models instead of using the mean, regardless of methods of weighting component forecast mod

In this paper we frame the problem of obtaining estimates from the sur- vey on the employment status of graduates in Italy as a Small Area Estimation problem because of unit nonresponse. We propose to use generalized linear mixed models and to include two variables that can be considered proxies of the response propensity among the set of covariates to make the MAR assumption more tenable. Estimates for degree programmes are obtained as (semi-parametric) empirical best predictions.

After a review of the class of discrete latent variable models in terms of formulation and estimation methods, recent advances and perspectives regarding these models are illustrated. We consider in detail the stochastic block model for social networks and models for spatio-temporal data. Among these developments, we discuss, in particular, the analysis of longitudinal compositional data about expenditures of the Spanish regions over several decades.

The present research explores the heterogeneity in trajectories of subjective well-being of Japanese citizens using longitudinal data collected with the Preference Parameters Study from 2003 to 2018. The analysis is carried out through the hidden Markov model that assumes a latent process underlying the individual perception of happiness observed through a categorical response variable. The first-order Markov chain is parameterized in terms of initial and transition probabilities depending on time-constant and time-varying socioeconomic and demographic variables. Maximum likelihood estimation of model parameters accounts for longitudinal sampling weights and missing responses under the Missing-At-Random assumption. Through this model-based clustering approach, we discover three clusters of individuals showing different dynamics across the life course, each of which represents, namely “not so happy”, “moderately happy”, and “very happy” individuals. We find that males tend to be less happy than females, and a U-shaped association between happiness and age is not detected. Each state has a substantial persistence over time, meaning that initial happiness perceptions play an important role in the life course inequalities of subjective well-being.

We illustrate a causal latent transition model to evaluate the effects of educational programs administered to pupils in the 6th and 7th grades during their middle school period. The programs are conducted in an Italian region and focus on improving non-cognitive abilities. The interest is in evaluating the effects on the skills acquired in the 8th grade in Italian and Mathematics. The model can be cast in the hidden Markov literature and is formulated as an extension of Rubin's causal model based on potential versions of discrete time-varying latent variables.

Background: Short-term forecasts of infectious disease contribute to situational awareness and capacity planning. Based on best practice in other fields and recent insights in infectious disease epidemiology, one can maximise forecasts’ predictive performance by combining independent models into an ensemble. Here we report the performance of ensemble predictions of COVID-19 cases and deaths across Europe from March 2021 to March 2022. Methods: We created the European COVID-19 Forecast Hub, an online open-access platform where modellers upload weekly forecasts for 32 countries with results publicly visualised and evaluated. We created a weekly ensemble forecast from the equally-weighted average across individual models’ predictive quantiles. We measured forecast accuracy using a baseline and relative Weighted Interval Score (rWIS). We retrospectively explored ensemble methods, including weighting by past performance. Results: We collected weekly forecasts from 48 models, of which we evaluated 29 models alongside the ensemble model. The ensemble had a consistently strong performance across countries over time, performing better on rWIS than 91% of forecasts for deaths (N=763 predictions from 20 models), and 83% forecasts for cases (N=886 predictions from 23 models). Performance remained stable over a 4-week horizon for death forecasts but declined with longer horizons for cases. Among ensemble methods, the most influential choice came from using a median average instead of the mean, regardless of weighting component models. Conclusions: Our results support combining independent models into an ensemble forecast to improve epidemiological predictions, and suggest that median averages yield better performance than methods based on means. We highlight that forecast consumers should place more weight on incident death forecasts than case forecasts at horizons greater than two weeks. Funding: European Commission, Ministerio de Ciencia, Innovación y Universidades, FEDER; Ag

In order to evaluate the effect of a policy or treatment with pre- and post-treatment outcomes, we propose an approach based on a transition model, which may be applied with multivariate outcomes and accounts for unobserved heterogeneity. This model is based on potential versions of discrete latent variables representing the individual characteristic of interest and may be cast in the hidden (latent) Markov literature for panel data. Therefore, it can be estimated by maximum likelihood in a relatively simple way. The approach extends the difference-in-difference method as it is possible to deal with multivariate outcomes. Moreover, causal effects may be expressed with respect to transition probabilities. The proposal is validated through a simulation study, and it is applied to evaluate educational programs administered to pupils in the sixth and seventh grades during their middle school period. These programs are carried out in an Italian region to improve non-cognitive skills (CSs). We study if they impact also on students’ CSs in Italian and Mathematics in the eighth grade, exploiting the pretreatment test scores available in the fifth grade. The main conclusion is that the educational programs aimed to develop noncognitive abilities help the best students to maintain their higher cognitive abilities over time.

We propose a hidden Markov model for multivariate continuous longitudinal responses with covariates that accounts for three different types of missing pattern: (I) partially missing outcomes at a given time occasion, (II) completely missing outcomes at a given time occasion (intermittent pattern), and (III) dropout before the end of the period of observation (monotone pattern). The missing-at-random (MAR) assumption is formulated to deal with the first two types of missingness, while to account for the informative dropout, we rely on an extra absorbing state. Estimation of the model parameters is based on the maximum likelihood method that is implemented by an expectation-maximization (EM) algorithm relying on suitable recursions. The proposal is illustrated by a Monte Carlo simulation study and an application based on historical data on primary biliary cholangitis.

Maximum likelihood estimation of discrete latent variable (DLV) models is usually performed by the expectation-maximization (EM) algorithm. A well-known drawback is related to the multimodality of the log-likelihood function so that the estimation algorithm can converge to a local maximum, not corresponding to the global one. We propose a tempered EM algorithm to explore the parameter space adequately for two main classes of DLV models, namely latent class and hidden Markov. We compare the proposal with the standard EM algorithm by an extensive Monte Carlo simulation study, evaluating both the ability to reach the global maximum and the computational time. We show the results of the analysis of discrete and continuous cross-sectional and longitudinal data referring to some applications of interest. All the results provide supporting evidence that the proposal outperforms the standard EM algorithm, and it significantly improves the chance to reach the global maximum. The advantage is relevant even considering the overall computing time.

The dynamic stochastic blockmodel is commonly used to analyze longitudinal network data when multiple snapshots are observed over time. The variational expectation-maximization (VEM) algorithm is typically employed for maximum likelihood inference to allocate nodes to groups dynamically. To address the problem of multiple local maxima, which may arise in this context, we propose modifying the VEM according to an evolutionary algorithm to explore the whole parameter space. A simulation study on dynamic networks and an application illustrate the proposal comparing the performance with that of the VEM algorithm.

Background and aims We analyze the possible predictive variables for Adverse Events (AEs) during sedation for gastrointestinal (GI) endoscopy. Methods We consider 23,788 GI endoscopies under sedation on adults between 2012 and 2019. A Zero-Inflated Poisson Regression Mixture (ZIPRM) model for count data with concomitant variables is applied, accounting for unobserved heterogeneity and evaluating the risks of multi-drug sedation. A multinomial logit model is also estimated to evaluate cardiovascular, respiratory, hemorrhagic, other AEs and stopping the procedure risk factors. Results In 7.55% of cases, one or more AEs occurred, most frequently cardiovascular (3.26%) or respiratory (2.77%). Our ZIPRM model identifies one population for non-zero counts. The AE-group reveals that age >75 years yields 46% more AEs than age <66 years; Body Mass Index (BMI) ≥27 27% more AEs than BMI <21; emergency 11% more AEs than routine. Any one-point increment in the American Society of Anesthesiologists (ASA) score and the Mallampati score determines respectively a 42% and a 16% increment in AEs; every hour prolonging endoscopy increases AEs by 41%. Regarding sedation with propofol alone (the sedative of choice), adding opioids to propofol increases AEs by 43% and adding benzodiazepines by 51%. Cardiovascular AEs are increased by age, ASA score, smoke, in-hospital, procedure duration, midazolam/fentanyl associated with propofol. Respiratory AEs are increased by BMI, ASA and Mallampati scores, emergency, in-hospital, procedure duration, midazolam/fentanyl associated with propofol. Hemorrhagic AEs are increased by age, in-hospital, procedure duration, midazolam/fentanyl associated with propofol. The risk of suspension of the endoscopic procedure before accomplishment is increased by female gender, ASA and Mallampati scores, and in-hospital, and it is reduced by emergency and procedure duration. Conclusions Age, BMI, ASA score, Mallampati score, in-hospital, procedure duration

We propose statistical autoregressive models to analyze the observed time series of count data referred to different categories. The main assumption is that observed frequencies correspond to margins of a sequence of unobserved contingency tables. Inference is based on a Bayesian approach and a suitable Markov chain Monte Carlo (MCMC) algorithm. We apply the proposal to Italian COVID-19 data (at national level and for Lombardy) considering different categories of patients further to susceptible individuals and deaths.

We consider hidden Markov and regime-switching copula models as approaches for state allocation in multiple time-series, where state allocation means the prediction of the latent state characterizing each time occasion based on the observed data. This dynamic clustering, performed under the two model specifications, takes the correlation structure of the time-series into account. Maximum likelihood estimation of the model parameters is carried out by the expectation-maximization algorithm. For illustration we use data on the market of cryptocurrencies characterized by periods of high turbulence in which interdependence among assets is marked.

We provide an analysis of the market data of the major cryptocurrencies by summing a multivariate hidden Markov process also known as the latent Markov process. We model jointly the daily log-returns of BTC, ETH, XRP, LTC, and BCH. The observed log-returns are assumed to be correlated according to a variance-covariance matrix conditionally on a latent Markov process of first-order having a discrete number of latent states. In order to compare states according to their volatility, we estimate the specific variance-covariance matrix of each state. Maximum likelihood estimation of the model parameters is carried out by the Expectation-Maximization algorithm. The latent states can be ordered according to expected average values of the log-returns and their estimated volatility. We consider different model specifications in terms of number of latent states, which are identified in terms of expected log-returns and level of volatility. Under each considered scenario we also predict the latent state by the maximum a posteriori rule.

We review the discrete latent variable approach, which is very popular in statistics and related fields. It allows us to formulate interpretable and flexible models that can be used to analyze complex datasets in the presence of articulated dependence structures among variables. Specific models including discrete latent variables are illustrated, such as finite mixture, latent class, hidden Markov, and stochastic block models. Algorithms for maximum likelihood and Bayesian estimation of these models are reviewed, focusing, in particular, on the expectation–maximization algorithm and the Markov chain Monte Carlo method with data augmentation. Model selection, particularly concerning the number of support points of the latent distribution, is discussed. The approach is illustrated by summarizing applications avail- able in the literature; a brief review of the main software packages to handle discrete latent variable models is also provided. Finally, some possible developments in this literature are suggested.

We propose a Hidden Markov (HM) model for continuous longitudinal data with missing responses and dropout, thus extending the finite mixture model of multivariate Gaussian distributions. As known, the HM models assume the existence of an unobservable process, which follows a Markov chain with a discrete number of hidden states, affecting the distribution of the observed outcomes. We consider multivariate continuous responses that, for the same time occasion, are assumed to be correlated, according to a specific variance-covariance matrix, even conditionally on the latent states. For the analysis of such data, missing observations represent a relevant problem since dropout or non-monotone missing data patterns could occur. We propose an approach for inference with missing data by exploiting the steps of the Expectation-Maximization (EM) algorithm on the basis of suitable recursions. The resulting EM algorithm provides exact maximum likelihood estimates of model parameters under the missing-at-random (MAR) assumption, where the missing patterns are independent of the missing responses given all the observed data. The resulting HM model accounts for different types of missing pattern: (i) partially missing outcomes at a given time occasion; (ii) completely missing outcomes at a given time occasion (intermittent pattern); (iii) dropout before the end of the period of observation (monotone pattern). The estimation algorithm is also employed when there are available covariates supposed to affect the distribution of the latent process and, in particular, the initial and the transition probabilities of the Markov chain. In this way, it is possible to identify latent or unobserved clusters of units with homogeneous behavior and understand the influence of the covariates on the dynamic allocation of the individuals between states over time. The approach is illustrated by a Monte Carlo simulation study involving different scenarios. We also report an application based on the w

The Latent Class (LC) model is one of the most well-known latent variable models; it is very popular for the analysis of categorical response variables, and it is typically used to cluster subjects, by assuming the existence of individual-specific latent variables having a discrete distribution. A Hidden (or Latent) Markov (HM) model represents a generalization of the LC model to the case of longitudinal data. It assumes the existence of a discrete latent process generally following a first-order Markov chain, corresponding to subpopulations, usually referred to as latent states. As typically happens for discrete latent variable models, despite maximum likelihood estimation of both LC and HM model parameters can be rather simply performed using the Expectation-Maximization (EM) algorithm, a well-known drawback of this estimation method is related to the multimodality of the log-likelihood function. The consequence is that the estimation algorithm could converge to one of the local maxima, not corresponding to the global optimum. In order to face the multimodality problem described above, we propose a Tempered EM (T-EM) algorithm, which is able to explore the parameter space adequately. It consists in rescaling the objective function depending on a parameter known as the temperature, which controls global and local maxima prominence. High temperatures allow us to explore wide regions of the parameter space, avoiding the maximization algorithm being trapped in non-global maxima; low temperatures, instead, guarantee a sharp optimization in a local region of the parameter space. By properly tuning the sequence of temperature values, the target function is gradually attracted towards the global maximum, escaping local sub-optimal solutions. We rely on an accurate Monte Carlo simulation study to compare the proposal with the standard EM algorithm, evaluating both the ability to hit the global maximum and the computational time of the proposed algorithm. We also show the r