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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 results for both LC and HM models, using the proposal on discrete and continuous cross-sectional and longitudinal data in connection with some applications of interest. We conclude that the proposal outperforms the standard EM algorithm, significantly improving the chance to reach the global maximum in the overwhelming majority of considered cases. The advantage is relevant even considering the overall computing time.

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 maximize the log-likelihood with respect to v given the previous update of R. The proposal is validated through a simulation study showing that the estimators have good finite sample properties. We consider data on daily log-returns over four years of five cryptos Bitcoin, Bitcoin Cash, Ethereum, Litecoin, and Ripple as an application.

We consider maximum likelihood estimation of the Latent Class (LC) model, which is formulated through individual discrete latent variables. We explore tempering techniques to overcome the problem of multimodality of the log-likelihood function. A Tempered Expectation-Maximization (T-EM) algorithm is proposed, which can adequately explore the parameter space and reach the global maximum more frequently than the standard EM algorithm. We assess the performance of the proposed approach by a Monte Carlo simulation study and an application based on data about anxiety and depression in oncological patients.

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.

Sin dai primi giorni della diffusione dell’epidemia COVID-19 in Italia, stiamo sviluppando e confrontando diversi modelli per l’analisi di serie storiche di conteggi. In particolare, ci siamo focalizzati su modelli in grado di fornire previsioni stabili in tempo reale, pur disponendo di esigue osservazioni che vengono aggiornate giorno per giorno e che possono presentare problemi dovuti a questioni legate alla rilevazione dei dati. In particolare, è noto che in alcune situazioni e a causa dell’emergenza, i dati riferiti a più giorni di rilevazione sono stati accumulati in uno stesso giorno di pubblicazione creando delle irregolarità per le serie storiche osservate. Inizialmente abbiamo utilizzato un modello di Poisson ed un modello basato sulla distribuzione Binomiale Negativa per analizzare le serie storiche univariate dei pazienti malati o in una specifica condizione in Italia o in alcune regioni, in particolare in Lombardia. Tali modelli sono stati formulati includendo un trend temporale attraverso dei polinomi e delle spline e anche includendo delle componenti autoregressive del primo e del secondo ordine. Inoltre, per tenere conto dell’effetto dei provvedimenti emanati dal Governo e dalle regioni, sono state introdotte delle covariate di intervento o dei nodi nel caso di utilizzo di spline per il trend temporale. In seguito, ci siamo resi conto che l’interesse della collettività, così come dei decisori pubblici, è quello di conoscere quante persone si ammalano e, nello stesso tempo, quante persone hanno bisogno delle diverse tipologie di assistenza, con particolare riferimento alla terapia intensiva. Abbiamo quindi pensato ad un approccio multivariato basato su un modello Multinomiale autoregressivo che considera simultaneamente le diverse tipologie di pazienti ed include anche i deceduti e i guariti come categorie di osservazione. Questo modello permette di studiare le transizioni tra diverse categorie esclusive, giorno per giorno, e di prevedere in modo attendibile, almeno nel breve termine, il fabbisogno di posti letto in ospedale ed in terapia intensiva. È interessante notare che il modello assume una struttura Markoviana avente degli stati assorbenti, come ovviamente quello dei decessi, pur essendo stimato sulla base dei dati a livello aggregato, ovvero delle distribuzioni marginali delle tabelle di contingenza relative al numero di pazienti che transitano da una categoria all’altra in giorni consecutivi. Si presta inoltre ad essere interpretato come modello epidemiologico in linea con i più comuni modelli SEIR (Susceptible – Exposed – Infected – Recovered) ed è in grado di fornire una stima di indicatori di diffusione dell’infezione collegati con il numero di riproduzione di base (R0, numero medio di persone contagiate da una persona infetta). I parametri vengono stimati utilizzando l’approccio Bayesiano attraverso un algoritmo di tipo Markov chain Monte Carlo che permette di ricavare, in forma simulata, la distribuzione a posteriori di questi parametri. L’algoritmo che abbiamo implementato è basato su due passi che vengono iterati ripetutamente e che contemplano l’utilizzo di regole di accettazione di tipo Metropolis-Hastings. L’inferenza Bayesiana è particolarmente vantaggiosa se si intende utilizzare anche le informazioni a priori, quando disponibili, derivanti da altri paesi dove l’epidemia si è sviluppata in precedenza. Nel caso di COVID-19, in particolare, è vantaggioso utilizzare i dati della Cina da dove è partita la diffusione del virus. È inoltre possibile fornire intervalli di credibilità per i parametri ed effettuare confronti tra modelli con e senza effetti di intervento per valutare l’efficacia degli stessi. Gli esercizi di cross-validation che abbiamo effettuato in questi giorni hanno permesso di evidenziare interessanti risultati in termini di affidabilità previsionale nel breve termine. Auspichiamo che la proposta del modello Multinomiale autoregressivo possa essere utilizzata come strumento di ausilio per coloro che si trovano ad affrontare questa emergenza ed intendiamo continuare a perfezionare la proposta per renderla fruibile a tutta la comunità scientifica, anche mettendo a disposizione un apposito pacchetto di facile utilizzo nell’ambiente R. From the beginning of the spread of the novel coronavirus that emerged in Lombardy (Italy), at the end of February 2020, we are developing and comparing different models for time series with count data. In particular, we are considering models able to provide stable predictions with real time data, having only few observations updated day by day. The official data may have problems due to the collection process that is made from the local healthcare public company of the local area for each of the 21 Italian regions. In particular, during this emergency some date referred to more days of collection have been officialised in a single day, causing irregularity on the observed time series. We used a Poisson model and a model based on a Negative Binomial distribution to consider the univariate time series of the official COVID-19 data provided by the Civil Protection Agency and in particular of the patients in a specific condition such as in intensive care at national or at regional level. The models have been formulated with a temporal trend through polynomials and splines. We fitted the models considering also autoregressive components of first and second order. We introduced intervention effects through suitable covariates or nodes for the splines as to account for the effects of the disease control made by the local and national authorities on the temporal trend. With the time passing we realized that a main interest of the community, and of the policy makers is to know daily the number of ill patients, and at the same time, how many people need to be hospitalized and need especially intensive care. We developed a multivariate Multinomial autoregressive model to account simultaneously for different typologies of patients and as observational categories also for those deceased and recovered. The proposed model allows us to study the transitions between different mutually exclusive typologies of individuals day by day and it provides reliable predictions of the future daily counts in each category. We notice that the model assumes a Markovian structure with absorbing states, as that of deceased, even if it is estimated according to aggregate data. Therefore, we employ the marginals of the contingency table with respect to the patients transit from one category to another. The proposed model is also suitable to be interpreted as an epidemiological model in line with the more common SIRD (Susceptible-Infective-Recovered-Dead) models. It is able to provide an estimate of the reproducibility number (R0, average number of people infected by a person with the virus). The model parameters are estimated through a Bayesian approach with the algorithm Markov Chain Monte Carlo which allows us to dispose of the simulated posterior probabilities of the model parameters. The implemented algorithm is based on two steps which are iteratively repeated where at the second step the acceptance rule defined by the proposal of Metropolis-Hastings is considered. The Bayesian approach is particularly suitable when prior probabilities are available. In the context of COVID-19, we use data provided by China from where the spreading process of COVID-19 started. This approach allows us to dispose of credibility intervals for the parameters and to compare models with and without interventions to evaluate the efficacy of the restrictive measures due to context-specific interventions. A substantial very good prediction within a short term is demonstrated by exercises of Bayesian leave-one-out cross-validation performed during these days. The proposal of the Multinomial autoregressive model is an attempt to provide an instrument to support decisions during the emergency and we intend to continue to develop the model to make it suitable and available for the scientific community also by providing a friendly R package.

We use data derived from the World Bank and UNESCO Institute for Statis- tics, which are referred to 217 countries followed for 18 years, from 2000 to 2017, to study the persistence of economic conditions at country level over time. We consider several variables, such as GDP per capita, educational levels, life expectancy at birth, infant mortality rate, school enrollment, government expenditure in education, and the value of the Gini index that measures the inequality level of the distribution of income through household survey data (Schultz, 1998). These variables are related to the human development index proposed by the United Nations Development Programme (1990) for measuring the well-being at country level. The proposed approach allow us to charac- terize disparities among countries in a dynamic fashion so as to evaluate the changes in inequality over time by estimating a multivariate hidden Markov (HM) model (Bartolucci, Farcomeni, and Pennoni, 2013; Zucchini, MacDonald, and Langrock, 2016), also known as latent Markov model. In particular we pro- vide a dynamic clustering of the countries on the basis of the observed economic variables and, in turn, this permits to analyze inequalities in the development level. The resulting clustering is obtained by relying on a model tailored for longitudinal data rather than by predetermined threshold values as currently adopted by many international organizations to classify countries. The HM model assumes the existence of an unobservable process, which follows a Markov chain with a discrete number of latent states, affecting the distribution of the observed variables. In particular, we consider multivariate continuous re- sponses that, for the same time occasion, are assumed to be correlated accord- ing to a specific variance-covariance matrix, even conditionally on the latent states. In such a context, maximum likelihood estimation of model parameters is straightforward by means of the Expectation-Maximization (EM) algorithm (Dempster, Laird, and Rubin, 1977). Missing observations represent a relevant problem in this analysis, where values for some countries may be not available. Therefore, in this work, we also propose an approach for inference with missing data based on modifying the standard steps of the EM algorithm in a suitable way. In particular, we assume a structure of missing at random (MAR) re- sponses (Little and Rubin, 2002) where the missing patterns are independent of the missing responses given all the observed data. The resulting EM algorithm provides exact maximum likelihood estimates of the model parameters. Since the resulting latent states may be ordered according to the average values of the economic variables, inequalities among countries are detected for each year on the basis of the estimated posterior probabilities of the countries of be- longing to each latent state. In this way, we also explore patterns of disparities among countries and track their evolution from a dynamic perspective.

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.

L’Italia è localizzata al centro del bacino Mediterraneo, uno tra i centri di biodiversità a livello globale più minacciati. La sua eterogeneità ambientale e climatica ha permesso la differenziazione di un ampio numero di specie vegetali (oltre 8,200 taxa), che tuttavia sono sottoposte ad un crescente numero di minacce principalmente legate ai cambiamenti di dinamiche socio-economiche. Pertanto, urgenti misure di conservazione sono necessarie per fermare la perdita di diversità vegetale e conservare il patrimonio naturale del nostro paese. Questo volume è il prodotto finale di un progetto iniziato nel 2012, finanziato dal Ministero dell’Ambiente e della Tutela del Territorio e del Mare, portato avanti dalla Società Botanica Italiana. Complessivamente, i criteri per la redazione delle liste rosse elaborati dalla IUCN sono stati applicati a 2.488 taxa della flora vascolare autoctona del nostro paese, al fine di valutare il loro attuale stato di conservazione e mettere in evidenza le principali minacce che incombono su di esse. Dopo la pubblicazione del primo volume (2013), che include la valutazione di tutte le specie italiane incluse nelle normative ratificate a livello nazionale (policy species), quali le specie elencate negli allegati della Direttiva 92/43/UE “Habitat” e negli allegati della Convenzione di Berna, in questo contributo è stato valutato un ulteriore gruppo di 2.191 taxa, incluse tutte le specie endemiche italiane non ancora sottoposte ad assessment. I risultati pubblicati nel presente volume hanno rilevato che 37 taxa sono estinti, estinti in natura o verosimilmente estinti a livello regionale (inclusi 11 taxa endemici), mentre 420 taxa (inclusi 228 taxa endemici) sono stati assegnati ad una categoria di rischio. Inoltre, 338 taxa (inclusi 248 taxa endemici) sono stati inseriti nella categoria DD, ovvero specie per cui i dati a disposizione sono insufficienti per una valutazione. La flora vascolare italiana è minacciata soprattutto dalla modificazione degli habitat naturali e semi-naturali, dovuta al disturbo antropico e, specialmente, ad agricoltura, turismo e sviluppo residenziale. Il numero più alto di taxa estinti o in declino è infatti localizzato nelle zone costiere e di pianura, dove gli impatti antropici e la distruzione degli ecosistemi sono più evidenti. La valutazione di circa un terzo dei taxa vegetali spontanei del nostro paese costituisce un importante passo verso la conservazione della flora italiana. Tuttavia, per mettere in atto una strategia di conservazione efficace del paesaggio naturale italiano e in particolare della flora vascolare, ulteriori ricerche ed approfondimenti sono necessari. Sarebbe infatti urgente risolvere alcune criticità tassonomiche, effettuare monitoraggi sulle specie minacciate e sviluppare strategie specifiche di conservazione in situ ed ex situ e piani di azione specie specifici.

Il capitolo ha l’intento di illustrare in modo divulgativo l’utilizzo dei modelli statistici per analizzare, l’andamento dei casi di pazienti affetti da COVID-19. In particolare si considera la serie degli individui attualmente positivi, che comprende coloro che sono in isolamento domiciliare, i ricoverati con sintomi e in terapia intensiva. I dati utilizzati nel presente lavoro sono quelli forniti giornalmente a livello ufficiale dal Dipartimento della Protezione Civile sull’andamento del COVID-19 nella regione Umbria e nell’intera penisola italiana. La prima classe di modelli che viene presa in esame è basata sulla distribuzione di Poisson, che è stata proposta dal matematico francese Siméon Denis Poisson (1781-1840) e poi estesa recentemente per tenere conto della correlazione temporale dei dati. La seconda classe di modelli è basata sulla distribuzione Binomiale Negativa che, pur essendo meno utilizzata della distribuzione di Poisson presenta delle peculiarità di interesse ed è stata formulata, in una versione preliminare, dal matematico Blaise Pascal (1623-1662). Infatti una formulazione particolare di questo modello è nota anche come distribuzione di Pascal. Questi modelli permettono di effettuare delle previsioni e dei confronti tra regioni o nazioni in quanto servono a regolarizzare i dati raccolti giornalmente. Nel presente capitolo si confronta l’andamento della prevalenza stimata per l’Umbria e per l’Italia.

Seventeen European endemic plant species were considered extinct, but improved taxonomic and distribution knowledge as well as ex situ collecting activities brought them out of the extinct status. These species have now been reported into a conservation framework that may promote legal protection and in situ and ex situ conservation. © 2021, The Author(s), under exclusive licence to Springer Nature Limited part of Springer Nature.

Hidden Markov models represent a popular tool for the analysis of longitudinal data, allowing the dynamic clustering of sample units on the basis of a set of repeated responses. In the literature on longitudinal data analysis, these models are typically used in the presence of multivariate categorical data, that is, when more categorical responses are observed at each time occasion. These formulations rely on the assumption of local independence, according to which the responses are conditionally independent given the latent states. Such assumption also simplifies the treatment of missing responses when the missing-at-random assumption is plausible. Here, we deal with the case of continuous multivariate responses in which, as in a Gaussian mixture models, it is natural to assume that the continuous responses for the same time occasion are correlated, according to a specific variance-covariance matrix, even conditionally on the latent states. Although maximum likelihood estimation of this model is straightforward in standard cases using the Expectation-Maximization algorithm, we focus on its estimation when: (i) suitable constraints on the variance-covariance matrix are assumed; (ii) there are missing responses. The constraints we refer to are commonly adopted in the literature of Gaussian finite mixture models. Regarding the assumptions on the generation of missing data we focus on the missing-at-random assumption and we also account for possible individual covariates that may directly affect the responses (in addition to the latent states). In particular, we propose an Expectation Maximization (EM) algorithm that provides exact maximum likelihood estimates and also computes standard errors for the parameter estimates. The proposed approach is illustrated by a simulation study, to evaluate the computational load, and through a real case analysis. We also show how the proposal may be useful in a context of time-series analysis with an application to financial data. An R implementation of the proposed algorithm is made available by the authors within the LMest package.

A review of certain latent variable models that may be effectively used to build evaluation systems is proposed. The review is focused on the item response theory models, the latent class models and the latent Markov models. Certain latent Markov versions are illustrated in detail from the perspective of evaluation when multilevel or non-experimental longitudinal data are available.

The results of a project for the identification of species-specific monitoring protocols for the Italian plant species protected under the Habitats Directive (Annexes II/IV/V) are presented. The project led to the development of 118 monitoring factsheets, providing an operational guidance for 107 vascular taxa, 10 bryophytes and 1 lichen taxon. Each factsheet includes information on the species (distribution, biology, ecology, conservation status, threats, etc.) and the description of field methodologies for the detection of the two main reporting parameters, i.e. population size and habitat quality. Practical information to plan field activities are also given. Protocols were designed to address the requirements of the European reporting system with the aim to standardize future monitoring activities, optimize efforts at national scale and overcome some current problems related to data heterogeneity and discrepancies from the EC standards. More than 60 botanists collaborated to identify the best practices and to design an operational field survey format through several stages of discussion and sharing. The protocols, developed by ISPRA and Scientific Societies and shared with the Italian institutions responsible for the Directive application, were published in a dedicated National handbook. The work provides a first uniform technical basis for future national monitoring plans.

Studies to date about plants alien to Italy have had limited focus on the ecology of this component of the flora. Ellenberg's indicator values are a useful tool to delineate the relationship between plants and environment, recognizing for each species a functional role as biological indicator; these values have been proposed for estimating the influence of the main environmental factors in determining flora and vegetation changes on a specific surface area. This contribution includes a list of 1206 taxa of plants naturalized in at least one administrative region or casual in at least three regions of Italy. In addition, some methodological considerations on the attribution and use of Ellenberg's indicator values and a comparison with the average indices for the native Italian flora are reported.

Statistical methods currently employed for marketing are based on strong parametric assumptions to correct for endogeneity. These assumptions may com- promise the results when applied to data in which the research expects endogeneity, but this is not the case. By considering the recent advances in the literature of causal models for panel data we propose to employ a latent Markov model which is tailored to estimate causal effects of a marketing campaign and to deal dynamically with endogeneity. Our proposal does not require the strong assumption of strictly absence of unmeasured confounders. The endogeneity correction imposes no restrictions on the data and it does not bias results when the endogeneity is not in. We assess the effects of multiple mail campaigns conducted by a large European bank with the purpose of predicting customer acquisitions of financial products.

Many statistical methods currently employed to evaluate the effect of a marketing campaign in dealing with observational data advocate strong parametric assumptions to correct for endogeneity among the participants. In addition, the assumptions compromise the estimated values when applied to data in which the research expects endogeneity but this is not realized. Based on the recent advances in the literature of causal models dealing with data collected across time, we propose a dynamic version of the inverse-probability-of-treatment weighting within the latent Markov model. The proposal, which is based on a weighted maximum likelihood approach, accounts for endogeneity without imposing strong restrictions. The likelihood function is maximized through the Expectation-Maximization algorithm which is suitably modified to account for the inverse probability weights. Standard errors for the parameters estimates are obtained by a nonparametric bootstrap method. We show the effects of multiple mail campaigns conducted by a large European bank with the purpose to influence their customers to the acquisitions of the addressed financial products