Luigi Spezia

Bayesian inference in non-homogeneous Markov mixtures of periodic autoregressions with state-dependent exogenous variables

The Bayesian analysis of a non-homogeneous Markov mixture of periodic autoregressions with state-dependent exogenous variables is proposed to investigate a non-linear and non-Normal time series. It is performed within a Markov chain Monte Carlo framework, along four consecutive steps: the specification of the identifiability constraint; the selection of the exogenous variables which influence the observed process and the time-varying transition probabilities of the hidden Markov chain; the choice of the cardinality of the hidden Markov chain state-space and the autoregressive order; the estimation of the parameters. The selection of the exogenous variables is performed in the complex case of correlation between variables, by means of a new procedure. An application for relating the hourly mean concentrations of sulphur dioxide with six meteorological variables, recorded for three years by an air pollution testing station located in the lagoon of Venice (Italy), is presented. The reconstruction of the sequence of the hidden states, the restoration of the missing values occurring within the observed series, the description of the periodic component are also given.

Bayesian Variable Selection in Markov Mixture Models

Bayesian methods for variable selection and model choice have become increasingly popular in recent years, due to advances in Markov Chain Monte Carlo (MCMC) computational algorithms. Several methods have been proposed in literature in the case of linear and generalized linear models. In this article, we adapt some of the most popular algorithms to a class of nonlinear and non Gaussian time series models, i.e., the Markov Mixture Models (MMM). We also propose the “Metropolization” of the algorithm of Kuo and Mallick (1998), in order to tackle variable selection efficiently, both when the complexity of the model is high, as in MMM, and when the exogenous variables are strongly correlated. Numerical comparisons among the competing MCMC algorithms are also presented via simulation examples.

Bayesian analysis of non-homogeneous hidden Markov models

Bayesian estimation of the unknown parameters of a non-homogeneous Gaussian hidden Markov model is described here. The hidden Markov chain presents time-varying transition probabilities, depending on exogenous variables through a logistic function. Bayesian model choice is also proposed to select the unknown number of states of the hidden non-homogeneous Markov chain. Both the analyses are developed by using Markov chain Monte Carlo algorithms. Model selection and parameter estimation are performed after making the model identifiable, by selecting suitable constraints through a data-driven procedure. The methodology is illustrated by an empirical analysis of ozone data.

Bayesian Inference and Forecasting in Dynamic Neural Networks with Fully Markov Switching ARCH Noises

We deal with one-layer feed-forward neural network for the Bayesian analysis of nonlinear time series. Noises are modeled nonlinearly and nonnormally, by means of ARCH models whose parameters are all dependent on a hidden Markov chain. Parameter estimation is performed by sampling from the posterior distribution via Evolutionary Monte Carlo algorithm, in which two new crossover operators have been introduced. Unknown parameters of the model also include the missing values which can occur within the observed series, so, considering future values as missing, it is also possible to compute point and interval multi-step-ahead predictions.

Mapping species distributions in one dimension by non-homogeneous hidden Markov models: the case of freshwater pearl mussels in the River Dee

The investigation of species distributions in rivers involves data which are inherently sequential and unlikely to be fully independent. To take these characteristics into account, we develop a Bayesian hierarchical model for mapping the distribution of freshwater pearl mussels in the River Dee (Scotland). At the top of the hierarchy the likelihood is used to describe the sequence of sites in which mussels were observed or not. Given that false observations can occur, and that “not observed” means both that the species was not present and that it was not observed, a Markov prior is introduced at the second level of the hierarchy to represent the sequence of sites in which mussels are estimated to occur. The Markov prior allows modelling the spatial dependency between neighbouring sites. A third level in the hierarchy is given by the representation of the transition probabilities of the Markov chain in terms of site-specific explanatory variables, through a logistic regression. The selection of the explanatory variables which influence the Markov process is performed by means of a simulation-based procedure, in the complex case of association between covariates. Four features were found to be associated with reduced chance of finding a local mussel population: tributaries, bridges, dredging, and waste water treatment works. These results complement the results of a previous study, providing new evidence for the causes of the deterioration of a highly threatened species.

Periodic Markov switching autoregressive models for Bayesian analysis and forecasting of air pollution

Markov switching autoregressive models (MSARMs) are efficient tools to analyse nonlinear and non-Gaussian time series. A special MSARM with two harmonic components is proposed to analyse periodic time series. We present a full Bayesian analysis based on a Gibbs sampling algorithm for model choice and the estimations of the unknown parameters, missing data and predictive distributions. The implementation and modelling steps are developed by tackling the problem of the hidden states labeling by means of random permutation sampling and constrained permutation sampling. We apply MSARMs to study a data set about air pollution that presents periodicities since the hourly mean concentration of carbon monoxide varies according to the dynamics of the 24 day-hours and of the year. Hence, we introduce in the model both a hidden state-dependent daily component and a state-independent yearly component, giving rise to periodic MSARMs.

Example of Bayesian Uncertainty for Digital Soil Mapping

Any model for digital soil mapping suffers from different types of errors, including interpolation errors, so it is important to quantify the uncertainty associated with the maps produced. The most common approach is some form of regression kriging (RK) or variation involving geostatistical simulation. Another way of assessing the spatial uncertainty lies in the Bayesian approach where the uncertainty in the results is described by the posterior density. The aim of this paper is to present an example of a Bayesian approach for uncertainty estimation when mapping the topsoil organic matter content in the Grampian region of Scotland (UK, about 12,100 km2). The chosen approach uses (Bayesian) latent Gaussian models fitted using integrated nested Laplace approximation (INLA) and the stochastic partial differential equation (SPDE) models approach for coping with spatial correlation (INLA_SPDE). For practical comparison purposes, the results of INLA_SPDE were compared with the results of an extension of the scorpan kriging approach, i.e., (1) combining generalized additive models (GAM) with Gaussian simulations and (2) traditional RK. The results were assessed using in-sample and out-of-sample measures and compared for distribution similarity, spatial structure reproduction, computational load, and uncertainty ranges. We conclude that the Bayesian framework using INLA offers a viable alternative to existing methods and an improvement over traditional RK.

Modelling species abundance in a river by Negative Binomial hidden Markov models

The investigation of species abundance in rivers involves data which are inherently sequential and unlikely to be fully independent. To take these characteristics into account, a Bayesian hierarchical model within the class of hidden Markov models is proposed to map the distribution of freshwater pearl mussels in the River Dee (Scotland). In order to model the overdispersed series of mussel counts, the conditional probability function of each observation, given the hidden state, is assumed to be Negative Binomial. Both the transition probabilities of the hidden Markov chain and the state-dependent means of the observed process depend on covariates obtained from a hydromorphological survey. Bayesian inference, model choice, and covariate selection based on Markov chain Monte Carlo algorithms are presented. The stochastic selection of the explanatory variables which are associated with a reduced chance of finding a local mussel population provides new evidence for the causes of the deterioration of a highly threatened species.

Spatial hidden Markov models and species distributions

ABSTRACT A spatial hidden Markov model (SHMM) is introduced to analyse the distribution of a species on an atlas, taking into account that false observations and false non-detections of the species can occur during the survey, blurring the true map of presence and absence of the species. The reconstruction of the true map is tackled as the restoration of a degraded pixel image, where the true map is an autologistic model, hidden behind the observed map, whose normalizing constant is efficiently computed by simulating an auxiliary map. The distribution of the species is explained under the Bayesian paradigm and Markov chain Monte Carlo (MCMC) algorithms are developed. We are interested in the spatial distribution of the bird species Greywing Francolin in the south of Africa. Many climatic and land-use explanatory variables are also available: they are included in the SHMM and a subset of them is selected by the mutation operators within the MCMC algorithm.