Summary This study considers spatial dependence in the number of injury crashes reported on a road network. The aggregated crash counts are considered realisations of a Poisson random variable; thus, we model both over‐dispersion and serial correlation using the Poisson hidden Markov model (PHMM). PHMMs have typically been used for modelling temporal dependence, but they have rarely been used to model spatial dependence. Our interest, however, is specifically in relation to an underlying point process which is constrained to occur on a network. We illustrate the use of the PHMM with police‐reported data on injury road collisions on selected motorways in the United Kingdom over a 5‐year period (2010–2014). We use officially recorded estimates of traffic volume as an exposure variable. The aim is to identify highway segments which might have distinctly high crash rates. To do this, we first select an optimal model in terms of the number of latent states. As we use a Bayesian approach, we can assign a posterior classification probability to each segment in terms of membership of the various underlying risk states. Model fitting is conducted using the Markov chain Monte Carlo approach; we develop a new modified version of the Akaike Information Criterion and Bayesian Information Criterion, approximated from a Bayesian framework, to select the best model in terms of number of states.

This study investigates the spatial heterogeneity in the maximum monthly rainfall amounts reported by stations in Ireland from January 2018 to December 2020. The heterogeneity is modeled by the Bayesian normal mixture model with different ranks. The selection of the best model or the degree of heterogeneity is implemented using four criteria which are the modified Akaike information criterion, the modified Bayesian information criterion, the deviance information criterion, and the widely applicable information criterion. The estimation and model selection process is implemented using the Gibbs sampling. The results show that the maximum monthly rainfall amounts are accommodated in two and three components. The goodness of fit for the selected models is checked using the graphical plots including the probability density function and cumulative distribution function. This article also contributes via the spatial determination of return level or rainfall amounts at risk with different return periods using the prediction intervals constructed from the posterior predictive distribution.

This article aims at identifying the high risk provinces in Iraq using a finite Poisson mixture. Through this methodology, the levels of relative risk is determined through identifying the number of components. In this article we do not investigate spatial correlation among regions and assume that the levels of risk observed in different regions are independent each other. The estimation of the model parameters and the model selection are performed using the Bayesian approach which allow to allocate each province to an identified risk level. We consider the data of the Coronavirus disease (COVID-19) infections in 18 provinces in Iraq and determining the levels of relative risks of this pandemic. The results are spatially shown in map which illustrates that the best Bayesian model fitted the data is 3 components model (high, medium and low risk).

PurposeOne of the most important sources of energy in the world, due to its great impact on the global economy, is the crude oil. Due to the instability of oil prices which exhibit extreme fluctuations during periods of different times of market uncertainty, it became hard to the governments to predict accurately the prices of crude oil in order to build their financial budgets. Therefore, this study aims to analyse and model crude oil price using the hidden Markov process (HMM).Design/methodology/approachTraditional mathematical approaches of time series may be not give accurate results to measure and analyse the crude oil price, since the latter has an unstable and fluctuating nature, hence, its prediction forms a challenge task. A novel methodology that is so-called the HMM is proposed that takes into account the heterogeneity in prices as well as their hidden state-based behaviour.FindingsUsing the Bayesian approach, several estimated models with different ranks are fitted to a non-homogeneous data of Iraqi crude oil prices from January 2010 into December 2021. The model selection criteria and measures of the prediction performance of each model are applied to choose the best model. Movements of crude oil prices exhibit extreme fluctuations during periods of different times of market uncertainty. The processes of model estimation and the model selection were conducted in Python V.3.10, and it is available from the first author on request.Originality/valueUsing the Bayesian approach, several estimated models with different ranks are fitted to a non-homogeneous data of Iraqi crude oil prices from January 2010 to December 2021.

In Bayesian model selection, the deviance information criterion (DIC) has become a widely used criterion. It is however not defined for the hidden Markov models (HMMs). In particular, the main challenge of applying the DIC for HMMs is that the observed likelihood function of such models is not available in closed form. A closed form for the observed likelihood function can be obtained either by summing all possible hidden states of the complete likelihood using the so-called the forward recursion, or via integrating out the hidden states in the conditional likelihood. Hence, we propose two versions of the DIC to the model choice problem in HMMs context, namely, the recursive deviance-based DIC and the conditional likelihood-based DIC. In this paper, we compare several normal HMMs after they are estimated by Bayesian MCMC method. We conduct a simulation study based on synthetic data generated under two assumptions, namely diversity in the heterogeneity level and also the number of states. We show that the recursive deviance-based DIC performs well in selecting the correct model compared with the conditional likelihood-based DIC that prefers the more complicated models. A real application involving the waiting time of Old Faithful Geyser data was also used to check those criteria. All the simulations were conducted in Python v.2.7.10, available from first author on request.