Decomposition of Seasonality and Long‐term Trend in Seismological Data: A Bayesian Modelling of Earthquake Detection Capability

This study demonstrates the decomposition of seasonality and long‐term trend in seismological data observed at irregular time intervals. The decomposition was applied to the estimation of earthquake detection capability using cubic B‐splines and a Bayesian approach, which is similar to the seasonal adjustment model frequently used to analyse economic time‐series data. We employed numerical simulation to verify the method and then applied it to real earthquake datasets obtained in and around the northern Honshu island, Japan. With this approach, we obtained the seasonality of the detection capability related to the annual variation of wind speed and the long‐term trend corresponding to the recent improvement of the seismic network in the studied region.     

A Conditional Autoregressive Gaussian Process for Irregularly Spaced Multivariate Data with Application to Modelling Large Sets of Binary Data

A Gaussian conditional autoregressive (CAR) formulation is presented that permits the modelling of the spatial dependence and the dependence between multivariate random variables at irregularly spaced sites so capturing some of the modelling advantages of the geostatistical approach. The model benefits not only from the explicit availability of the full conditionals but also from the computational simplicity of the precision matrix determinant calculation using a closed form expression involving the eigenvalues of a precision matrix submatrix. The introduction of covariates into the model adds little computational complexity to the analysis and thus the method can be straightforwardly extended to regression models. The model, because of its computational simplicity, is well suited to application involving the fully Bayesian analysis of large data sets involving multivariate measurements with a spatial ordering. An extension to spatio-temporal data is also considered. Here, we demonstrate use of the model in the analysis of bivariate binary data where the observed data is modelled as the sign of the hidden CAR process. A case study involving over 450 irregularly spaced sites and the presence or absence of each of two species of rain forest trees at each site is presented; Markov chain Monte Carlo (MCMC) methods are implemented to obtain posterior distributions of all unknowns. The MCMC method works well with simulated data and the tree biodiversity data set.     

Multivariate time series prediction using a hybridization of VARMA models and Bayesian networks

In this paper, a new hybrid model of vector autoregressive moving average (VARMA) models and Bayesian networks is proposed to improve the forecasting performance of multivariate time series. In the proposed model, the VARMA model, which is a popular linear model in time series forecasting, is specified to capture the linear characteristics. Then the errors of the VARMA model are clustered into some trends by K-means algorithm with Krzanowski–Lai cluster validity index determining the number of trends, and a Bayesian network is built to learn the relationship between the data and the trend of its corresponding VARMA error. Finally, the estimated values of the VARMA model are compensated by the probabilities of their corresponding VARMA errors belonging to each trend, which are obtained from the Bayesian network. Compared with VARMA models, the experimental results with a simulation study and two multivariate real-world data sets indicate that the proposed model can effectively improve the prediction performance.     

Bayesian computation for Log-Gaussian Cox processes: a comparative analysis of methods

The Log-Gaussian Cox process is a commonly used model for the analysis of spatial point pattern data. Fitting this model is difficult because of its doubly stochastic property, that is, it is a hierarchical combination of a Poisson process at the first level and a Gaussian process at the second level. Various methods have been proposed to estimate such a process, including traditional likelihood-based approaches as well as Bayesian methods. We focus here on Bayesian methods and several approaches that have been considered for model fitting within this framework, including Hamiltonian Monte Carlo, the Integrated nested Laplace approximation, and Variational Bayes. We consider these approaches and make comparisons with respect to statistical and computational efficiency. These comparisons are made through several simulation studies as well as through two applications, the first examining ecological data and the second involving neuroimaging data.     

A Bayesian structural-change analysis via the stochastic approximation Monte Carlo and Gibbs sampler

In this article, we propose a Bayesian approach to estimate the multiple structural change-points in a level and the trend when the number of change-points is unknown. Our formulation of the structural-change model involves a binary discrete variable that indicates the structural change. The determination of the number and the form of structural changes are considered as a model selection issue in Bayesian structural-change analysis. We apply an advanced Monte Carlo algorithm, the stochastic approximation Monte Carlo (SAMC) algorithm, to this structural-change model selection issue. SAMC effectively functions for the complex structural-change model estimation, since it prevents entrapment in local posterior mode. The estimation of the model parameters in each regime is made using the Gibbs sampler after each change-point is detected. The performance of our proposed method has been investigated on simulated and real data sets, a long time series of US real gross domestic product, US uses of force between 1870 and 1994 and 1-year time series of temperature in Seoul, South Korea.     

Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo

Abstract. We investigate simulation methodology for Bayesian inference in Lévy‐driven stochastic volatility (SV) models. Typically, Bayesian inference from such models is performed using Markov chain Monte Carlo (MCMC); this is often a challenging task. Sequential Monte Carlo (SMC) samplers are methods that can improve over MCMC; however, there are many user‐set parameters to specify. We develop a fully automated SMC algorithm, which substantially improves over the standard MCMC methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston model with an independent, additive, variance gamma process in the returns equation. The driving gamma process can capture the stylized behaviour of many financial time series and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data. We demonstrate that it is possible to draw exact inference, in the sense of no time‐discretization error, from the Bayesian SV model.     

Bayesian diffusion process models with time-varying parameters

The diffusion process is a widely used statistical model for many natural dynamic phenomena but its inference is very complicated because complete data describing the diffusion sample path is not necessarily available. In addition, data is often collected with substantial uncertainty and it is not uncommon to have missing observations. Thus, the observed process will be discrete over a finite time period and the marginal likelihood given by this discrete data is not always available. In this paper, we consider a class of nonstationary diffusion process models with not only the measurement error but also discretely time-varying parameters which are modeled via a state space model. Hierarchical Bayesian inference for such a diffusion process model with time-varying parameters is applied to financial data.     

Mobile safety cameras: estimating casualty reductions and the demand for secondary healthcare

We consider a fully Bayesian analysis of road casualty data at 56 designated mobile safety camera sites in the Northumbria Police Force area in the UK. It is well documented that regression to the mean (RTM) can exaggerate the effectiveness of road safety measures and, since the 1980s, an empirical Bayes (EB) estimation framework has become the gold standard for separating real treatment effects from those of RTM. In this paper we suggest some diagnostics to check the assumptions underpinning the standard estimation framework. We also show that, relative to a fully Bayesian treatment, the EB method is over-optimistic when quantifying the variability of estimates of casualty frequency. Implementing a fully Bayesian analysis via Markov chain Monte Carlo also provides a more flexible and complete inferential procedure. We assess the sensitivity of estimates of treatment effectiveness, as well as the expected monetary value of prevention owing to the implementation of the safety cameras, to different model specifications, which include the estimation of trend and the construction of informative priors for some parameters.     

Spatial–temporal interpolation of non methane hydrocarbon levels in Kuwait

This article handles the prediction of hourly concentrations ofnon methane hydrocarbon (NMHC) pollutants at 15 unmonitored sites in Kuwait using the data recorded from 6 monitored stations at successive time points. The trend model depends on hourly meteorological variables and seasonal effects. The stochasticcomponent of the trend model which has spatiotemporal features is modeled as autoregressive temporal process. A spatial predictive distribution for residuals of the AR model is developed for the unmonitored sites. By transforming the predicted residuals back to the original data scales, we impute Kuwait’s hourly NMHC field.     

Simulation-based model comparison methodology with application to road accident models

This article assumes the goal of proposing a simulation-based theoretical model comparison methodology with application to two time series road accident models. The model comparison exercise helps to quantify the main differences and similarities between the two models and comprises of three main stages: (1) simulation of time series through a true model with predefined properties; (2) estimation of the alternative model using the simulated data; (3) sensitivity analysis to quantify the effect of changes in the true model parameters on alternative model parameter estimates through analysis of variance, ANOVA. The proposed methodology is applied to two time series road accident models: UCM (unobserved components model) and DRAG (Demand for Road Use, Accidents and their Severity). Assuming that the real data-generating process is the UCM, new datasets approximating the road accident data are generated, and DRAG models are estimated using the simulated data. Since these two methodologies are usually assumed to be equivalent, in a sense that both models accurately capture the true effects of the regressors, we are specifically addressing the modeling of the stochastic trend, through the alternative model. Stochastic trend is the time-varying component and is one of the crucial factors in time series road accident data. Theoretically, it can be easily modeled through UCM, given its modeling properties. However, properly capturing the effect of a non-stationary component such as stochastic trend in a stationary explanatory model such as DRAG is challenging. After obtaining the parameter estimates of the alternative model (DRAG), the estimates of both true and alternative models are compared and the differences are quantified through experimental design and ANOVA techniques. It is observed that the effects of the explanatory variables used in the UCM simulation are only partially captured by the respective DRAG coefficients. This a priori, could be due to multicollinearity but the results of both simulation of UCM data and estimating of DRAG models reveal that there is no significant static correlation among regressors. Moreover, in fact, using ANOVA, it is determined that this regression coefficient estimation bias is caused by the presence of the stochastic trend present in the simulated data. Thus, the results of the methodological development suggest that the stochastic component present in the data should be treated accordingly through a preliminary, exploratory data analysis.     

A Bayesian kriged Kalman model for short-term forecasting of air pollution levels

Summary.  Short-term forecasts of air pollution levels in big cities are now reported in news-papers and other media outlets. Studies indicate that even short-term exposure to high levels of an air pollutant called atmospheric particulate matter can lead to long-term health effects. Data are typically observed at fixed monitoring stations throughout a study region of interest at different time points. Statistical spatiotemporal models are appropriate for modelling these data. We consider short-term forecasting of these spatiotemporal processes by using a Bayesian kriged Kalman filtering model. The spatial prediction surface of the model is built by using the well-known method of kriging for optimum spatial prediction and the temporal effects are analysed by using the models underlying the Kalman filtering method. The full Bayesian model is implemented by using Markov chain Monte Carlo techniques which enable us to obtain the optimal Bayesian forecasts in time and space. A new cross-validation method based on the Mahalanobis distance between the forecasts and observed data is also developed to assess the forecasting performance of the model implemented.     

Fusing point and areal level space–time data with application to wet deposition

Summary.  Motivated by the problem of predicting chemical deposition in eastern USA at weekly, seasonal and annual scales, the paper develops a framework for joint modelling of point- and grid-referenced spatiotemporal data in this context. The hierarchical model proposed can provide accurate spatial interpolation and temporal aggregation by combining information from observed point-referenced monitoring data and gridded output from a numerical simulation model known as the 'community multi-scale air quality model'. The technique avoids the change-of-support problem which arises in other hierarchical models for data fusion settings to combine point- and grid-referenced data. The hierarchical space–time model is fitted to weekly wet sulphate and nitrate deposition data over eastern USA. The model is validated with set-aside data from a number of monitoring sites. Predictive Bayesian methods are developed and illustrated for inference on aggregated summaries such as quarterly and annual sulphate and nitrate deposition maps. The highest wet sulphate deposition occurs near major emissions sources such as fossil-fuelled power plants whereas lower values occur near background monitoring sites.     

Spatio-temporal model for crop yield forecasting

This paper proposes a linear mixed model (LMM) with spatial effects, trend, seasonality and outliers for spatio-temporal time series data. A linear trend, dummy variables for seasonality, a binary method for outliers and a multivariate conditional autoregressive (MCAR) model for spatial effects are adopted. A Bayesian method using Gibbs sampling in Markov Chain Monte Carlo is used for parameter estimation. The proposed model is applied to forecast rice and cassava yields, a spatio-temporal data type, in Thailand. The data have been extracted from the Office of Agricultural Economics, Ministry of Agriculture and Cooperatives of Thailand. The proposed model is compared with our previous model, an LMM with MCAR, and a log transformed LMM with MCAR. We found that the proposed model is the most appropriate, using the mean absolute error criterion. It fits the data very well in both the fitting part and the validation part for both rice and cassava. Therefore, it is recommended to be a primary model for forecasting these types of spatio-temporal time series data.     

Bayesian inference for non-stationary spatial covariance structure via spatial deformations

Summary. In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, i.e. the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin. However, in environmental problems, such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data. The paper proposes a Bayesian model to address the anisot- ropy problem. Following Sampson and Guttorp, we define the correlation function of the spatial process by reference to a latent space, denoted by D , where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by G . We adopt a Bayesian approach in which the mapping between G and D is represented by an unknown function d (·). A Gaussian process prior distribution is defined for d (·). Unlike the Sampson–Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Markov chain Monte Carlo methods are used to obtain samples from the posterior distributions. Two examples are discussed: a simulated data set and the solar radiation data set that also was analysed by Sampson and Guttorp.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.     

Bayesian variable selection for longitudinal substance abuse treatment data subject to informative censoring

Summary.  Measuring the process of care in substance abuse treatment requires analysing repeated client assessments at critical time points during treatment tenure. Assessments are frequently censored because of early departure from treatment. Most analyses accounting for informative censoring define the censoring time to be that of the last observed assessment. However, if missing assessments for those who remain in treatment are attributable to logistical reasons rather than to the underlying treatment process being measured, then the length of stay in treatment might better characterize censoring than would time of measurement. Bayesian variable selection is incorporated in the conditional linear model to assess whether time of measurement or length of stay better characterizes informative censoring. Marginal posterior distributions of the trajectory of treatment process scores are obtained that incorporate model uncertainty. The methodology is motivated by data from an on-going study of the quality of care in in-patient substance abuse treatment.     

Forward Simulation Markov Chain Monte Carlo with Applications to Stochastic Epidemic Models

For many stochastic models, it is difficult to make inference about the model parameters because it is impossible to write down a tractable likelihood given the observed data. A common solution is data augmentation in a Markov chain Monte Carlo (MCMC) framework. However, there are statistical problems where this approach has proved infeasible but where simulation from the model is straightforward leading to the popularity of the approximate Bayesian computation algorithm. We introduce a forward simulation MCMC (fsMCMC) algorithm, which is primarily based upon simulation from the model. The fsMCMC algorithm formulates the simulation of the process explicitly as a data augmentation problem. By exploiting non‐centred parameterizations, an efficient MCMC updating schema for the parameters and augmented data is introduced, whilst maintaining straightforward simulation from the model. The fsMCMC algorithm is successfully applied to two distinct epidemic models including a birth–death–mutation model that has only previously been analysed using approximate Bayesian computation methods.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.     

Bayesian Multivariate Spatial Interpolation with Data Missing by Design

In a network of s _g sites, responses like levels of airborne pollutant concentrations may be monitored over time. The sites need not all measure the same set of response items and unmeasured items are considered as data missing by design . We propose a hierarchical Bayesian approach to interpolate the levels of, say, k responses at s _u other locations called ungauged sites and also the unmeasured levels of the k responses at the gauged sites. Our method involves two steps. First, when all hyperparameters are assumed to be known, a predictive distribution is derived. In turn, an interpolator, its variance and a simultaneous interpolation region are obtained. In step two, we propose the use of an empirical Bayesian approach to estimate the hyperparameters through an EM algorithm. We base our theory on a linear Gaussian model and the relationship between a multivariate normal and matrix T -distribution. Our theory allows us to pool data from several existing networks that measure different subsets of response items for interpolation.