A hierarchical model for space–time surveillance data on meningococcal disease incidence

Summary. We describe a model-based approach to analyse space–time surveillance data on meningococcal disease. Such data typically comprise a number of time series of disease counts, each representing a specific geographical area. We propose a hierarchical formulation, where latent parameters capture temporal, seasonal and spatial trends in disease incidence. We then add—for each area—a hidden Markov model to describe potential additional (autoregressive) effects of the number of cases at the previous time point. Different specifications for the functional form of this autoregressive term are compared which involve the number of cases in the same or in neighbouring areas. The two states of the Markov chain can be interpreted as representing an 'endemic' and a 'hyperendemic' state. The methodology is applied to a data set of monthly counts of the incidence of meningococcal disease in the 94 départements of France from 1985 to 1997. Inference is carried out by using Markov chain Monte Carlo simulation techniques in a fully Bayesian framework. We emphasize that a central feature of our model is the possibility of calculating—for each region and each time point—the posterior probability of being in a hyperendemic state, adjusted for global spatial and temporal trends, which we believe is of particular public health interest.     

Self‐normalization for Spatial Data

This paper considers inference for both spatial lattice data with possibly irregularly shaped sampling region and non‐lattice data, by extending the recently proposed self‐normalization (SN) approach from stationary time series to the spatial setup. A nice feature of the SN method is that it avoids the choice of tuning parameters, which are usually required for other non‐parametric inference approaches. The extension is non‐trivial as spatial data has no natural one‐directional time ordering. The SN‐based inference is convenient to implement and is shown through simulation studies to provide more accurate coverage compared with the widely used subsampling approach. We also illustrate the idea of SN using a real data example.     

Spatio-temporal change-point modeling

There is by now a substantial literature on spatio-temporal modeling. However, to date, there exists essentially no literature which addresses the issue of process change from a certain time. In fact, if we look at change points for purely time series data, the customary form is to propose a model involving a mean or level shift. We see little attempting to capture a change in association structure. Part of the concern is how to specify flexible ways to bridge the association across the time point and still ensure that a proper joint distribution has been defined for all of the data. Introducing a spatial component evidently adds further complication. We want to allow for a change-point reflecting change in both temporal and spatial association. In this paper we propose a constructive, flexible model formulation through additive specifications. We also demonstrate how computational concerns benefit from the availability of temporal order. Finally, we illustrate with several simulated datasets to examine the capability of the model to detect different types of structural changes.     

Two-scale spatial models for binary data

A spatial lattice model for binary data is constructed from two spatial scales linked through conditional probabilities. A coarse grid of lattice locations is specified, and all remaining locations (which we call the background) capture fine-scale spatial dependence. Binary data on the coarse grid are modelled with an autologistic distribution, conditional on the binary process on the background. The background behaviour is captured through a hidden Gaussian process after a logit transformation on its Bernoulli success probabilities. The likelihood is then the product of the (conditional) autologistic probability distribution and the hidden Gaussian–Bernoulli process. The parameters of the new model come from both spatial scales. A series of simulations illustrates the spatial-dependence properties of the model and likelihood-based methods are used to estimate its parameters. Presence–absence data of corn borers in the roots of corn plants are used to illustrate how the model is fitted.     

A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

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.     

A First-Order Spatial Integer-Valued Autoregressive SINAR(1, 1) Model

Binomial thinning operator has a major role in modeling one-dimensional integer-valued autoregressive time series models. The purpose of this article is to extend the use of such operator to define a new stationary first-order spatial non negative, integer-valued autoregressive SINAR(1, 1) model. We study some properties of this model like the mean, variance and autocorrelation function. Yule-Walker estimator of the model parameters is also obtained. Some numerical results of the model are presented and, moreover, this model is applied to a real data set.     

Bayesian Clustering of Many Garch Models

We consider the estimation of a large number of GARCH models, of the order of several hundreds. Our interest lies in the identification of common structures in the volatility dynamics of the univariate time series. To do so, we classify the series in an unknown number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster contains therefore similar series. We do not know a priori which series belongs to which cluster. The model is a finite mixture of distributions, where the component weights are unknown parameters and each component distribution has its own conditional mean and variance. Inference is done by the Bayesian approach, using data augmentation techniques. Simulations and an illustration using data on U.S. stocks are provided.     

Bayesian Clustering of Many Garch Models

We consider the estimation of a large number of GARCH models, of the order of several hundreds. Our interest lies in the identification of common structures in the volatility dynamics of the univariate time series. To do so, we classify the series in an unknown number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster contains therefore similar series. We do not know a priori which series belongs to which cluster. The model is a finite mixture of distributions, where the component weights are unknown parameters and each component distribution has its own conditional mean and variance. Inference is done by the Bayesian approach, using data augmentation techniques. Simulations and an illustration using data on U.S. stocks are provided.     

The generalized linear chirp process

The linear chirp process is an important class of time series for which the instantaneous frequency changes linearly in time. Linear chirps have been used extensively to model a variety of physical signals such as radar, sonar, and whale clicks (see 1, 5 and 6). We introduce the stochastic linear chirp model and then define the generalized linear chirp (GLC) process as a special case of the G-stationary process studied by Jiang et al. (2006) to model data with time-varying frequencies. We then define GLC(p,q) processes and show that the relationship between stochastic linear chirp processes and GLC(p,q) processes is analogous to that between harmonic and ARMA models. The new methods are then applied to both simulated and actual data sets.     

Functional regression concurrent model with spatially correlated errors: application to rainfall ground validation

In this paper, we give an extension of the functional regression concurrent model to the case of spatially correlated errors. We propose estimating the spatial correlation structure by using functional geostatistics. The estimation of the regression parameters is carried out by feasible generalized least squares. This modeling approach is motivated by the problem of validating rainfall data retrieved from satellite sensors. In this sense, we use the methodology to study the relationship between satellite and ground rainfall time series recorded in 82 weather stations from Department of Valle del Cauca, Colombia. The model obtained allows predicting pentadal rainfall curves in many sites of the region of interest by using as input the satellite information. A residual analysis shows a good performance of the methodology proposed.     

Blur-generated non-separable space–time models

Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by 'blurring' the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.     

Empirical information criteria for time series forecasting model selection

In this article, we propose a new empirical information criterion (EIC) for model selection which penalizes the likelihood of the data by a non-linear function of the number of parameters in the model. It is designed to be used where there are a large number of time series to be forecast. However, a bootstrap version of the EIC can be used where there is a single time series to be forecast. The EIC provides a data-driven model selection tool that can be tuned to the particular forecasting task.

We compare the EIC with other model selection criteria including Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC). The comparisons show that for the M3 forecasting competition data, the EIC outperforms both the AIC and BIC, particularly for longer forecast horizons. We also compare the criteria on simulated data and find that the EIC does better than existing criteria in that case also.     

Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data

In this paper, we propose a hidden Markov model for the analysis of the time series of bivariate circular observations, by assuming that the data are sampled from bivariate circular densities, whose parameters are driven by the evolution of a latent Markov chain. The model segments the data by accounting for redundancies due to correlations along time and across variables. A computationally feasible expectation maximization (EM) algorithm is provided for the maximum likelihood estimation of the model from incomplete data, by treating the missing values and the states of the latent chain as two different sources of incomplete information. Importance-sampling methods facilitate the computation of bootstrap standard errors of the estimates. The methodology is illustrated on a bivariate time series of wind and wave directions and compared with popular segmentation models for bivariate circular data, which ignore correlations across variables and/or along time.     

Multivariate non-linear time series modelling of exposure and risk in road safety research

Summary.  A multivariate non-linear time series model for road safety data is presented. The model is applied in a case-study into the development of a yearly time series of numbers of fatal accidents (inside and outside urban areas) and numbers of kilometres driven by motor vehicles in the Netherlands between 1961 and 2000. The model accounts for missing entries in the disaggregated numbers of kilometres driven although the aggregated numbers are observed throughout. We consider a multivariate non-linear time series model for the analysis of these data. The model consists of dynamic unobserved factors for exposure and risk that are related in a non-linear way to the number of fatal accidents. The multivariate dimension of the model is due to its inclusion of multiple time series for inside and outside urban areas. Approximate maximum likelihood methods based on the extended Kalman filter are utilized for the estimation of unknown parameters. The latent factors are estimated by extended smoothing methods. It is concluded that the salient features of the observed time series are captured by the model in a satisfactory way.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

Robust Estimation of a Spatiotemporal Model with Structural Change

A spatiotemporal model is postulated and estimated using a procedure that infuses the forward search algorithm and maximum likelihood estimation into the backfitting framework. The forward search algorithm filters the effect of temporary structural change in the estimation of covariate and spatial parameters. Simulation studies illustrate capability of the method in producing robust estimates of the parameters even in the presence of structural change. The method provides good model fit even for small sample sizes in short time series data and good predictions for a wide range of lengths of contamination periods and levels of severity of contamination.     

Bayesian Time Series Analysis of Structural Changes in Level and Trend

In this article we consider the problem of detecting changes in level and trend in time series model in which the number of change-points is unknown. The approach of Bayesian stochastic search model selection is introduced to detect the configuration of changes in a time series. The number and positions of change-points are determined by a sequence of change-dependent parameters. The sequence is estimated by its posterior distribution via the maximum a posteriori (MAP) estimation. Markov chain Monte Carlo (MCMC) method is used to estimate posterior distributions of parameters. Some actual data examples including a time series of traffic accidents and two hydrological time series are analyzed.     

Dynamic models for spatiotemporal data

We propose a model for non-stationary spatiotemporal data. To account for spatial variability, we model the mean function at each time period as a locally weighted mixture of linear regressions. To incorporate temporal variation, we allow the regression coefficients to change through time. The model is cast in a Gaussian state space framework, which allows us to include temporal components such as trends, seasonal effects and autoregressions, and permits a fast implementation and full probabilistic inference for the parameters, interpolations and forecasts. To illustrate the model, we apply it to two large environmental data sets: tropical rainfall levels and Atlantic Ocean temperatures.     

An endemic–epidemic beta model for time series of infectious disease proportions

Time series of proportions of infected patients or positive specimens are frequently encountered in disease control and prevention. Since proportions are bounded and often asymmetrically distributed, conventional Gaussian time series models only apply to suitably transformed proportions. Here we borrow both from beta regression and from the well-established HHH model for infectious disease counts to propose an endemic–epidemic beta model for proportion time series. It accommodates the asymmetric shape and heteroskedasticity of proportion distributions and is consistent for complementary proportions. Coefficients can be interpreted in terms of odds ratios. A multivariate formulation with spatial power-law weights enables the joint estimation of model parameters from multiple regions. In our application to a flu activity index in the USA, we find that the endemic–epidemic beta model provides a better fit than a seasonal ARIMA model for the logit-transformed proportions. Furthermore, a multivariate approach can improve regional forecasts and reduce model complexity in comparison to univariate beta models stratified by region.