Bayesian Geostatistical Design

Abstract.  This paper describes the use of model-based geostatistics for choosing the set of sampling locations, collectively called the design, to be used in a geostatistical analysis. Two types of design situation are considered. These are retrospective design, which concerns the addition of sampling locations to, or deletion of locations from, an existing design, and prospective design, which consists of choosing positions for a new set of sampling locations. We propose a Bayesian design criterion which focuses on the goal of efficient spatial prediction whilst allowing for the fact that model parameter values are unknown. The results show that in this situation a wide range of inter-point distances should be included in the design, and the widely used regular design is often not the best choice.     

A Sequential Point Process Model and Bayesian Inference for Spatial Point Patterns with Linear Structures

Abstract. We introduce a flexible spatial point process model for spatial point patterns exhibiting linear structures, without incorporating a latent line process. The model is given by an underlying sequential point process model. Under this model, the points can be of one of three types: a ‘background point’ an ‘independent cluster point’ or a ‘dependent cluster point’. The background and independent cluster points are thought to exhibit ‘complete spatial randomness’, whereas the dependent cluster points are likely to occur close to previous cluster points. We demonstrate the flexibility of the model for producing point patterns with linear structures and propose to use the model as the likelihood in a Bayesian setting when analysing a spatial point pattern exhibiting linear structures. We illustrate this methodology by analysing two spatial point pattern datasets (locations of bronze age graves in Denmark and locations of mountain tops in Spain).     

Non-stationary partition modeling of geostatistical data for malaria risk mapping

The most common assumption in geostatistical modeling of malaria is stationarity, that is spatial correlation is a function of the separation vector between locations. However, local factors (environmental or human-related activities) may influence geographical dependence in malaria transmission differently at different locations, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling may lead to inaccurate estimates of the standard errors for both the covariate effects and the predictions. In this paper, a model based on random Voronoi tessellation that takes into account non-stationarity was developed. In particular, the spatial domain was partitioned into sub-regions (tiles), a stationary spatial process was assumed within each tile and between-tile correlation was taken into account. The number and configuration of the sub-regions are treated as random parameters in the model and inference is made using reversible jump Markov chain Monte Carlo simulation. This methodology was applied to analyze malaria survey data from Mali and to produce a country-level smooth map of malaria risk.     

Nonparametric prediction of spatial multivariate data

This paper investigates a nonparametric spatial predictor of a stationary multidimensional spatial process observed over a rectangular domain. The proposed predictor depends on two kernels in order to control both the distance between observations and that between spatial locations. The uniform almost complete consistency and the asymptotic normality of the kernel predictor are obtained when the sample considered is an alpha-mixing sequence. Numerical studies were carried out in order to illustrate the behaviour of our methodology both for simulated data and for an environmental data set.     

A spatiotemporal model for Mexico City ozone levels

Summary.  We consider hourly readings of concentrations of ozone over Mexico City and propose a model for spatial as well as temporal interpolation and prediction. The model is based on a time-varying regression of the observed readings on air temperature. Such a regression requires interpolated values of temperature at locations and times where readings are not available. These are obtained from a time-varying spatiotemporal model that is coupled to the model for the ozone readings. Two location-dependent harmonic components are added to account for the main periodicities that ozone presents during a given day and that are not explained through the covariate. The model incorporates spatial covariance structure for the observations and the parameters that define the harmonic components. Using the dynamic linear model framework, we show how to compute smoothed means and predictive values for ozone. We illustrate the methodology on data from September 1997.     

Moment estimation for statistics from marked point processes

In spatial statistics the data typically consist of measurements of some quantity at irregularly scattered locations; in other words, the data form a realization of a marked point process. In this paper, we formulate subsampling estimators of the moments of general statistics computed from marked point process data, and we establish their L ₂-consistency. The variance estimator in particular can be used for the construction of confidence intervals for estimated parameters. A practical data-based method for choosing a subsampling parameter is given and illustrated on a data set. Finite sample simulation examples are also presented.     

Bayesian spatial models with repeated measurements: with application to the herbaceous data analysis

This paper proposes a new statistical spatial model to analyze and predict the coverage percentage of the upland ground flora in the Missouri Ozark Forest Ecosystem Project (MOFEP). The flora coverage percentages are collected from clustered locations, which requires a new spatial model other than the traditional kriging method. The proposed model handles this special data structure by treating the flora coverage percentages collected from the clustered locations as repeated measurements in a Bayesian hierarchical setting. The correlation among the observations from the clustered locations are considered as well. The total vegetation coverage data in MOFEP is analyzed in this study. An Markov chain Monte Carlo algorithm based on the shrinkage slice sampler is developed for simulation from the posterior densities. The total vegetation coverage is modeled by three components, including the covariates, random spatial effect and correlated random errors. Prediction of the total vegetation coverage at unmeasured locations is developed.     

Spatial modeling of extreme rainfall in northeast Thailand

It is well recognized that the generalized extreme value (GEV) distribution is widely used for any extreme events. This notion is based on the study of discrete choice behavior; however, there is a limit for predicting the distribution at ungauged sites. Hence, there have been studies on spatial dependence within extreme events in continuous space using recorded observations. We model the annual maximum daily rainfall data consisting of 25 locations for the period from 1982 to 2013. The spatial GEV model that is established under observations is assumed to be mutually independent because there is no spatial dependency between the stations. Furthermore, we divide the region into two regions for a better model fit and identify the best model for each region. We show that the regional spatial GEV model reflects the spatial pattern well compared with the spatial GEV model over the entire region as the local GEV distribution. The advantage of spatial extreme modeling is that more robust return levels and some indices of extreme rainfall can be obtained for observed stations as well as for locations without observed data. Thus, the model helps to determine the effects and assessment of vulnerability due to heavy rainfall in northeast Thailand.     

From distance sampling to spatial capture–recapture

Distance sampling and capture–recapture are the two most widely used wildlife abundance estimation methods. capture–recapture methods have only recently incorporated models for spatial distribution and there is an increasing tendency for distance sampling methods to incorporated spatial models rather than to rely on partly design-based spatial inference. In this overview we show how spatial models are central to modern distance sampling and that spatial capture–recapture models arise as an extension of distance sampling methods. Depending on the type of data recorded, they can be viewed as particular kinds of hierarchical binary regression, Poisson regression, survival or time-to-event models, with individuals’ locations as latent variables and a spatial model as the latent variable distribution. Incorporation of spatial models in these two methods provides new opportunities for drawing explicitly spatial inferences. Areas of likely future development include more sophisticated spatial and spatio-temporal modelling of individuals’ locations and movements, new methods for integrating spatial capture–recapture and other kinds of ecological survey data, and methods for dealing with the recapture uncertainty that often arise when “capture” consists of detection by a remote device like a camera trap or microphone.     

Modelling extreme wind speeds in regions prone to hurricanes

Extreme wind speeds can arise as the result of a simple pressure differential, or a complex dynamic system such as a tropical storm. When sets of record values comprise a mixture of two or more different types of event, the standard models for extremes based on a single limiting distribution are not applicable. We develop a mixture model for extreme winds arising from two distinct processes. Working with sequences of annual maximum speeds obtained at hurricane prone locations in the USA, we take a Bayesian approach to inference, which allows the incorporation of prior information obtained from other sites. We model the extremal behaviour for the contrasting wind climates of Boston and Key West, and show that the standard models can give misleading results at such locations.     

The posterior distribution of the fixed and random effects in a mixed-effects linear model

The subject of this paper is Bayesian inference about the fixed and random effects of a mixed-effects linear statistical model with two variance components. It is assumed that a priori the fixed effects have a noninformative distribution and that the reciprocals of the variance components are distributed independently (of each other and of the fixed effects) as gamma random variables. It is shown that techniques similar to those employed in a ridge analysis of a response surface can be used to construct a one-dimensional curve that contains all of the stationary points of the posterior density of the random effects. The “ridge analysis” (of the posterior density) can be useful (from a computational standpoint) in finding the number and the locations of the stationary points and can be very informative about various features of the posterior density. Depending on what is revealed by the ridge analysis, a multivariate normal or multivariate-t distribution that is centered at a posterior mode may provide a satisfactory approximation to the posterior distribution of the random effects (which is of the poly-t form).     

BAYESIAN ANALYSIS OF VECTOR ARFIMA PROCESSES

A general framework is presented for Bayesian inference of multivariate time series exhibiting long-range dependence. The series are modelled using a vector autoregressive fractionally integrated moving-average (VARFIMA) process, which can capture both short-term correlation structure and long-range dependence characteristics of the individual series, as well as interdependence and feedback relationships between the series. To facilitate a sampling-based Bayesian approach, the exact joint posterior density is derived for the parameters, in a form that is computationally simpler than direct evaluation of the likelihood, and a modified Gibbs sampling algorithm is used to generate samples from the complete conditional distribution associated with each parameter. The paper also shows how an approximate form of the joint posterior density may be used for long time series. The procedure is illustrated using sea surface temperatures measured at three locations along the central California coast. These series are believed to be interdependent due to similarities in local atmospheric conditions at the different locations, and previous studies have found that they exhibit ‘long memory’ when studied individually. The approach adopted here permits investigation of the effects on model estimation of the interdependence and feedback relationships between the series.     

Spatial bootstrapped microeconometrics: Forecasting for out-of-sample geo-locations in big data

Spatial econometric models estimated on the big geo-located point data have at least two problems: limited computational capabilities and inefficient forecasting for the new out-of-sample geo-points. This is because of spatial weights matrix W defined for in-sample observations only and the computational complexity. Machine learning models suffer the same when using kriging for predictions; thus this problem still remains unsolved. The paper presents a novel methodology for estimating spatial models on big data and predicting in new locations. The approach uses bootstrap and tessellation to calibrate both model and space. The best bootstrapped model is selected with the PAM (Partitioning Around Medoids) algorithm by classifying the regression coefficients jointly in a nonindependent manner. Voronoi polygons for the geo-points used in the best model allow for a representative space division. New out-of-sample points are assigned to tessellation tiles and linked to the spatial weights matrix as a replacement for an original point what makes feasible usage of calibrated spatial models as a forecasting tool for new locations. There is no trade-off between forecast quality and computational efficiency in this approach. An empirical example illustrates a model for business locations and firms' profitability.     

Spatial prediction and temporal backcasting for environmental fields having monotone data patterns

The authors develop a methodology for predicting unobserved values in a conditionally lognormal random spatial field like those commonly encountered in environmental risk analysis. These unobserved values are of two types. The first come from spatial locations where the field has never been monitored, the second, from currently monitored sites which have been only recently installed. Thus the monitoring data exhibit a monotone pattern, resembling a staircase whose highest step comes from the oldest monitoring sites. The authors propose a hierarchical Bayesian approach using the lognormal sampling distribution, in conjunction with a conjugate generalized Wishart distribution. This prior distribution allows different degrees of freedom to be fitted for individual steps, taking into account the differential amounts of information available from sites at the different steps in the staircase. The resulting hierarchical model is a predictive distribution for the unobserved values of the field. The method is demonstrated by application to the ambient ozone field for the southwestern region of British Columbia.     

Bayesian spatio‐temporal models based on discrete convolutions

Abstract: The authors consider a class of models for spatio‐temporal processes based on convolving independent processes with a discrete kernel that is represented by a lower triangular matrix. They study two families of models. In the first one, spatial Gaussian processes with isotropic correlations are convoluted with a kernel that provides temporal dependencies. In the second family, AR(p) processes are convoluted with a kernel providing spatial interactions. The covariance structures associated with these two families are quite rich. Their covariance functions that are stationary and separable in space and time as well as time dependent nonseparable and nonisotropic ones.     

Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach

In this paper we investigate the impact of model mis-specification, in terms of the dependence structure in the extremes of a spatial process, on the estimation of key quantities that are of interest to hydrologists and engineers. For example, it is often the case that severe flooding occurs as a result of the observation of rainfall extremes at several locations in a region simultaneously. Thus, practitioners might be interested in estimates of the joint exceedance probability of some high levels across these locations. It is likely that there will be spatial dependence present between the extremes, and this should be properly accounted for when estimating such probabilities. We compare the use of standard models from the geostatistics literature with max-stables models from extreme value theory. We find that, in some situations, using an incorrect spatial model for our extremes results in a significant under-estimation of these probabilities which – in flood defence terms – could lead to substantial under-protection.     

Efficient Bayesian Multivariate Surface Regression

Methods for choosing a fixed set of knot locations in additive spline models are fairly well established in the statistical literature. The curse of dimensionality makes it nontrivial to extend these methods to nonadditive surface models, especially when there are more than a couple of covariates. We propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient Markov chain Monte Carlo algorithm that updates all the knot locations jointly. We use shrinkage prior to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. Simulated data and an application to firm leverage data show that the approach is computationally efficient, and that allowing for freely estimated knot locations can offer a substantial improvement in out‐of‐sample predictive performance.     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

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.