Custom Epoch Estimation for Surveys

The paper provides a method for generating epoch estimates for time series survey data, allowing for different periods of time (or even point estimates) according to user demand. The method uses a modified kriging estimator, which suppresses the contribution of sampling error variability in order to guarantee that custom epoch estimates have an interpolation property. For the veteran population variable of the American Community Survey, we utilize a simple Brownian Motion model of the population process and derive the modified kriging estimator for this case. The tuning parameters of this population model can be calibrated to the data via simple formulas. We illustrate the application of this method to the generation of point estimates of veteran population, an important objective for Veterans Affairs.     

Residual Kriging for Functional Spatial Prediction of Salinity Curves

Recently, several methodologies to perform geostatistical analysis of functional data have been proposed. All of them assume that the spatial functional process considered is stationary. However, in practice, we often have nonstationary functional data because there exists an explicit spatial trend in the mean. Here, we propose a methodology to extend kriging predictors for functional data to the case where the mean function is not constant through the region of interest. We consider an approach based on the classical residual kriging method used in univariate geostatistics. We propose a three steps procedure. Initially, a functional regression model is used to detrend the mean. Then we apply kriging methods for functional data to the regression residuals to predict a residual curve at a non-data location. Finally, the prediction curve is obtained as the sum of the trend and the residual prediction. We apply the methodology to salinity data corresponding to 21 salinity curves recorded at the Ciénaga Grande de Santa Marta estuary, located in the Caribbean coast of Colombia. A cross-validation analysis was carried out to track the performance of the proposed methodology.     

Transect-spacing design of ice cores on the Antarctic continent

Chemical analyses of ice cores, drilled deep into an ice sheet, provide a historical record of the earth's atmosphere that dates back as far as 400,000–500,000 years. Although the atmosphere mixes quite well, it is recognized that spatial variability associated with ice-core locations should be allowed for. In this article, spatial statistical methodology is applied to the design question of finding the best spacing of ice-core locations on a partial transect of Antarctica.     

A Mixed Model Approach for Geoadditive Hazard Regression

Abstract.  Mixed model based approaches for semiparametric regression have gained much interest in recent years, both in theory and application. They provide a unified and modular framework for penalized likelihood and closely related empirical Bayes inference. In this article, we develop mixed model methodology for a broad class of Cox-type hazard regression models where the usual linear predictor is generalized to a geoadditive predictor incorporating non-parametric terms for the (log-)baseline hazard rate, time-varying coefficients and non-linear effects of continuous covariates, a spatial component, and additional cluster-specific frailties. Non-linear and time-varying effects are modelled through penalized splines, while spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field prior. Generalizing existing mixed model methodology, inference is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. In a simulation we study the performance of the proposed method, in particular comparing it with its fully Bayesian counterpart using Markov chain Monte Carlo methodology, and complement the results by some asymptotic considerations. As an application, we analyse leukaemia survival data from northwest England.     

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.     

Analyzing non-stationarity in cement stone pit by median polish interpolation: a case study

The raw materials utilized in the manufacture of cement comprise mainly of lime, silica, alumina and iron oxide. Spatial evaluation of these main chemical constituents of cement has crucial importance for providing effective production. Because these components are composed of some raw materials such as limestone and marl, the spatial relationships in a calcareous marl stone pit was taken into consideration. In practice, spatial field data taken from a cement quarry may include some variations and trends. For modeling and removing spatial trend in a cement raw material quarry as well as providing unbiased estimates, median polish kriging was used. By using the variation of the data itself, some approximations and interpolations were carried out. It was recorded that the method obtained outlier-resistant estimation of spatial trend without needing an external exploratory variable. In addition, it provided very effective estimations and additional information for analyzing spatial non-stationary data.     

Point and block prediction in log-Gaussian random fields: The non-constant mean case

This work considers the problems of point and block prediction in log-Gaussian random fields for the case when the mean of the log-process is not constant and depends linearly on unknown parameters. First, we propose a new point predictor that is optimal within a certain family of predictors, which extend a result in De Oliveira [2006. On optimal point and block prediction in log-Gaussian random fields. Scand. J. Statist. 33, 523–540.] that holds in the case when the mean of the log-process is constant. Second, we show that the results in De Oliveira [2006. On optimal point and block prediction in log-Gaussian random fields. Scand. J. Statist. 33, 523–540.] regarding optimal block prediction cannot be extended to the case when the mean of the log-process is not constant. Specifically, we show that the two families of block predictors considered by De Oliveira lack an optimal predictor. Finally, we numerically compare the predictive efficiency of the proposed point and block predictors.     

Multivariate spatial interpolation and exposure to air pollutants

We develop and apply an approach to the spatial interpolation of a vector-valued random response field. The Bayesian approach we adopt enables uncertainty about the underlying models to be représentés in expressing the accuracy of the resulting interpolants. The methodology is particularly relevant in environmetrics, where vector-valued responses are only observed at designated sites at successive time points. The theory allows space-time modelling at the second level of the hierarchical prior model so that uncertainty about the model parameters has been fully expressed at the first level. In this way, we avoid unduly optimistic estimates of inferential accuracy. Moreover, the prior model can be upgraded with any available new data, while past data can be used in a systematic way to fit model parameters. The theory is based on the multivariate normal and related joint distributions. Our hierarchical prior models lead to posterior distributions which are robust with respect to the choice of the prior (hyperparameters). We illustrate our theory with an example involving monitoring stations in southern Ontario, where monthly average levels of ozone, sulphate, and nitrate are available and between-station response triplets are interpolated. In this example we use a recently developed method for interpolating spatial correlation fields.     

Geostatistical analysis of disease data: a case study of tuberculosis incidence in Iran

The main objective of this study is to introduce two advanced statistical methods and to consider geographical distribution of tuberculosis incidence in Iran. With the knowledge that environmental and climatic conditions in each region are affective for the incidence and spread of the disease, the study has been taken into consideration. The disease incidences in different counties are realizations of spatial data, therefore we apply the Poisson kriging and ordinary kriging for prediction of tuberculosis incidence rates map in Iran. To identify high risk areas using statistical map of disease, our results show that tuberculosis incidences are not uniformly distributed in whole of the country and estimated risk is high in the eastern parts. Assessing geographical distribution of a disease is essential for health officials to recognize high-risk areas, and improve case management and resource allocation.