Bayesian Maximum Entropy Approach Research Articles

Maximum entropy approach for sediment porosity and source strength of transiting ships

Remote sensing using passive sonar in the ocean is a challenging problem due to variations in the geoacoustic structure of the seabed and unknown source location and strength. One way to improve remote sensing is to perform an optimization for geoacoustic and source parameters. We use a Bayesian maximum entropy (BME) approach with a viscous-grain shearing parameterization for two sediment layers. The statistical optimization provides probability distributions for porosity and thickness of the sediment layers as well as ship speed, closest point of approach, and the source strength for the Wales-Heitmeyer empirical source level spectrum. We use this approach on spectrograms of transiting ships collected on a vertical line array during the 2017 seabed characterization experiment. We compare the resulting parameter distributions from distinct ships as well as previous estimates of geoacoustic values and source properties. This research shows that the BME approach obtains estimates for porosity and source strength that have narrow posterior probability distributions. [Work supported by the Office of Naval Research.]

Application and Analysis of XCO2 Data from OCO Satellite Using a Synthetic DINEOF–BME Spatiotemporal Interpolation Framework

Carbon dioxide (CO2) is one of the main greenhouse gases leading to global warming, and the ocean is the largest carbon reservoir on the earth that plays an important role in regulating CO2 concentration on a global scale. The column-averaged dry-air mole fraction of atmospheric CO2 (XCO2) is a key parameter in describing ocean carbon content. In this paper, the Data Interpolation Empirical Orthogonal Function (DINEOF) and the Bayesian Maximum Entropy (BME) methods are combined to interpolate XCO2 data of Orbiting Carbon Observatory 2 (OCO-2) and Orbiting Carbon Observatory 3 (OCO-3) from January to December 2020 occurring within the geographical range of 15–45°N and 120–150°E. At the first stage of our proposed analysis, spatiotemporal information was used by the DINEOF method to perform XCO2 interpolation that improved data coverage; at the second stage, the DINEOF-generated interpolation results were regarded as soft data and were subsequently assimilated using the BME method to obtain improved XCO2 interpolation values. The performance of the synthetic DINEOF–BME interpolation method was evaluated by means of a five-fold cross-validation method. The results showed that the Mean Absolute Error (MAE), the Root Mean Square Error (RMSE), and the Bias of the DINEOF-based OCO-2 and OCO-3 interpolations were 2.106 ppm, 3.046 ppm, and 1.035 ppm, respectively. On the other hand, the MAE, RMSE, and Bias of the cross-validation results obtained by the DINEOF–BME were 1.285 ppm, 2.422 ppm, and −0.085 ppm, respectively, i.e., smaller than the results obtained by DINEOF. In addition, based on the in situ measured XCO2 data provided by the Total Carbon Column Observing Network (TCCON), the original OCO-2 and OCO-3 data were combined and compared with the interpolated products of the synthetic DINEOF–BME framework. The accuracy of the original OCO-2 and OCO-3 products is lower than the DINEOF–BME-generated XCO2 products in terms of MAE (1.751 ppm vs. 2.616 ppm), RMSE (2.877 ppm vs. 3.566 ppm) and Bias (1.379 ppm vs 1.622 ppm), the spatiotemporal coverage of XCO2 product also improved dramatically from 16% to 100%. Lastly, this study demonstrated the feasibility of the synthetic DINEOF–BME approach for XCO2 interpolation purposes and the ability of the BME method to be successfully combined with other techniques.

Spatiotemporal modelling of soil organic matter changes in Jiangsu, China between 1980 and 2006 using INLA-SPDE

The growing human population and demand for food have significantly impacted soil resources. Understanding the spatiotemporal change of soil conditions is important to support food production, environmental sustainability, and climate change adaptation. Nevertheless, spatiotemporal prediction of soil properties could be seriously influenced by the uncertainties of the data and model. Integrated Nested Laplace Approximation (INLA) with the Stochastic Partial Differential Equation (SPDE) was proposed as a general model that can account for the uncertainties in spatiotemporal soil modelling and prediction. INLA-SPDE has significant advantages in computation efficiency over commonly-used geostatistical methods with Markov Chain Monte Carlo. However, until now, only few pedometrics studies used it for soil spatial modelling. This study demonstrates an application of INLA-SPDE within a hierarchical spatiotemporal model for soil organic matter based on soil survey data collected in Jiangsu, China, during three periods, i.e., 1979–1982, 2000 and 2006–2007. Compared with updating digital soil maps using the Bayesian Maximum Entropy approach, the prediction generated using INLA-SPDE is more accurate. For example, the root mean square error using INLA-SPDE (i.e., 6.57 g kg−1) was reduced by 20% compared to the updating approach (i.e., 8.39 g kg−1). Moreover, accounting for sources of uncertainties made the prediction using INLA-SPDE more certain. Nevertheless, the uncertainty in the temporal prediction of soil change is still large due to the scarcity of data across the sampling periods. The INLA-SPDE model predicts much detailed spatiotemporal changes along the sampling periods. Therefore, this study recommends the use of INLA-SPDE within a hierarchical model as an effective method for studying spatiotemporal soil change.

Spatiotemporal analysis of wind speed via the Bayesian maximum entropy approach

Spatiotemporal mapping of wind speed data is of great importance on determining the wind energy potential, construction zones, projecting future values and planning resources. From this point of view, monthly and yearly mean wind speed maps of Turkey are generated using the Bayesian maximum entropy (BME) approach. The BME, a nonlinear geostatistical approach, is the only method which uses not only data on the process but also all information about the data in a spatiotemporal mapping. During processing data, physical laws, hypotheses, experiences, scientific theories, high order space/time moments, various types of uncertain information, outputs of models, etc. are incorporated to the process. Hence, it maximizes knowledge about the variable processed because it uses exact and auxiliary data which belong to the regarding variable. In this study, Turkey daily mean wind speed data (m/s) measured at 10 m between the years 2010 and 2015 have been used. 0.05°, 0.1°, 0.3°, 0.5° and 1.0° have been chosen for spatial ranges and 1 day, week, month have been used for temporal ranges to determine the most appropriate intervals in the yearly forecast after lots of trials. Different kinds of local means are used so as to find proper kriging method. In the forecast process, hard data (raw data), hard data with soft data (auxiliary data), detrended hard with soft data are implemented, separately. All studies show that using hard with soft data gives the best forecast results and the most appropriate kriging method is determined as the ordinary kriging. Similarly to the other techniques which involve kriging methods, variances of errors are taken as a performance criterion. When the soft data are used, variances of the forecast errors reduce by half. In addition, all forecast results stay the interval of confidence levels and error variances of forecast results are quite low. Moreover, the BME forecast is employed for each month. Differently from the yearly forecast, the application of the detrended hard and soft data is found the most convenient experiment because it decreases the error values. From this point forth, it can be said that the usage of the soft data increases the accuracy of the forecasts. In addition, unknown wind speed values due to the lack of the stations are calculated and even wind speed values over the seas are identified. The study is the first and only spatiotemporal wind speed study used the BME approach for Turkey. It is thought that the mean wind speed map for Turkey can be considered as a current wind atlas of Turkey.

Ocean pollution assessment by integrating physical law and site‐specific data

AbstractThis work introduces the Bayesian maximum entropy (BME) perspective in the space–time characterization and prediction of ocean pollution. The proposed approach can outperform the standard numerical techniques that can also solve the partial differential equation representing the ocean pollution law but either cannot integrate in the solution any case‐specific information (provided by the hard or soft data available) or they do not provide a complete stochastic characterization of the ocean pollution phenomenon. This happens because the BME‐based solution of the ocean pollution law can both take advantage of any case‐specific data available and derive the complete probability density function throughout the space–time domain of interest, which can be then used to derive more than one kind of ocean pollution predictors (including the mean, the median, and the mode) at every point of this domain. Valuable insight into the proposed approach is gained by means of a simulation (synthetic) experiment that considers the advection–reaction law in the light of hard data and three different types of soft information in a controlled environment so a useful sensitivity analysis of the results can be performed (which is not usually possible in a real‐world case study). The uncertainty assessment of this experiment demonstrated that BME leads to improved ocean pollution predictions compared with the standard numerical technique. In principle, the higher the quality and informativeness of the site‐specific data, the more realistic should be the BME solution of the ocean pollution law under conditions of in situ uncertainty (this result also points out the importance of collecting data as informative as possible to maximize prediction accuracy). Moreover, the uncertainty assessment of the pollutant concentration predictions demonstrated that the BME approach makes good use of the soft data leading to improved ocean pollution predictions compared with predictions that do not involve soft data.

Blending Multiple Nitrogen Dioxide Data Sources for Neighborhood Estimates of Long-Term Exposure for Health Research.

Exposure to traffic related nitrogen dioxide (NO2) air pollution is associated with adverse health outcomes. Average pollutant concentrations for fixed monitoring sites are often used to estimate exposures for health studies, however these can be imprecise due to difficulty and cost of spatial modeling at the resolution of neighborhoods (e.g., a scale of tens of meters) rather than at a coarse scale (around several kilometers). The objective of this study was to derive improved estimates of neighborhood NO2 concentrations by blending measurements with modeled predictions in Sydney, Australia (a low pollution environment). We implemented the Bayesian maximum entropy approach to blend data with uncertainty defined using informative priors. We compiled NO2 data from fixed-site monitors, chemical transport models, and satellite-based land use regression models to estimate neighborhood annual average NO2. The spatial model produced a posterior probability density function of estimated annual average concentrations that spanned an order of magnitude from 3 to 35 ppb. Validation using independent data showed improvement, with root mean squared error improvement of 6% compared with the land use regression model and 16% over the chemical transport model. These estimates will be used in studies of health effects and should minimize misclassification bias.

Bayesian maximum entropy approach and its applications: a review

The present paper reviews the conceptual framework and development of the Bayesian Maximum Entropy (BME) approach. BME has been considered as a significant breakthrough and contribution to applied stochastics by introducing an improved, knowledge-based modeling framework for spatial and spatiotemporal information. In this work, one objective is the overview of distinct BME features. By offering a foundation free of restrictive assumptions that limit comparable techniques, an ability to integrate a variety of prior knowledge bases, and rigorous accounting for both exact and uncertain data, the BME approach was coined as introducing modern spatiotemporal geostatistics. A second objective is to illustrate BME applications and adoption within numerous different scientific disciplines. We summarize examples and real-world studies that encompass the perspective of science of the total environment, including atmosphere, lithosphere, hydrosphere, and ecosphere, while also noting applications that extend beyond these fields. The broad-ranging application track suggests BME as an established, valuable tool for predictive spatial and space–time analysis and mapping. This review concludes with the present status of BME, and tentative paths for future methodological research, enhancements, and extensions.

A spatiotemporal dengue fever early warning model accounting for nonlinear associations with hydrological factors: a Bayesian maximum entropy approach

Dengue fever is identified as one of the most widespread vector-borne diseases in the world. In the last decade, high incidence of dengue fever has been observed in southern Taiwan on an annual basis. There is an urgent need to develop a dengue fever early warning model for the area. Previous studies showed that dengue fever transmission has complex space–time dynamics highly depending upon the activities of human and vectors and their interactions, as well as hydrological processes. This study developed a 1-week-ahead dengue fever warning system in southern Taiwan considering weekly-based nonlinear temporal lagged associations between dengue cases and hydrological factors across space and time by investigating the disease database during 1998–2011. The proposed model is based on an integration of distributed lag nonlinear model and spatiotemporal dependence structure under an epistemic framework of Bayesian maximum entropy method. This study identified that minimum temperature and maximum 24-h rainfall are the most significant to dengue fever incidences. Their associations to dengue fever risks are presented with respect to both hydrological measurement levels and temporal lags. A nested spatiotemporal covariance of transmissions was used to account for the disease diffusion across space and time. Results show that the proposed approach can provide early warnings of the dengue fever occurrences in both initial and peak epidemic stages in 2012, and obtain the spatiotemporal distribution of dengue fever epidemics. These features can be useful for local governmental agencies to seek more effective strategies for dengue fever prevention and control.

Spatiotemporal patterns of non-point source nitrogen loss in an agricultural catchment

Non-point source nitrogen loss poses a risk to sustainable aquatic ecosystems. However, non-point sources, as well as impaired river segments with high nitrogen concentrations, are difficult to monitor and regulate because of their diffusive nature, budget constraints, and resource deficiencies. For the purpose of catchment management, the Bayesian maximum entropy approach and spatial regression models have been used to explore the spatiotemporal patterns of non-point source nitrogen loss. In this study, a total of 18 sampling sites were selected along the river network in the Hujiashan Catchment. Over the time period of 2008–2012, water samples were collected 116 times at each site and analyzed for non-point source nitrogen loss. The morphometric variables and soil drainage of different land cover types were studied and considered potential factors affecting nitrogen loss. The results revealed that, compared with the approach using the Euclidean distance, the Bayesian maximum entropy approach using the river distance led to an appreciable 10.1% reduction in the estimation error, and more than 53.3% and 44.7% of the river network in the dry and wet seasons, respectively, had a probability of non-point source nitrogen impairment. The proportion of the impaired river segments exhibited an overall decreasing trend in the study catchment from 2008 to 2012, and the reduction in the wet seasons was greater than that in the dry seasons. High nitrogen concentrations were primarily found in the downstream reaches and river segments close to the residential lands. Croplands and residential lands were the dominant factors affecting non-point source nitrogen loss, and explained up to 70.7% of total nitrogen in the dry seasons and 54.7% in the wet seasons. A thorough understanding of the location of impaired river segments and the dominant factors affecting total nitrogen concentration would have considerable importance for catchment management.

Bayesian Data Fusion Applied to Soil Drainage Classes Spatial Mapping

Soil drainage classes spatial mapping is of great interest since drainage has direct effects on crop productivity and hydrological modelling. However, the prediction of this categorical variable often requires a laborious and expensive sampling over large areas. There is thus a need for a methodology that is able to combine several sources of information to improve the prediction. Bayesian maximum entropy (BME) has become a complete framework in the context of space–time prediction. This method proposes solutions to combine several sources of data no matter what the nature of information is. However, the various attempts that were made for adapting the BME methodology to categorical variables and mixed random fields faced some limitations, as a high computational burden. The main objective of this paper is to overcome this limitation by generalizing the Bayesian data fusion (BDF) theoretical framework to categorical variables, which is a simplification of the BME method through the conditional independence hypothesis. The BDF methodology for categorical variables is first described and then applied to a practical case study: the estimation of soil drainage classes using a soil map and point observations around Mechelen (Belgium). The BDF approach is compared to BME along with more classical approaches, as indicator cokriging (ICK) and logistic regression. Estimators are compared using various indicators, namely the percentage of correctly classified locations and the average highest probability. Although BDF methodology for categorical variables is a simplification of BME approach, both methods lead to very close results and have strong advantages compared to ICK and logistic regression.

BME prediction of continuous geographical properties using auxiliary variables

Using auxiliary information to improve the prediction accuracy of soil properties in a physically meaningful and technically efficient manner has been widely recognized in pedometrics. In this paper, we explored a novel technique to effectively integrate sampling data and auxiliary environmental information, including continuous and categorical variables, within the framework of the Bayesian maximum entropy (BME) theory. Soil samples and observed auxiliary variables were combined to generate probability distributions of the predicted soil variable at unsampled points. These probability distributions served as soft data of the BME theory at the unsampled locations, and, together with the hard data (sample points) were used in spatial BME prediction. To gain practical insight, the proposed approach was implemented in a real-world case study involving a dataset of soil total nitrogen (TN) contents in the Shayang County of the Hubei Province (China). Five terrain indices, soil types, and soil texture were used as auxiliary variables to generate soft data. Spatial distribution of soil total nitrogen was predicted by BME, regression kriging (RK) with auxiliary variables, and ordinary kriging (OK). The results of the prediction techniques were compared in terms of the Pearson correlation coefficient (r), mean error (ME), and root mean squared error (RMSE). These results showed that the BME predictions were less biased and more accurate than those of the kriging techniques. In sum, the present work extended the BME approach to implement certain kinds of auxiliary information in a rigorous and efficient manner. Our findings showed that the BME prediction technique involving the transformation of variables into soft data can improve prediction accuracy considerably, compared to other techniques currently in use, like RK and OK.

A Bayesian trans-dimensional approach for the fusion of multiple geophysical datasets

We propose a Bayesian fusion approach to integrate multiple geophysical datasets with different coverage and sensitivity. The fusion strategy is based on the capability of various geophysical methods to provide enough resolution to identify either subsurface material parameters or subsurface structure, or both. We focus on electrical resistivity as the target material parameter and electrical resistivity tomography (ERT), electromagnetic induction (EMI), and ground penetrating radar (GPR) as the set of geophysical methods. However, extending the approach to different sets of geophysical parameters and methods is straightforward. Different geophysical datasets are entered into a trans-dimensional Markov chain Monte Carlo (McMC) search-based joint inversion algorithm. The trans-dimensional property of the McMC algorithm allows dynamic parameterisation of the model space, which in turn helps to avoid bias of the post-inversion results towards a particular model. Given that we are attempting to develop an approach that has practical potential, we discretize the subsurface into an array of one-dimensional earth-models. Accordingly, the ERT data that are collected by using two-dimensional acquisition geometry are re-casted to a set of equivalent vertical electric soundings. Different data are inverted either individually or jointly to estimate one-dimensional subsurface models at discrete locations. We use Shannon's information measure to quantify the information obtained from the inversion of different combinations of geophysical datasets. Information from multiple methods is brought together via introducing joint likelihood function and/or constraining the prior information. A Bayesian maximum entropy approach is used for spatial fusion of spatially dispersed estimated one-dimensional models and mapping of the target parameter. We illustrate the approach with a synthetic dataset and then apply it to a field dataset We show that the proposed fusion strategy is successful not only in enhancing the subsurface information but also as a survey design tool to identify the appropriate combination of the geophysical tools and show whether application of an individual method for further investigation of a specific site is beneficial. (C) 2013 Elsevier B.V. All rights reserved.

Comment on "Hierarchical Bayesian space-time interpolation versus spatio-temporal BME approach" by Hussain et al. (2010)

This is a clarifying comment regarding the article “Hierarchical Bayesian space-time interpolation versus spatiotemporal BME approach” (Hussain et al., 2010). In the article, the authors present a very interesting methodological study to compare the results obtained by the Bayesian Maximum Entropy (BME) and hierarchical Bayesian interpolation (HBI) techniques in a spatiotemporal context. The study concludes in declaring HBI a more accurate contestant that also produces lower prediction error, compared to BME. BME began its development as a novel approach and an improvement in space-time analysis in the early 1990’s (Christakos, 1990, 1992, 2000). BME has a distinctive edge compared to the classical linear model predictive approaches because its foundations are free from restrictions and assumptions typically present in competing methods (e.g., data distribution assumptions, consideration of soft data as hard values, etc.), and it can incorporate a broader spectrum of information such as uncertain observations, probabilistic data, physical laws, etc. Naturally, like any consistent theory, BME includes pre-existing theories as its special cases. The article authors are clear, accurate and explicit in stating that they apply BME kriging in their study, that is, they apply the limited case of BME when it is mathematically reduced to the kriging classical technique. The above remark is at the heart of my comment: The article describes inaccurately the comparison as HBI versus BME, because the conclusions place the critique on the BME methodology in its entirety without having used any of the exclusive BME features (e.g., use of soft data, physical laws, non-linear prediction). To the informed reader, the article essentially illustrates a comparison between HBI and kriging applied through BME software. I think this clarification is necessary to prevent readers that are less familiarized with

A Bayesian Maximum Entropy approach to address the change of support problem in the spatial analysis of childhood asthma prevalence across North Carolina

The spatial analysis of data observed at different spatial observation scales leads to the change of support problem (COSP). A solution to the COSP widely used in linear spatial statistics consists in explicitly modeling the spatial autocorrelation of the variable observed at different spatial scales. We present a novel approach that takes advantage of the nonlinear Bayesian Maximum Entropy (BME) extension of linear spatial statistics to address the COSP directly without relying on the classical linear approach. Our procedure consists in modeling data observed over large areas as soft data for the process at the local scale. We demonstrate the application of our approach to obtain spatially detailed maps of childhood asthma prevalence across North Carolina (NC). Because of the high prevalence of childhood asthma in NC, the small number problem is not an issue, so we can focus our attention solely to the COSP of integrating prevalence data observed at the county level together with data observed at a targeted local scale equivalent to the scale of school districts. Our spatially detailed maps can be used for different applications ranging from exploratory and hypothesis-generating analyses to targeting intervention and exposure mitigation efforts.

Spatial analysis and modelling of land use distributions in Belgium

When statistical analyses of land use drivers are performed, they rarely deal explicitly with spatial autocorrelation. Most studies are undertaken on autocorrelation-free data samples. By doing this, a great deal of information that is present in the dataset is lost. This paper presents a spatially explicit, cross-sectional analysis of land use drivers in Belgium. It is shown that purely regressive logistic models only identify trends or global relationships between socio-economic or physico-climatic drivers and the precise location of each land use type. However, when the goal of a study is to obtain the best statistical model fit of land use distribution, a purely autoregressive model is appropriate. It is shown that this type of model deals appropriately with spatial autocorrelation as measured by the lack of autocorrelation in the deviance residuals of the model. More specifically, three types of autoregressive models are compared: (1) a set of binomial logistic regression models (one for each modelled land use) accounting only for the proportion of the modelled land use within the neighbourhood of a cell; (2) a multinomial autologistic regression that accounts for the composition of a cell’s neighbourhood; and (3) a state-of-the-art Bayesian Maximum Entropy (BME) based model that accounts fully for the spatial organization of the land uses within the neighbourhood of a cell. The comparative analysis shows that the BME approach has no advantages over the other methods, for our specific application, but that accounting for the composition of a cell’s neighbourhood is essential in obtaining an optimal fit.

Modeling a Syphilis Outbreak Through Space and Time Using the Bayesian Maximum Entropy Approach

The aim of the study is to describe changes in the spatial distribution of syphilis before, during, and after an outbreak in Baltimore, MD, by using Bayesian maximum entropy (BME), a modern geostatistical technique for space-time analysis and mapping. BME was used to conduct simple and composite space-time analyses of the density of syphilis infection based on primary, secondary and early latent syphilis cases reported to the Baltimore City Health Department between January 1, 1994, and December 31, 2002. Spatiotemporal covariance plots indicated that the distribution of the density of syphilis cases showed both spatial and temporal dependence. Temporally dependent disease maps suggested that syphilis increased within two geographic core areas of infection and spread outward. A new core area of infection was established to the northwest. As the outbreak waned, density diminished and receded in all core areas. Morbidity remained elevated in the two original central and new northwestern core areas after the outbreak. Density of syphilis infection was a simple informative measure easily compared across years. The BME approach was useful for quantitatively and qualitatively describing the spatial development and spread of syphilis. Our results are specific to Baltimore; however, the BME approach is generalizable to other settings and diseases.

SPATIAL DISTRIBUTION OF WHEEZING PREVALENCE AND AIRBORNE LEAD LEVELS ACROSS RURAL AND URBAN AREAS OF NORTH CAROLINA

ISEE-453 Introduction: The effect of lead on asthma is questionable. Recently there have been a few laboratory based papers indicating that lead may have an effect on the respiratory immune system. Aim: We used Bayesian Maximum Entropy (BME) to investigate a potential relationship between lead and asthma or wheezing at a population based level. Methods: Spatial analyses can be a useful tool in investigating relationships between air pollutants common in urban environments and their possible health effects. We used BME, a modern spatiotemporal geostatistics method, to analyze the spatial distribution of wheezing prevalence in middle school age children, and lead (Pb), as monitored by the Environmental Protection Agency (EPA). It has been previously reported that the BME approach allows for versatile spatiotemporal analysis and mapping, which can be used to integrate hard (accurate) and soft (imprecise) information about wheezing prevalence, as well as the composite space/time variability of criteria air pollutants. BME gives estimates of spatial distribution that are more accurate than those obtained using the regression-based methods of classical geostatistics. Spatial maps of lead concentrations and spatial maps of asthma and wheezing prevalence were generated using the BME framework. Wheezing prevalence data were collected with the North Carolina School Asthma Survey, a population-based survey of children (n=122,289) completed in the 1999-2000 school year. Exposures mapped include the annual median concentration of lead over an X year period. Results/Discussion: Lead results, as well as pollutant levels, will be presented. The high ozone, PM 2.5 and PM10 levels in the Charlotte area in 1999 may be due to urbanization and increased traffic. This methodology could be useful for tracking environmental health research in the US, and for hypothesis generating. These associations had not been previously detected. The pattern of high levels of SOx in rural areas of North Carolina also needs further investigation. By rigorously processing soft information and space/time variability, the BME method allowed for detection of spatial associations between wheezing prevalence and several EPA criteria air pollutants.

Spatial prediction of categorical variables with the Bayesian Maximum Entropy approach: the Ooypolder case study

SummaryCategorical variables such as water table status are often predicted using the indicator kriging (IK) formalism. However, this method is known to suffer from important limitations that are most frequently solved by ad hoc solutions and approximations. Recently, the Bayesian Maximum Entropy (BME) approach has proved its ability to predict categorical variables efficiently and in a flexible way. In this paper, we apply this approach to the Ooypolder data set for the prediction of the water table classes from a sample data set. BME is compared with IK using global as well as local criteria. The inconsistencies of the IK predictor are emphasized and it is shown how BME permits avoiding them.

SPATIAL DISTRIBUTION OF WHEEZING PREVALENCE AND AIR POLLUTION ACROSS NORTH CAROLINA

ISEE-143 Introduction: Spatial analyses can be a useful tool in investigating relationships between air pollutants common in urban environments and their possible health effects. Spatial maps can show the distribution across space of (1) the prevalence of a health outcome and (2) the concentration in the air of a potentially toxic environmental pollutant. We used the Bayesian Maximum Entropy (BME), a modern spatiotemporal geostatistics method, to analyze the spatial distribution of wheezing prevalence in middle school age children, and six criteria air pollutants monitored by the Environmental Protection Agency (EPA). The BME approach allows for versatile spatiotemporal analysis and mapping which can be used to integrate hard (accurate) and soft (imprecise) information about wheezing prevalence, as well as the composite space/time variability of criteria air pollutants (Serre et al., 2002). BME gives estimates of spatial distribution that are more accurate than those obtained using the regression-based methods of classical geostatistics. Methods: Spatial maps of six EPA criteria pollutants concentrations and spatial maps of asthma and wheezing prevalence were generated using the BME framework. Wheezing prevalence data were collected with the North Carolina School Asthma Survey, a population-based survey of children (n=122,289) completed in the 1999–2000 school year. Exposures mapped include the annual median concentration of each of the six EPA criteria air pollutants (ozone, NOx, SOx, PM10 and PM2.5, and lead). Results: The visual comparison of the color plots of ozone, PM10 and PM 2.5 with the color plot of the current wheezing data for year 1999 suggest that the high prevalence of current wheezing seen in the eastern part of the western piedmont (around the cities of Charlotte and Winston-Salem) may correspond to the high concentration of ozone, PM10 and PM 2.5 in those areas. High concentrations of SOx seem to correspond to areas with high prevalence of current wheezing responses in rural areas such as the western most mountain area and the rural coastal area. Discussion: The high ozone, PM 2.5 and PM10 levels in the Charlotte area in 1999 may be due to urbanization and increased traffic. These associations had not been previously detected. The pattern of high levels of SOx in rural areas of NC also needs further investigation. By rigorously processing soft information and space/time variability, the BME method allowed for detection of spatial associations between wheezing prevalence and several EPA criteria air pollutants.

Spatial prediction of categorical variables: the Bayesian maximum entropy approach

Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.