Sheng-Wei Wang

Global uncertainty assessments by high dimensional model representations (HDMR)

A general set of quantitative model assessment and analysis tools, termed high-dimensional model representations (HDMR), have been introduced recently for high dimensional input–output systems. HDMR are a particular family of representations where each term in the representation reflects the independent and cooperative contributions of the inputs upon the output. When data are randomly sampled, a RS(random sampling)-HDMR can be constructed, which is an efficient tool to provide a fully global statistical analysis of a model. The individual RS-HDMR component functions have a direct statistical correlation interpretation. This relation permits the model output variance σ2 to be decomposed into its input contributions σ2=∑iσi2+∑i<jσij2+⋯ due to the independent variable action σi2, the pair correlation action σij2, etc. The information gained from this decomposition can be valuable for attaining a physical understanding of the origins of output uncertainty as well as suggesting additional laboratory/field studies or model refinements to best improve the quality of the model. To reduce sampling effort, the RS-HDMR component functions are approximately represented by orthonormal polynomials. Only one randomly sampled set of input–output data is needed to determine all σi, σij, etc. and a few hundred samples may give reliable results. This paper presents its methodology and applications on an atmospheric photochemistry model and a trace metal bioremediation model.

Probabilistic modeling of dietary arsenic exposure and dose and evaluation with 2003-2004 NHANES data.

Background Dietary exposure from food to toxic inorganic arsenic (iAs) in the general U.S. population has not been well studied. Objectives The goal of this research was to quantify dietary As exposure and analyze the major contributors to total As (tAs) and iAs. Another objective was to compare model predictions with observed data. Methods Probabilistic exposure modeling for dietary As was conducted with the Stochastic Human Exposure and Dose Simulation–Dietary (SHEDS-Dietary) model, based on data from the National Health and Nutrition Examination Survey. The dose modeling was conducted by combining the SHEDS-Dietary model with the MENTOR-3P (Modeling ENvironment for TOtal Risk with Physiologically Based Pharmacokinetic Modeling for Populations) system. Model evaluation was conducted via comparing exposure and dose-modeling predictions against duplicate diet data and biomarker measurements, respectively, for the same individuals. Results The mean modeled tAs exposure from food is 0.38 μg/kg/day, which is approximately 14 times higher than the mean As exposures from the drinking water. The mean iAs exposure from food is 0.05 μg/kg/day (1.96 μg/day), which is approximately two times higher than the mean iAs exposures from the drinking water. The modeled exposure and dose estimates matched well with the duplicate diet data and measured As biomarkers. The major food contributors to iAs exposure were the following: vegetables (24%); fruit juices and fruits (18%); rice (17%); beer and wine (12%); and flour, corn, and wheat (11%). Approximately 10% of tAs exposure from foods is the toxic iAs form. Conclusions The general U.S. population may be exposed to tAs and iAs more from eating some foods than from drinking water. In addition, this model evaluation effort provides more confidence in the exposure assessment tools used.

High dimensional model representations generated from low dimensional data samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x1,x2,...,xn} with n∼102 or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.

A source-to-dose assessment of population exposures to fine PM and ozone in Philadelphia, PA, during a summer 1999 episode

A novel source-to-dose modeling study of population exposures to fine particulate matter (PM2.5) and ozone (O3) was conducted for urban Philadelphia. The study focused on a 2-week episode, 11–24 July 1999, and employed the new integrated and mechanistically consistent source-to-dose modeling framework of MENTOR/SHEDS (Modeling Environment for Total Risk studies/Stochastic Human Exposure and Dose Simulation). The MENTOR/SHEDS application presented here consists of four components involved in estimating population exposure/dose: (1) calculation of ambient outdoor concentrations using emission-based photochemical modeling, (2) spatiotemporal interpolation for developing census-tract level outdoor concentration fields, (3) calculation of microenvironmental concentrations that match activity patterns of the individuals in the population of each census tract in the study area, and (4) population-based dosimetry modeling. It was found that the 50th percentiles of calculated microenvironmental concentrations of PM2.5 and O3 were significantly correlated with census-tract level outdoor concentrations, respectively. However, while the 95th percentiles of O3 microenvironmental concentrations were strongly correlated with outdoor concentrations, this was not the case for PM2.5. By further examining the modeled estimates of the 24-h aggregated PM2.5 and O3 doses, it was found that indoor PM2.5 sources dominated the contributions to the total PM2.5 doses for the upper 5 percentiles, Environmental Tobacco Smoking (ETS) being the most significant source while O3 doses due to time spent outdoors dominated the contributions to the total O3 doses for the upper 5 percentiles. The MENTOR/SHEDS system presented in this study is capable of estimating intake dose based on activity level and inhalation rate, thus completing the source-to-dose modeling sequence. The MENTOR/SHEDS system also utilizes a consistent basis of source characterization, exposure factors, and human activity patterns in conducting population exposure assessment of multiple co-occurring air pollutants, and this constitutes a primary distinction from previous studies of population exposure assessment, where different exposure factors and activity patterns would be used for different pollutants. Future work will focus on incorporating the effects of commuting patterns on population exposure/dose assessments as well as on extending the MENTOR/SHEDS applications to seasonal/annual studies and to other areas in the U.S.

Simulating bioremediation of uranium-contaminated aquifers; uncertainty assessment of model parameters

Bioremediation of trace metals and radionuclides in groundwater may require the manipulation of redox conditions via the injection of a carbon source. For example, after nitrate has been reduced, soluble U(VI) can be reduced simultaneously with other electron acceptors such as Fe(III) or sulfate to U(IV), which may precipitate as a solid (uraninite). To simulate the numerous biogeochemical processes that will occur during the bioremediation of trace-metal-contaminated aquifers, a time-dependent one-dimensional reactive transport model has been developed. The model consists of a set of coupled mass balance equations, accounting for advection, hydrodynamic dispersion, and a kinetic formulation of the biological or chemical transformations affecting an organic substrate, electron acceptors, corresponding reduced species, and trace metal contaminants of interest, uranium in this study. This set of equations is solved numerically, using a finite difference approximation. The redox conditions of the domain are characterized by estimating the pE, based on the concentration of the dominant terminal electron acceptor and its corresponding reduced species. This pE and the concentrations of relevant species are then used by a modified version of MINTEQA2, which calculates the speciation/sorption and precipitation/dissolution of the species of interest under equilibrium conditions. Kinetics of precipitation/dissolution processes are described as being proportional to the difference between the actual and calculated equilibrium concentration. A global uncertainty assessment, determined by Random Sampling High Dimensional Model Representation (RS-HDMR), was performed to attain a phenomenological understanding of the origins of output variability and to suggest input parameter refinements as well as to provide guidance for field experiments to improve the quality of the model predictions. By decomposing the model output variance into its different input contributions, RS-HDMR can identify the model inputs with the most influence on various model outputs, as well as their behavior pattern on the model output. Simulations are performed to illustrate the effect of biostimulation on the fate of uranium in a saturated aquifer, and to identify the key processes that need to be characterized with the highest accuracy prior to designing a uranium bioremediation scheme.

Correlation method for variance reduction of Monte Carlo integration in RS‐HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high‐dimensional functions. A practical form of the technique, RS‐HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS‐HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS‐HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS‐HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.

High-dimensional model representations generated from low order terms—lp-RS-HDMR

High‐dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. RS‐HDMR is a particular form of HDMR based on random sampling (RS) of the input variables. The component functions in an HDMR expansion are optimal choices tailored to the n‐variate function f(x) being represented over the desired domain of the n‐dimensional vector x. The high‐order terms (usually larger than second order, or equivalently beyond cooperativity between pairs of variables) in the expansion are often negligible. When it is necessary to go beyond the first and the second order RS‐HDMR, this article introduces a modified low‐order term product (lp)‐RS‐HDMR method to approximately represent the high‐order RS‐HDMR component functions as products of low‐order functions. Using this method the high‐order truncated RS‐HDMR expansions may be constructed without directly computing the original high‐order terms. The mathematical foundations of lp‐RS‐HDMR are presented along with an illustration of its utility in an atmospheric chemical kinetics model.

Dietary Arsenic Exposure: Xue et al. Respond

In our article (Xue et al. 2010), we cited Petito Boyce et al. (2008) based on their major conclusion stated at the end of their abstract that, “typical and high-end background exposures to inorganic arsenic in U.S. populations do not present elevated risks of carcinogenicity.” We agree with Petito Boyce et al. that we “missed an opportunity to provide additional support for” our overall conclusions, and very much appreciate that they have offered this detailed comparison showing the agreement between our modeling results. Our discussion of Petito Boyce et al. (2008)’s conclusions was intended to bolster the need to develop a more comprehensive analysis of the sources of inorganic arsenic exposure, not to suggest that their exposure analysis was incomplete or inaccurate.

Comparative evaluation of computationally ecient uncertainty propagation methods through application to regional-scale air quality models

This work presents the comparative evaluation of two computationally ecient uncertainty propagation techniques: the Stochastic Response Surface Method (SRSM) and the High Dimensional Model Representation (HDMR) method. The evaluation is performed in relation to the applicability to these methods to complex numerical models, specifically those dealing with simulating regional-scale air quality. The air quality model used in the application case study is a Eulerian type three-dimensional grid-based model, and involves a large set of non-linear partial and ordinary dierential equations to describe atmospheric transport and chemistry, thus making it impractical to use traditional Monte Carlo based techniques for performing uncertainty analysis. The application case study focuses on studying uncertainties in ozone levels estimated by a regulatory air quality model due to uncertainties in biogenic emissions of ozone precursors. Preliminary results show that 95th confidence interval for the peak ozone levels spans a range of over ±15% from the mean value, indicating significant uncertainties with respect to the health impact and regulatory compliance. Both the SRSM and HDMR methods provide similar estimates, thus serving to cross-validate each other, while requiring a small number of model simulations.