Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.

Kernels and Designs for Modelling Invariant Functions: From Group Invariance to Additivity

We focus on kernels incorporating different kinds of prior knowledge on functions to be approximated by Kriging. A recent result on random fields with paths invariant under a group action is generalised to combinations of composition operators, and a characterisation of kernels leading to random fields with additive paths is obtained as a corollary. A discussion follows on some implications on design of experiments, and it is shown in the case of additive kernels that the so-called class of “axis designs” outperforms Latin hypercubes in terms of the IMSE criterion.

A Nonstationary Space-Time Gaussian Process Model for Partially Converged Simulations

In the context of expensive numerical experiments, a promising solution for alleviating the computational costs consists of using partially converged simulations instead of exact solutions. The gain in computational time is at the price of precision in the response. This work addresses the issue of fitting a Gaussian process model to partially converged simulation data for further use in prediction. The main challenge consists of the adequate approximation of the error due to partial convergence, which is correlated in both design variables and time directions. Here, we propose fitting a Gaussian process in the joint space of design parameters and computational time. The model is constructed by building a nonstationary covariance kernel that reflects accurately the actual structure of the error. Practical solutions are proposed for solving parameter estimation issues associated with the proposed model. The method is applied to a computational fluid dynamics test case and shows significant improvement in prediction compared to a classical kriging model.

Quantile-Based Optimization of Noisy Computer Experiments with Tunable Precision

This article addresses the issue of kriging-based optimization of stochastic simulators. Many of these simulators depend on factors that tune the level of precision of the response, the gain in accuracy being at a price of computational time. The contribution of this work is two-fold: first, we propose a quantile-based criterion for the sequential design of experiments, in the fashion of the classical expected improvement criterion, which allows an elegant treatment of heterogeneous response precisions. Second, we present a procedure for the allocation of the computational time given to each measurement, allowing a better distribution of the computational effort and increased efficiency. Finally, the optimization method is applied to an original application in nuclear criticality safety. This article has supplementary material available online. The proposed criterion is available in the R package DiceOptim.

Precipitation isoscape of high reliefs: interpolation scheme designed and tested for monthly resolved precipitation oxygen isotope records of an Alpine domain

Stable oxygen isotope composition of atmospheric precipitation (δ18Op) was scrutinized from 39 stations distributed over Switzerland and its border zone. Monthly amount-weighted δ18Op values averaged over the 1995–2000 period showed the expected strong linear altitude dependence (−0.15 to −0.22‰ per 100 m) only during the summer season (May–September). Steeper gradients (~ −0.56 to −0.60‰ per 100 m) were observed for winter months over a low elevation belt, while hardly any altitudinal difference was seen for high elevation stations. This dichotomous pattern could be explained by the characteristically shallower vertical atmospheric mixing height during winter season and provides empirical evidence for recently simulated effects of stratified atmospheric flow on orographic precipitation isotopic ratios. This helps explain "anomalous" deflected altitudinal water isotope profiles reported from many other high relief regions. Grids and isotope distribution maps of the monthly δ18Op have been calculated over the study region for 1995–1996. The adopted interpolation method took into account both the variable mixing heights and the seasonal difference in the isotopic lapse rate and combined them with residual kriging. The presented data set allows a point estimation of δ18Op with monthly resolution. According to the test calculations executed on subsets, this biannual data set can be extended back to 1992 with maintained fidelity and, with a reduced station subset, even back to 1983 at the expense of faded reliability of the derived δ18Op estimates, mainly in the eastern part of Switzerland. Before 1983, reliable results can only be expected for the Swiss Plateau since important stations representing eastern and south-western Switzerland were not yet in operation.

Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using concepts from random set theory, we propose to adapt the Vorob’ev expectation and deviation to capture the variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob’ev deviation can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front.

Hazardous air pollutants and childhood lymphohematopoietic cancer in Southeast Texas, 1995--2004

Southeast Texas, including Houston, has a large presence of industrial facilities and has been documented to have poorer air quality and significantly higher cancer rates than the remainder of Texas. Given citizens’ concerns in this 4th largest city in the U.S., Mayor Bill White recently partnered with the UT School of Public Health to determine methods to evaluate the health risks of hazardous air pollutants (HAPs). Sexton et al. (2007) published a report that strongly encouraged analytic studies linking these pollutants with health outcomes. In response, we set out to complete the following aims: 1. determine the optimal exposure assessment strategy to assess the association between childhood cancer rates and increased ambient levels of benzene and 1,3-butadiene (in an ecologic setting) and 2. evaluate whether census tracts with the highest levels of benzene or 1,3-butadiene have higher incidence of childhood lymphohematopoietic cancer compared with census tracts with the lowest levels of benzene or 1,3-butadiene, using Poisson regression. The first aim was achieved by evaluating the usefulness of four data sources: geographic information systems (GIS) to identify proximity to point sources of industrial air pollution, industrial emission data from the U.S. EPA’s Toxic Release Inventory (TRI), routine monitoring data from the U.S. EPA Air Quality System (AQS) from 1999-2000 and modeled ambient air levels from the U.S. EPA’s 1999 National Air Toxic Assessment Project (NATA) ASPEN model. Further, once these four data sources were evaluated, we narrowed them down to two: the routine monitoring data from the AQS for the years 1998-2000 and the 1999 U.S. EPA NATA ASPEN modeled data. We applied kriging (spatial interpolation) methodology to the monitoring data and compared the kriged values to the ASPEN modeled data. Our results indicated poor agreement between the two methods. Relative to the U.S. EPA ASPEN modeled estimates, relying on kriging to classify census tracts into exposure groups would have caused a great deal of misclassification. To address the second aim, we additionally obtained childhood lymphohematopoietic cancer data for 1995-2004 from the Texas Cancer Registry. The U.S. EPA ASPEN modeled data were used to estimate ambient levels of benzene and 1,3-butadiene in separate Poisson regression analyses. All data were analyzed at the census tract level. We found that census tracts with the highest benzene levels had elevated rates of all leukemia (rate ratio (RR) = 1.37; 95% confidence interval (CI), 1.05-1.78). Among census tracts with the highest 1,3-butadiene levels, we observed RRs of 1.40 (95% CI, 1.07-1.81) for all leukemia. We detected no associations between benzene or 1,3-butadiene levels and childhood lymphoma incidence. This study is the first to examine this association in Harris and surrounding counties in Texas and is among the first to correlate monitored levels of HAPs with childhood lymphohematopoietic cancer incidence, evaluating several analytic methods in an effort to determine the most appropriate approach to test this association. Despite recognized weakness of ecologic analyses, our analysis suggests an association between childhood leukemia and hazardous air pollution.^

Spatial analysis of cardiovascular disease mortality and air pollution in Harris County, Texas

The association between fine particulate matter air pollution (PM2.5) and cardiovascular disease (CVD) mortality was spatially analyzed for Harris County, Texas, at the census tract level. The objective was to assess how increased PM2.5 exposure related to CVD mortality in this area while controlling for race, income, education, and age. An estimated exposure raster was created for Harris County using Kriging to estimate the PM2.5 exposure at the census tract level. The PM2.5 exposure and the CVD mortality rates were analyzed in an Ordinary Least Squares (OLS) regression model and the residuals were subsequently assessed for spatial autocorrelation. Race, median household income, and age were all found to be significant (p<0.05) predictors in the model. This study found that for every one μg/m3 increase in PM2.5 exposure, holding age and education variables constant, an increase of 16.57 CVD deaths per 100,000 would be predicted for increased minimum exposure values and an increase of 14.47 CVD deaths per 100,000 would be predicted for increased maximum exposure values. This finding supports previous studies associating PM2.5 exposure with CVD mortality. This study further identified the areas of greatest PM2.5 exposure in Harris County as being the geographical locations of populations with the highest risk of CVD (i.e., predominantly older, low-income populations with a predominance of African Americans). The magnitude of the effect of PM2.5 exposure on CVD mortality rates in the study region indicates a need for further community-level studies in Harris County, and suggests that reducing excess PM2.5 exposure would reduce CVD mortality.^

Ozone and fine particulate matter association with CVD and COPD emergency room visits in Harris County, Texas: Spatiotemporal analysis

This study represents a secondary analysis of the merging of emergency room visits and daily ozone and PM2.5. Although the adverse health effects of ozone and fine particulate matter have been documented in the literature, evidence regarding the health risks of these two pollutants in Harris County, Texas, is limited. Harris County (Houston) has sufficiently unique characteristics that analysis of these relationships in this setting and with the ozone and industry issues in Houston is informative. The objective of this study was to investigate the association between the joint exposure to ozone and fine particulate matter, and emergency room diagnoses of chronic obstructive pulmonary disease and cardiovascular disease in Harris County, Texas, from 2004 to 2009, with zero and one day lags. ^ The study variables were daily emergency room visits for Harris County, Texas, from 2004 to 2009, temperature, relative humidity, east wind component, north wind component, ozone, and fine particulate matter. Information about each patient's age, race, and gender was also included. The two dichotomous outcomes were emergency room visits diagnoses for chronic obstructive pulmonary disease and cardiovascular disease. Estimates of ozone and PM2.5 were interpolated using kriging, in which estimates of the two pollutants were predicted from monitoring data for every case residence zip code for every day of the six years, over 3 million estimates (one of each pollutant for each case in the database). ^ Logistic regressions were conducted to estimate odds ratios of the two outcomes. Three analyses were conducted: one for all records, another for visits during the four months of April and September of 2005 and 2009, and a third one for visits from zip codes that are close to PM2.5 monitoring stations (east area of Harris County). The last two analyses were designed to investigate special temporal and spatial characteristics of the associations. ^ The dataset included all ER visits surveyed by Safety Net from 2004 to 2009, exceeding 3 million visits for all causes. There were 95,765 COPD and 96,596 CVD cases during this six year period. A 1-μg/m3 increase in PM2.5 on the same day was associated with a 1.0% increase in the odds of chronic obstructive pulmonary disease emergency room diagnoses, a 0.4% increase in the odds of cardiovascular disease emergency room diagnoses, and a 0.2% increase in the odds of cardiovascular disease emergency room diagnoses on the following day. A 1-ppb increase in ozone was associated with a 0.1% increase in the odds of chronic obstructive pulmonary disease emergency room diagnoses on the same day. These four percentages add up to 1.7% of ER visits. That is, over the period of six years, one unit increase for both ozone and PM2.5 (joint increase), resulted in about 55,286 (3,252,102 * 0.017) extra ER visits for CVD or COPD, or 9,214 extra ER visits per year. ^ After adjustment for age, race, gender, day of the week, temperature, relative humidity, east wind component, north wind component, and wind speed, there were statistically significant associations between emergency room chronic obstructive pulmonary disease diagnosis in Harris County, Texas, with joint exposure to ozone and fine particulate matter for the same day; and between emergency room cardiovascular disease diagnosis and exposure to PM2.5 of the same day and the previous day. ^ Despite the small association between the two air pollutants and the health outcomes, this study points to important findings. Namely, the need to identify reasons for the increase of CVD and COPD ER visits over the course of the project, the statistical association between humidity (or whatever other variables for which it may serve as a surrogate) and CVD and COPD cases, and the confirmatory finding that males and blacks have higher odds for the two outcomes, as consistent with other studies. ^ An important finding of this research suggests that the number and distribution of PM2.5 monitors in Harris County - although not evenly spaced geographically—are adequate to detect significant association between exposure and the two outcomes. In addition, this study points to other potential factors that contribute to the rising incidence rates of CVD and COPD ER visits in Harris County such as population increases, patient history, life style, and other pollutants. Finally, results of validation, using a subset of the data demonstrate the robustness of the models.^

Modelización y generación de mapas del comportamiento espacial de Sporisorium reilianum ((Kühn) Langdon y Fullerton) en Estado de México, México

El presente estudio se realizó con la finalidad de modelizar la distribución espacial del carbón de la espiga del maíz causada por Sporisorium reilianum durante 2006 en el Estado de México y su visualización a través de la generación mapas de densidad. El muestreo se realizó en 100 parcelas georreferenciadas por cada localidad analizada. La incidencia de la enfermedad (porcentaje de plantas enfermas) se determinó al establecer cinco puntos parcela, en cada punto se contabilizaron 100 plantas. Se realizó el análisis geoestadístico para estimar el semivariograma experimental, una vez obtenido, se ajustó a un modelo teórico (esférico, exponencial o gaussiano) a través de los programas Variowin 2.2., su ajuste se validó a través de la validación cruzada. Posteriormente, se elaboraron mapas de agregación de la enfermedad con el método de interpolación geoestadística o krigeado. Los resultados indicaron que la enfermedad se presentó en 20 localidades de 19 municipios del Estado de México; todas las localidades presentaron un comportamiento espacial agregado de la enfermedad, 16 localidades se ajustaron al modelo esférico, dos al modelo exponencial y dos localidades se ajustaron al modelo gaussiano. En todos los modelos se lograron establecer mapas de agregación que permitirá adecuar las acciones de manejo en términos de puntos o sitios específicos.

Modelización espacial de huevos de Thrips (Thysanoptera: Frankliniella occidentalis) en tomate de cáscara por medio de técnicas geoestadísticas

El tomate de cáscara (Physalis ixocarpa Brot.) es un cultivo alimenticio de gran importancia económica en México. Sin embargo, es afectado por diversas plagas y enfermedades tales como los Thrips (Thysanoptera: Frankliniella occidentalis) y el virus de la marchitez manchada del tomate (TSWV) que llegan a causar hasta un 80% de pérdidas. El objetivo del presente trabajo fue modelizar la distribución espacial de huevos de Thrips mediante técnicas geoestadísticas y obtener, en consecuencia, mapas de incidencia por medio del Kriging. Se georreferenciaron 121 puntos de muestreo en cada una de las parcelas comerciales de los municipios de Luvianos, Jocotitlán e Ixtlahuaca, a través del método de transectos en tres etapas fenológicas del cultivo. Se contabilizó el número de huevos de Thrips en cada punto de muestreo. Los resultados mostraron que las poblaciones de huevos deThrips presentan una distribución agregada, identificándose varios centros de conglomeración a través de los mapas obtenidos. Los semivariogramas obtenidos de la distribución espacial se ajustaron principalmente a los modelos gaussianos y esféricos. La distribución de huevos de Thrips se presentó en centros de agregación dentro de las parcelas estudiadas, lo cual permitirá establecer estrategias y medidas de control o mitigación en términos de sitios específicos de infestación de huevos de Thrips.

Probability inundation map for a scenario combining storm surge and sea-level rise

Coastal managers require reliable spatial data on the extent and timing of potential coastal inundation, particularly in a changing climate. Most sea level rise (SLR) vulnerability assessments are undertaken using the easily implemented bathtub approach, where areas adjacent to the sea and below a given elevation are mapped using a deterministic line dividing potentially inundated from dry areas. This method only requires elevation data usually in the form of a digital elevation model (DEM). However, inherent errors in the DEM and spatial analysis of the bathtub model propagate into the inundation mapping. The aim of this study was to assess the impacts of spatially variable and spatially correlated elevation errors in high-spatial resolution DEMs for mapping coastal inundation. Elevation errors were best modelled using regression-kriging. This geostatistical model takes the spatial correlation in elevation errors into account, which has a significant impact on analyses that include spatial interactions, such as inundation modelling. The spatial variability of elevation errors was partially explained by land cover and terrain variables. Elevation errors were simulated using sequential Gaussian simulation, a Monte Carlo probabilistic approach. 1,000 error simulations were added to the original DEM and reclassified using a hydrologically correct bathtub method. The probability of inundation to a scenario combining a 1 in 100 year storm event over a 1 m SLR was calculated by counting the proportion of times from the 1,000 simulations that a location was inundated. This probabilistic approach can be used in a risk-aversive decision making process by planning for scenarios with different probabilities of occurrence. For example, results showed that when considering a 1% probability exceedance, the inundated area was approximately 11% larger than mapped using the deterministic bathtub approach. The probabilistic approach provides visually intuitive maps that convey uncertainties inherent to spatial data and analysis.

Environmental variables and multivariate seabed classification maps via k means clustering of Potter Cove, Antarctica, link to input and result files in GeoTIFF format

This study subdivides the Potter Cove, King George Island, Antarctica, into seafloor regions using multivariate statistical methods. These regions are categories used for comparing, contrasting and quantifying biogeochemical processes and biodiversity between ocean regions geographically but also regions under development within the scope of global change. The division obtained is characterized by the dominating components and interpreted in terms of ruling environmental conditions. The analysis includes in total 42 different environmental variables, interpolated based on samples taken during Australian summer seasons 2010/2011 and 2011/2012. The statistical errors of several interpolation methods (e.g. IDW, Indicator, Ordinary and Co-Kriging) with changing settings have been compared and the most reasonable method has been applied. The multivariate mathematical procedures used are regionalized classification via k means cluster analysis, canonical-correlation analysis and multidimensional scaling. Canonical-correlation analysis identifies the influencing factors in the different parts of the cove. Several methods for the identification of the optimum number of clusters have been tested and 4, 7, 10 as well as 12 were identified as reasonable numbers for clustering the Potter Cove. Especially the results of 10 and 12 clusters identify marine-influenced regions which can be clearly separated from those determined by the geological catchment area and the ones dominated by river discharge.

Whale sightings, group sizes and krill biomass in West Greenland in 2005

This study combined data on fin whale Balaenoptera physalus, humpback whale Megaptera novaeangliae, minke whale B. acutorostrata, and sei whale B. borealis sightings from large-scale visual aerial and ship-based surveys (248 and 157 sightings, respectively) with synoptic acoustic sampling of krill Meganyctiphanes norvegica and Thysanoessa sp. abundance in September 2005 in West Greenland to examine the relationships between whales and their prey. Krill densities were obtained by converting relationships of volume backscattering strengths at multiple frequencies to a numerical density using an estimate of krill target strength. Krill data were vertically integrated in 25 m depth bins between 0 and 300 m to obtain water column biomass (g/m**2) and translated to density surfaces using ordinary kriging. Standard regression models (Generalized Additive Modeling, GAM, and Generalized Linear Modeling, GLM) were developed to identify important explanatory variables relating the presence, absence, and density of large whales to the physical and biological environment and different survey platforms. Large baleen whales were concentrated in 3 focal areas: (1) the northern edge of Lille Hellefiske bank between 65 and 67°N, (2) north of Paamiut at 63°N, and (3) in South Greenland between 60 and 61° N. There was a bimodal pattern of mean krill density between depths, with one peak between 50 and 75 m (mean 0.75 g/m**2, SD 2.74) and another between 225 and 275 m (mean 1.2 to 1.3 g/m**2, SD 23 to 19). Water column krill biomass was 3 times higher in South Greenland than at any other site along the coast. Total depth-integrated krill biomass was 1.3 x 10**9 (CV 0.11). Models indicated the most important parameter in predicting large baleen whale presence was integrated krill abundance, although this relationship was only significant for sightings obtained on the ship survey. This suggests that a high degree of spatio-temporal synchrony in observations is necessary for quantifying predator-prey relationships. Krill biomass was most predictive of whale presence at depths >150 m, suggesting a threshold depth below which it is energetically optimal for baleen whales to forage on krill in West Greenland.

A Case Study of Geometric Modelling via 3-D Point Interpolation for the Bathymetry of the Rabasa Lakes (Alicante, Spain)

The interpolation of points by means of Information Technology programs appears as a technical tool of some relevancy in the hydrogeology in general and in the study of the humid zones in particular. Our approach has been the determination of the 3-D geometry of the humid zones of major depth of the Rabasa Lakes. To estimate the topography of the lake bed, we proceed to acquire information in the field by means of sonar and GPS equipment. A total of 335 points were measured both on the perimeter and in the lake bed. In a second stage, this information was used in a kriging program to obtain the bathymetry of the wetland. This methodology is demonstrated as one of the most reliable and cost-efficient for the 3-D analysis of this type of water masses. The bathymetric study of the zone allows us to characterize the mid- and long-term hydrological evolution of the lakes by means of depth-area-volume curves.