Combining categorical and continuous spatial information within the Bayesian maximum entropy paradigm

Due to the fast pace increasing availability and diversity of information sources in environmental sciences, there is a real need of sound statistical mapping techniques for using them jointly inside a unique theoretical framework. As these information sources may vary both with respect to their nature (continuous vs. categorical or qualitative), their spatial density as well as their intrinsic quality (soft vs. hard data), the design of such techniques is a challenging issue. In this paper, an efficient method for combining spatially non-exhaustive categorical and continuous data in a mapping context is proposed, based on the Bayesian maximum entropy paradigm. This approach relies first on the definition of a mixed random field, that can account for a stochastic link between categorical and continuous random fields through the use of a cross-covariance function. When incorporating general knowledge about the first- and second-order moments of these fields, it is shown that, under mild hypotheses, their joint distribution can be expressed as a mixture of conditional Gaussian prior distributions, with parameters estimation that can be obtained from entropy maximization. A posterior distribution that incorporates the various (soft or hard) continuous and categorical data at hand can then be obtained by a straightforward conditionalization step. The use and potential of the method is illustrated by the way of a simulated case study. A comparison with few common geostatistical methods in some limit cases also emphasizes their similarities and differences, both from the theoretical and practical viewpoints. As expected, adding categorical information may significantly improve the spatial prediction of a continuous variable, making this approach powerful and very promising.     

The Bayesian maximum entropy method for lognormal variables

The Bayesian maximum entropy (BME) method can be used to predict the value of a spatial random field at an unsampled location given precise (hard) and imprecise (soft) data. It has mainly been used when the data are non-skewed. When the data are skewed, the method has been used by transforming the data (usually through the logarithmic transform) in order to remove the skew. The BME method is applied for the transformed variable, and the resulting posterior distribution transformed back to give a prediction of the primary variable. In this paper, we show how the implementation of the BME method that avoids the use of a transform, by including the logarithmic statistical moments in the general knowledge base, gives more appropriate results, as expected from the maximum entropy principle. We use a simple illustration to show this approach giving more intuitive results, and use simulations to compare the approaches in terms of the prediction errors. The simulations show that the BME method with the logarithmic moments in the general knowledge base reduces the errors, and we conclude that this approach is more suitable to incorporate soft data in a spatial analysis for lognormal data.     

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.     

Defining probability of saturation with indicator kriging on hard and soft data

In humid, well-vegetated areas, such as in the northeastern US, runoff is most commonly generated from relatively small portions of the landscape becoming completely saturated, however, little is known about the spatial and temporal behavior of these saturated regions. Indicator kriging provides a way to use traditional water table data to quantify probability of saturation to evaluate predicted spatial distributions of runoff generation risk, especially for the new generation of water quality models incorporating saturation excess runoff theory. When spatial measurements of a variable are transformed to binary indicators (i.e., 1 if above a given threshold value and 0 if below) and the resulting indicator semivariogram is modeled, indicator kriging produces the probability of the measured variable to exceed the threshold value. Indicator kriging gives quantified probability of saturation or, consistent with saturation excess runoff theory, runoff generation risk with depth to water table as the variable and the threshold set near the soil surface. The probability of saturation for a 120 m × 180 m hillslope based upon 43 measurements of depth to water table is investigated with indicator semivariograms for six storm events. The indicator semivariograms show high spatial structure in saturated regions with large antecedent rainfall conditions. The temporal structure of the data is used to generate interpolated (soft) data to supplement measured (hard) data. This improved the spatial structure of the indicator semivariograms for lower antecedent rainfall conditions. Probability of saturation was evaluated through indicator kriging incorporating soft data showing, based on this preliminary study, highly connected regions of saturation as expected for the wet season (April through May) in the Catskill Mountain region of New York State. Supplementation of hard data with soft data incorporates physical hydrology of the hillslope to capture significant patterns not available when using hard data alone for indicator kriging. With the need for water quality models incorporating appropriate runoff generation risk estimates on the rise, this manner of data will lay the groundwork for future model evaluation and development.     

Accounting for the uncertainty in the local mean in spatial prediction by Bayesian Maximum Entropy

Bayesian Maximum Entropy (BME) has been successfully used in geostatistics to calculate predictions of spatial variables given some general knowledge base and sets of hard (precise) and soft (imprecise) data. This general knowledge base commonly consists of the means at each of the locations considered in the analysis, and the covariances between these locations. When the means are not known, the standard practice is to estimate them from the data; this is done by either generalized least squares or maximum likelihood. The BME prediction then treats these estimates as the general knowledge means, and ignores their uncertainty. In this paper we develop a prediction that is based on the BME method that can be used when the general knowledge consists of the covariance model only. This prediction incorporates the uncertainty in the estimated local mean. We show that in some special cases our prediction is equal to results from classical geostatistics. We investigate the differences between our approach and the standard approach for predicting in this common practical situation.     

Space-time mapping of soil salinity using probabilistic bayesian maximum entropy

The mapping of saline soils is the first task before any reclamation effort. Reclamation is based on the knowledge of soil salinity in space and how it evolves with time. Soil salinity is traditionally determined by soil sampling and laboratory analysis. Recently, it became possible to complement these hard data with soft secondary data made available using field sensors like electrode probes. In this study, we had two data sets. The first includes measurements of field salinity (EC_a) at 413 locations and 19 time instants. The second, which is a subset of the first (13 to 20 locations), contains, in addition to EC_a, salinity determined in the laboratory (EC_2.5). Based on a procedure of cross-validation, we compared the prediction performance in the space-time domain of 3 methods: kriging using either only hard data (HK) or hard and mid interval soft data (HMIK), and Bayesian maximum entropy (BME) using probabilistic soft data. We found that BME was less biased, more accurate and giving estimates, which were better correlated with the observed values than the two kriging techniques. In addition, BME allowed one to delineate with better detail saline from non-saline areas.     

Spatial assessment of fishing effort around European marine reserves: implications for successful fisheries management

We examined the spatial dynamic of artisanal fishing fleets around five European marine protected areas (MPAs) to derive general implications for the evaluation of MPAs as fisheries management tools. The coastal MPAs studied were located off France, Malta and Spain and presented a variety of spatial designs and processes of establishment. We developed a standardized methodology to define factors influencing effort allocation and to produce fishing effort maps by merging GIS with geostatistical modelling techniques. Results revealed that in most cases the factors “distance to the no-take”, “water depth”, and “distance to the port” had a significant influence on effort allocation by the fishing fleets. Overall, we found local concentration of fishing effort around the MPA borders. Thus, neglecting the pattern of fishing effort distribution in evaluating MPA benefits, such as spillover of biomass, could hamper sound interpretation of MPAs as fisheries management tools.     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

Application of the BME approach to soil texture mapping

In order to derive accurate space/time maps of soil properties, soil scientists need tools that combine the usually scarce hard data sets with the more easily accessible soft data sets. In the field of modern geostatistics, the Bayesian maximum entropy (BME) approach provides new and powerful means for incorporating various forms of physical knowledge (including hard and soft data, soil classification charts, land cover data from satellite pictures, and digital elevation models) into the space/time mapping process. BME produces the complete probability distribution at each estimation point, thus allowing the calculation of elaborate statistics (even when the distribution is not Gaussian). It also offers a more rigorous and systematic method than kriging for integrating uncertain information into space/time mapping. In this work, BME is used to estimate the three textural fractions involved in a texture map. The first case study focuses on the estimation of the clay fraction, whereas the second one considers the three textural fractions (sand, silt and clay) simultaneously. The BME maps obtained are informative (important soil characteristics are identified, natural variations are well reproduced, etc.). Furthermore, in both case studies, the estimates obtained by BME were more accurate than the simple kriging (SK) estimates, thus offering a better picture of soil reality. In the multivariate case, classification error rate analysis in terms of BME performs considerably better than in terms of kriging. Analysis in terms of BME can offer valuable information to be used in sampling design, in optimizing the hard to soft data ratio, etc.     

Enviromental genotoxicity evaluation: Bayesian approach for a mixture statistical model

 The data analyzed in this paper are part of the results described in Bueno et al. (2000). Three cytogenetics endpoints were analyzed in three populations of a species of wild rodent – Akodon montensis – living in an industrial, an agricultural, and a preservation area at the Itajaí Valley, State of Santa Catarina, Brazil. The polychromatic/normochromatic ratio, the mitotic index, and the frequency of micronucleated polychromatic erythrocites were used in an attempt to establish a genotoxic profile of each area. It was assumed that the three populations were in the same conditions with respect to the influence of confounding factors such as animal age, health, nutrition status, presence of pathogens, and intra- and inter-populational genetic variability. Therefore, any differences found in the endpoints analyzed could be attributed to the external agents present in each area. The statistical models used in this paper are mixtures of negative-binomials and Poisson variables. The Poisson variables are used as approximations of binomials for rare events. The mixing distributions are beta densities. The statistical analyzes are under the bayesian perspective, as opposed to the frequentist ones often considered in the literature, as for instance in Bueno et al. (2000).     

Revisiting prior distributions,Part II: Implications of the physical prior in maximum entropy analysis

The well-known “Maximum Entropy Formalism” offers a powerful framework for deriving probability density functions given a relevant knowledge base and an adequate prior. The majority of results based on this approach have been derived assuming a flat uninformative prior, but this assumption is to a large extent arbitrary (any one-to-one transformation of the random variable will change the flat uninformative prior into some non-constant function). In a companion paper we introduced the notion of a natural reference point for dimensional physical variables, and used this notion to derive a class of physical priors that are form-invariant to changes in the system of dimensional units. The present paper studies effects of these priors on the probability density functions derived using the maximum entropy formalism. Analysis of real data shows that when the maximum entropy formalism uses the physical prior it yields significantly better results than when it is based on the commonly used flat uninformative prior. This improvement reflects the significance of the incorporating additional information (contained in physical priors), which is ignored when flat priors are used in the standard form of the maximum entropy formalism. A potentially serious limitation of the maximum entropy formalism is the assumption that sample moments are available. This is not the case in many macroscopic real-world problems, where the knowledge base available is a finite sample rather than population moments. As a result, the maximum entropy formalism generates a family of “nested models” parameterized by the unknown values of the population parameters. In this work we combine this formalism with a model selection scheme based on Akaike’s information criterion to derive the maximum entropy model that is most consistent with the available sample. This combination establishes a general inference framework of wide applicability in scientific/engineering problems.     

Testing the maximum entropy production approach for estimating evapotranspiration from closed canopy shrubland in a low‐energy humid environment

Quantifying and partitioning evapotranspiration (ET) into evaporation and transpiration is challenging but important for interpreting vegetation effects on the water balance. We applied a model based on the theory of maximum entropy production to estimate ET for shrubs for the first time in a low‐energy humid headwater catchment in the Scottish Highlands. In total, 53% of rainfall over the growing season was returned to the atmosphere through ET (59 ± 2% as transpiration), with 22% of rainfall ascribed to interception loss and understory ET. The remainder of rainfall percolated below the rooting zone. The maximum entropy production model showed good capability for total ET estimation, in addition to providing a first approximation for distinguishing evaporation and transpiration in such ecosystems. This study shows that this simple and low‐cost approach has potential for local to regional ET estimation with availability of high‐resolution hydroclimatic data. Limitations of the approach are also discussed.     

Using multiple‐variable indicator kriging to assess groundwater quality for irrigation in the aquifers of the Choushui River alluvial fan

This work attempted to locate clean and safe groundwater for irrigation use in the Choushui River alluvial fan. Multiple‐variable indicator kriging (MVIK) was adopted to evaluate numerous hydrochemical parameters for a standard of water quality for irrigation in Taiwan. Many hydrochemical parameters in groundwater were distinguished into three main categories—salinity/sodium hazard, nitrogen hazard and heavy metal hazard. Safe and potential hazardous regions of groundwater for irrigation were delineated according to different probabilities estimated by MVIK. The probabilistic results of the classifications gave an opportunity to explore the spatial uncertainty of the hazards and helped government administrators establish a sound policy associated with the development and management of groundwater resources. Analysis of the results indicate that the central distal‐fan and mid‐fan aquifers are the best places to extract clean and safe groundwater for irrigation, and the deep aquifer (exceeding 200 m depth) has wider regions with clean and safe groundwater for irrigation than shallow aquifers. The northern and southern aquifers, with multiple hazards, limit groundwater use for irrigation. Although the proximal‐fan aquifer is a zone of groundwater recharge, the high nitrogen content seriously affects the environment and is not suitable for irrigation use. Copyright © 2008 John Wiley & Sons, Ltd.     

Spatial correlation of shear-wave velocity in the San Francisco Bay Area sediments

Ground motions recorded within sedimentary basins are variable over short distances. One important cause of the variability is that local soil properties are variable at all scales. Regional hazard maps developed for predicting site effects are generally derived from maps of surficial geology; however, recent studies have shown that mapped geologic units do not correlate well with the average shear-wave velocity of the upper 30 m, V_s(30). We model the horizontal variability of near-surface soil shear-wave velocity in the San Francisco Bay Area to estimate values in unsampled locations in order to account for site effects in a continuous manner. Previous geostatistical studies of soil properties have shown horizontal correlations at the scale of meters to tens of meters while the vertical correlations are on the order of centimeters. In this paper we analyze shear-wave velocity data over regional distances and find that surface shear-wave velocity is correlated at horizontal distances up to 4 km based on data from seismic cone penetration tests and the spectral analysis of surface waves. We propose a method to map site effects by using geostatistical methods based on the shear-wave velocity correlation structure within a sedimentary basin. If used in conjunction with densely spaced shear-wave velocity profiles in regions of high seismic risk, geostatistical methods can produce reliable continuous maps of site effects.     

Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.     

Spatial organization of macroinvertebrate assemblages in the Lower Mekong Basin

In this study, we investigated patterns of spatial variation in macroinvertebrate assemblages in the Lower Mekong Basin (LMB) and examined their relationship with environmental factors. Cluster analysis was used to group macroinvertebrate samples and Linear Discriminant Analysis was performed to discriminate the major factors associated with the macroinvertebrate assemblages. Four clusters could be distinguished based on the dissimilarity between macroinvertebrate assemblages. The assemblages related to the tributaries and the upstream parts (cluster II) were characterized by a lower richness, abundance, diversity and a lower number of indicator taxa compared to the assemblage found downstream in the Mekong delta (cluster I). Aquatic insects and their indicator taxa (e.g. Caenodes sp., Dipseudopsis sp. and Gomphidae sp.), preferring a high-altitude environment with a high dissolved oxygen concentration and a high density of wood/shrub and evergreen forests, were the most predominant group in the assemblages occupying the tributaries and the upstream parts (cluster IIa). The assemblage found in the delta, consisting largely of molluscs and a moderate richness and abundance of worms, crustaceans and dipteran insects, was mainly represented by Corbicula leviuscula and C. moreletiana (molluscs), Namalycastis longicirris and Chaetogaster langi (worms), Corophium minutum and Grandidierella lignorum (crustaceans), and Cricotopus sp. and Clinotanypus sp. (dipteran insects). This assemblage was associated with a large watershed surface area, deep and wide rivers and a high water temperature. The intermediate assemblage (cluster IIb1) in-between could be discriminated based on land cover types including inundated, wetland and agricultural land, and was represented most by molluscs. Strikingly, the assemblage occupying the upstream parts (cluster IIa), which is related to intensified agriculture and a moderate conductivity, was characterized by a higher macroinvertebrate diversity compared to the mountainous and less impacted tributaries. This could mean that the natural stress is high in these systems for some taxa, leading to a lower overall taxonomic richness and abundance. Nevertheless, the number of taxa and the diversity of macroinvertebrates remained relatively high across the basin, especially in the delta assemblage. Therefore, the LMB deserves a particular attention for conservation.     

On the use of maximum entropy/autoregressive spectrum in harmonic analysis of time series

The maximum entropy (ME) spectrum, or its equivalent form of the autoregressive (AR) spectrum, has been used as a tool for harmonic analysis of time series in geophysics. This paper critically examines its usage in estimating the amplitude and the exponential decay rate of a harmonic function. The argument is based upon Prony's relation, which relates a complex-conjugate pair of poles for the AR model of the time series on one hand, to the complex frequency of one harmonic component in the time series on the other. It is found that: (i) the ME spectrum can be used as an estimator for the decay rate in a way similar to the Fourier spectral analysis; (ii) the ME spectrum contains no information whatsoever about the amplitude, contrary to what has been claimed and practiced in geophysical applications.     

水库诱发地震震级预测的统计研究

于收集到的全球102例水库及已发地震的资料,应用隶属函数方法综合分析了水库基本属性、震中区岩性、库坝区基本烈度和震中区断层类型等与水库诱发地震之间的关系,从统计学角度给出了水库诱发最大地震震级的判定方法．通过回溯检验和费舍尔判别检验给出了预测震级的相对误差和正确识别率,总体预测效果较好,可为将来水库的设防和最大地震震级的判定提供统计学上的依据．