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.     

A data assimilation approach for groundwater parameter estimation under Bayesian maximum entropy framework

Spatial heterogeneity in groundwater system introduces significant challenges in groundwater modeling and parameter calibration. In order to mitigate the modeling uncertainty, data assiilation...     

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.     

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.     

Bayesian maximum entropy and data fusion for processing qualitative data: theory and application for crowdsourced cropland occurrences in Ethiopia

Categorical data play an important role in a wide variety of spatial applications, while modeling and predicting this type of statistical variable has proved to be complex in many cases. Among other possible approaches, the Bayesian maximum entropy methodology has been developed and advocated for this goal and has been successfully applied in various spatial prediction problems. This approach aims at building a multivariate probability table from bivariate probability functions used as constraints that need to be fulfilled, in order to compute a posterior conditional distribution that accounts for hard or soft information sources. In this paper, our goal is to generalize further the theoretical results in order to account for a much wider type of information source, such as probability inequalities. We first show how the maximum entropy principle can be implemented efficiently using a linear iterative approximation based on a minimum norm criterion, where the minimum norm solution is obtained at each step from simple matrix operations that converges to the requested maximum entropy solution. Based on this result, we show then how the maximum entropy problem can be related to the more general minimum divergence problem, which might involve equality and inequality constraints and which can be solved based on iterated minimum norm solutions. This allows us to account for a much larger panel of information types, where more qualitative information, such as probability inequalities can be used. When combined with a Bayesian data fusion approach, this approach deals with the case of potentially conflicting information that is available. Although the theoretical results presented in this paper can be applied to any study (spatial or non-spatial) involving categorical data in general, the results are illustrated in a spatial context where the goal is to predict at best the occurrence of cultivated land in Ethiopia based on crowdsourced information. The results emphasize the benefit of the methodology, which integrates conflicting information and provides a spatially exhaustive map of these occurrence classes over the whole country.     

中国的地震活动性

基于华北地震统计区近600年地震资料,采用最大熵谱分析方法,定量解析了该区及各主要地震带的地震活动时间序列特征,为多个优势周期叠加的复杂强弱交替。以最优分割法划分了华北地震区及各主要地震带的地震幕,分析了其特点。结合当前华北地震区地震平静现象等,认为目前该区地震活动趋势正处于一个复杂的十字路口,华北地区随时可能进入一个新的地震活动幕（将持续10年左右）,然后进入新一轮地震平静期（将持续几十年）;也有另一种可能,华北地区现在实际已经进入了新的地震平静期。     

A new assessment approach for urban ecosystem health basing on maximum information entropy method

Information entropy is introduced to describe the interactions between diverse agents in urban ecosystems. Basing on maximum information entropy method, a holistic structural parameter and its dynamic equation are derived to reflect urban ecosystem health (UEH). In this way, a new UEH assessment model has been proposed. We then apply the model to assess the UEH of Beijing, Dalian, Shanghai, Wuhan, Xiamen and Guangzhou in China. It is shown that the holistic structural parameter, the radar chart, and the associated correlations from the model can reveal the health features of different cities. According to the calculated ranges of the holistic structural parameter, a new UEH assessment grade standard is suggested and applied to the UEH assessment of some typical cities in China. It is demonstrated that the new model and the new assessment grade standard are precise and readily operational, which can be widely used in other urban ecosystems.     

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.     

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).     

Two-dimensional equilibrium morphological modelling of a tidal inlet: an entropy based approach

The management of tidal inlets requires the accurate prediction of equilibrium morphologies. In areas where the flow from rivers is highly regulated, it is important to give decision makers the ability to determine optimal flow management schemes, in order to allow tidal inlets to function as naturally as possible, and minimise the risk of inlet closure. The River Murray Mouth in South Australia is one such problem area. Drought and the retention of water for irrigation and urban water consumption have limited the amount of water entering the estuary. As a result, sediment from the coastal environment is being deposited in the mouth of the estuary, reducing the effect of further coastal interactions. Currently, situations such as this are modelled using traditional process-based methods, where wave, current, sediment transport and sediment balance modules are linked together in a time-stepping process. The modules are reapplied and assessed until a stable morphology is formed. In this paper, new options for modelling equilibrium morphologies of tidal inlets are detailed, which alleviate some of the shortfalls of traditional process-based models, such as the amplification of small errors and reliance on initial conditions. The modelling problem is approached in this paper from a different angle and involves the use of entropy based objective functions, which are optimised in order to find equilibrium morphologies. In this way, characteristics of a system at equilibrium can be recognised and a stable system predicted without having to step through time. This paper also details the use of self-organisation based modelling methods, another non-traditional model application, where local laws and feedback result in the formation of a global stable equilibrium morphology. These methods represent a different approach to traditional models, without some of the characteristics that may add to their limitations. Responsible Editor: Alejandro Souza     

Simulation of urban expansion based on cellular automata and maximum entropy model

Urban expansion is a hot topic in land use/land cover change(LUCC) researches. In this paper, maximum entropy model and cellular automata(CA) model are coupled into a new CA model(Maxent-CA) for urban expansion. This model can help to obtain transition rules from single-period dataset. Moreover, it can be constructed and calibrated easily with several steps.Firstly, Maxent-CA model was built by using remote sensing data of China in 2000(basic data) and spatial variables(such as population density and Euclidean distance to cities). Secondly, the proposed model was calibrated by analyzing training samples,neighborhood structure and spatial scale. Finally, this model was verified by comparing logistic regression CA model and their simulation results. Experiments showed that suitable sampling ratio(sampling ratio equals the proportion of urban land in the whole region) and von Neumann neighborhood structure will help to yield better results. Spatial structure of simulation results becomes simple as spatial resolution decreases. Besides, simulation accuracy is significantly affected by spatial resolution.Compared to simulation results of logistic regression CA model, Maxent-CA model can avoid clusters phenomenon and obtain better results matching actual situation. It is found that the proposed model performs well in simulating urban expansion of China. It will be helpful for simulating even larger study area in the background of global environment changes.     

黄河尾闾段一维耦合水沙数学模型研究及其应用

小浪底水库于1999年运用以后,该河道经历了长时间持续冲刷过程.为掌握小浪底水库运用后黄河尾闾段洪水演进特点及河床冲淤规律,采用一维水沙数学模型研究是一条重要的途径.本研究首先采用浑水控制方程,建立了一维耦合水沙数学模型,并利用2003年利津-西河口段汛期实测水沙及汛前断面地形资料对该模型进行率定,计算的流量、水位及含沙量等过程与实测值吻合较好;然后采用2015年利津—汊3段汛期实测资料对该模型进行验证,结果显示水位与冲淤量计算值与实测值较为符合;最后基于2015年实测洪水过程,计算了若干组不同断面间距下的洪水演进及冲淤过程,分析了不同断面间距对沿程水位及河段冲淤量等计算结果的影响,结果表明:采用不同断面间距对水位计算结果影响较小,而对冲淤量计算结果会产生一定影响;在河段水沙及冲淤特性复杂的情况下,采用一维数学水沙模型计算时应考虑断面间距的选择.     

A comparison of windows and maximum entropy methods for MT transfer function estimation

The ability of the Maximum Entropy Spectral Estimate (MEM) to handle magnetotelluric signals was investigated. To this end simulations of naturally occurring signals and real data were spectral analyzed both by a standard FFT technique and by MEM. The method developed for the simulation of the time series, provides accurate amplitudes and phase, and turns out to be a good test procedure for spectral analysis methods, as they apply to MT signal processing. The final estimation of the apparent resistivity was better achieved by the FFT technique. Oscillations were observed in the estimation of the apparent resistivities by MEM.     

Cross-correlation of random fields: mathematical approach and applications

Random field cross‐correlation is a new promising technique for seismic exploration, as it bypasses shortcomings of usual active methods. Seismic noise can be considered as a reproducible, stationary in time, natural source. In the present paper we show why and how cross‐correlation of noise records can be used for geophysical imaging. We discuss the theoretical conditions required to observe the emergence of the Green's functions between two receivers from the cross‐correlation of noise records. We present examples of seismic imaging using reconstructed surface waves from regional to local scales. We also show an application using body waves extracted from records of a small‐scale network. We then introduce a new way to achieve surface wave seismic experiments using cross‐correlation of unsynchronized sources. At a laboratory scale, we demonstrate that body wave extraction may also be used to image buried scatterers. These works show the feasibility of passive imaging from noise cross‐correlation at different scales.     

Spatio-temporal modelling of hydro-meteorological derived risk using a Bayesian approach: a case study in Venezuela

Extreme environmental events have considerable impacts on society. Preparation to mitigate or forecast accurately these events is a growing concern for governments. In this regard, policy and decision makers require accurate tools for risk estimation in order to take informed decisions. This work proposes a Bayesian framework for a unified treatment and statistical modeling of the main components of risk: hazard, vulnerability and exposure. Risk is defined as the expected economic loss or population affected as a consequence of a hazard event. The vulnerability is interpreted as the loss experienced by an exposed population due to hazard events. The framework combines data of different spatial and temporal supports. It produces a sequence of temporal risk maps for the domain of interest including a measure of uncertainty for the hazard and vulnerability. In particular, the considered hazard (rainfall) is interpolated from point-based measured rainfall data using a hierarchical spatio-temporal Kriging model, whose parameters are estimated using the Bayesian paradigm. Vulnerability is modeled using zero-inflated distributions with parameters dependent on climatic variables at local and large scales. Exposure is defined as the total population settled in the spatial domain and is interpolated using census data. The proposed methodology was applied to the Vargas state of Venezuela to map the spatio-temporal risk for the period 1970–2006. The framework highlights both high and low risk areas given extreme rainfall events.     

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.     

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.     

New advances of seismic tomography and its applications to subduction zones and earthquake fault zones: A review

Abstract There have been significant advances in the theory and applications of seismic tomography in the last decade. These include the refinements in the model parameterization, 3-D ray tracing, inversion algorithm, resolution and error analyses, joint use of local, regional and teleseismic data, and the addition of converted and reflected waves in the tomographic inversion. Applications of the new generation tomographic methods to subduction zones have resulted in unprecedentedly clear images of the subducting oceanic lithosphere and magma chambers in the mantle wedge beneath active arc volcanoes, indicating that geodynamic systems associated with the arc magmatism and back-arc spreading are related to deep processes, such as the convective circulation in the mantle wedge and deep dehydration reactions in the subducting slab. High-resolution tomographic imagings of earthquake fault zones in Japan and California show that rupture nucleation and earthquake generating processes are closely related to the heterogeneities of crustal materials and inelastic processes in the fault zones, such as the migration of fluids. Evidence also shows that arc magmatism and slab dehydration may also contribute to the generation of large crustal earthquakes in subduction regions.