Comment on “Indicator principal component kriging” by V. Suro-Pérez and A. G. Journel

Conclusions The indicators covariance is equivalent to the bivariate distribution of the class index. The factor decomposition of the indicator covariance proposed by SPJ is an isofactorial model of the corresponding bivariate distribution. However, the choices of cumulative indicators and of principal component analysis produce unacceptable inconsistencies. In LL, a correspondence analysis of the bivariate distribution was used to produce more satisfactory empirical factors. These were used in the procedure of identification of discrete isofactorial models, with improved consistency, and the benefit of change of support models.     

Indicator kriging for unit vectors: Rock joint orientations

Indicator kriging (IK) is extended to analyze three-dimensional random unit vectors and evaluate the local probability distribution of rock joint orientations in geological formations. The pole vector representing joint orientations is regionalized and projected on a plane normal to the mean attitude of the joint family and centered at the mean. A two-dimensional cutoff system is developed to define the indicator variable, and corresponding indicator variograms and indicator kriging. The cutoff system defines probability regions similar to those of a bivariate distribution, concentric rings sliced into radial sectors. A case study made on an open pit mine proved positively the efficiency of IK and encourages its applications to localized probabilistic structural modeling for geotechnical or geohydrological analysis and oil and gas reservoir analysis.     

Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions

This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions.     

水利水电工程岩基抗剪强度参数二维分布模型构造的Copula方法

基于国内103个水利水电工程1 174组岩基抗剪强度试验数据,采用Copula函数研究岩基抗剪强度参数联合分布模型,探讨水利水电工程中岩基抗剪强度参数联合分布模型构建方法。利用最小二乘法求出岩基抗剪强度参数试验数据的相关统计参数,基于AIC准则识别出岩基抗剪强度参数边缘分布。选择4种Copula函数构造岩基抗剪强度参数二维分布模型,探讨了基于Copula函数的岩基抗剪强度参数二维分布模型的优越性。结果表明：水利水电工程岩基抗剪强度参数存在明显的统计负相关性。Copula方法能够构造具有任意边缘分布和任意相关结构的岩基抗剪强度参数联合分布模型,它为构造抗剪强度参数联合分布模型提供了一种简便的工具。已知岩基抗剪强度参数的边缘分布函数和相关系数不能唯一确定岩基抗剪强度参数的联合概率分布模型,在抗剪强度参数边缘分布函数和相关系数完全相同的前提下,不同Copula函数建立的抗剪强度参数联合概率分布模型差异显著。与常用的抗剪强度参数二维正态分布模型相比,基于Copula函数的抗剪强度参数二维分布模型具有较强的灵活性,它能更好地拟合原始观测数据。水利水电工程中惯用小值平均法确定标准值,当摩擦系数取较小值时,不同Copula函数构造的黏聚力的条件累积分布函数差异显著,这将对抗剪强度参数标准值的选取以及相应的设计方案具有明显的影响。     

两变量水文频率分布模型研究述评

水文变量多特征属性的频率分析,以及各种水文事件的遭遇及联合概率分布问题需要采用多变量概率分布模型解决。总结了当前应用最广泛的几种两变量概率分布模型,对各种模型的适用性和局限性做了详细分析,并介绍了一种新的两变量概率模型——Copula函数。现有模型大都基于变量之间的线性相关关系而建立,对于非线性、非对称的随机变量难以很好地描述;大部分模型假定各变量服从相同的边际分布或对变量间的相关性有严格的限定,从而限制了其应用。Copula函数所构造的两变量概率分布模型克服了现有模型的不足,它具有任意的边际分布,可以描述变量间非线性、非对称的相关关系。作为一种用于构造灵活的多变量联合分布的工具,Copula函数在水科学领域具有广阔的应用前景。     

Variograms of Order ω: A Tool to Validate a Bivariate Distribution Model

The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given.     

Nonparametric estimation of spatial distributions

The indicator approach, whereby the data are used through their rank order, allows a nonparametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualified, estimation of local and global spatial distributions.     

Positive definiteness is not enough

Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed.     

Comparing some approximate methods for building local confidence intervals for predicting regionalized variables

Approximate local confidence intervals can be produced by nonlinear methods designed to estimate indicator variables. The most precise of these methods, the conditional expectation, can only be used in practice in the multi-Gaussian context. Theoretically, less efficient methods have to be used in more general cases. The methods considered here are indicator kriging, probability kriging (indicator-rank co-kriging), and disjunctive kriging (indicator co-kriging). The properties of these estimators are studied in this paper in the multi-Gaussian context, for this allows a more detailed study than under more general models. Conditional distribution approximation is first studied. Exact results are given for mean squared errors and conditional bias. Then conditional quantile estimators are compared empirically. Finally, confidence intervals are compared from the points of view of bias and precision.     

Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.     

Bivariate simulation using copula and its application to probabilistic pile settlement analysis

This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best‐fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula‐based procedure can model and simulate the bivariate probability distribution of two curve‐fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best‐fit copula for modeling the dependence structure underlying two curve‐fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best‐fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve‐fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.     

A Characterization of Symmetric Isofactorial Models

The paper examines symmetric isofactorial models. A necessary and sufficient condition for a bivariate stationary random function to be isofactorial is given. Using this characterization, a procedure for checking whether an isofactorial model is appropriate is outlined. If data indicates that an isofactorial model is adequate, the procedure also provides a method for identifying the factors of the model. The paper concentrates on the case where Z(x) takes values 0, 1, 2,..., N and the general case is discussed briefly.     

A Mathematical View of Weights-of-Evidence,Conditional Independence,and Logistic Regression in Terms of Markov Random Fields

New light is shed on mathematical methods of potential modeling from the point of view of Markov random fields. In particular, weights-of-evidence and logistic regression models are discussed in terms of graphical models possessing Markov properties, where the notion of conditional independence is essential, and will be related to log-linear models. While weights-of-evidence with respect to indicator predictor variables and logistic regression with unrestricted predictor variables model conditional probabilities of an indicator random target variable, the subject of log-linear models is the joint probability of random variables. The relationship to log-linear models leads to a likelihood ratio test of conditional independence, rendering an omnibus test of conditional independence restricted by a normality assumption obsolete. Moreover, it reveals a hierarchy of methods comprising weights-of-evidence, logistic regression without interaction terms, and logistic regression including interaction terms, where each former method is a special case of the consecutive latter method. The assumptions of conditional independence of all predictor variables given the target variable lead to logistic regression without interaction terms. Violations of conditional independence are compensated exactly by corresponding interaction terms, no cumbersome approximate corrections are needed. Thus, including interaction terms into logistic regression models is an appropriate means to account for lacking conditional independence. Logistic regression exempts from the burden to worry about lack of conditional independence. Eventually, the relationship to log-linear models renders logistic regression with indicator predictor variables optimum for discrete predictor variables. Weights-of-evidence applies for indicator predictor variables only, logistic regression applies without restrictions of the type of predictor variables and approximates the proper distribution in the general case.     

Manifestation of remote sensing data and GIS on landslide hazard analysis using spatial-based statistical models

This paper presents landslide hazard analysis at Cameron area, Malaysia, using a geographic information system (GIS) and remote sensing data. Landslide locations were identified from interpretation of aerial photographs and field surveys. Topographical and geological data and satellite images were collected, processed, and constructed into a spatial database using GIS and image processing. The factors chosen that influence landslide occurrence are topographic slope, topographic aspect, topographic curvature, and distance to rivers, all from the topographic database; lithology and distance to faults were taken from the geologic database; land cover from TM satellite image; the vegetation index value was taken from Landsat images; and precipitation distribution from meteorological data. Landslide hazard area was analyzed and mapped using the landslide occurrence factors by frequency ratio and bivariate logistic regression models. The results of the analysis were verified using the landslide location data and compared with the probabilistic models. The validation results showed that the frequency ratio model (accuracy is 89.25%) is better in prediction of landslide than bivariate logistic regression (accuracy is 85.73%) model.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

Susceptibility mapping of gully erosion using GIS-based statistical bivariate models: a case study from Ali Al-Gharbi District,Maysan Governorate,southern Iraq

This work aims to evaluate the predictive capability of three bivariate statistical models, namely information value, frequency ratio, and evidential belief functions, in gully erosion susceptibility mapping in northeastern Maysan Governorate (Ali Al-Gharbi District) in southern Iraq. The gully inventory map, consisting of 21 gullies of different sizes, was prepared based on the interpretation of remotely sensed data supported by field survey. The gully inventory data (polygon format) were randomly partitioned into two sets: 14 gullies for build and training the bivariate model, and the remaining 7 gullies for validating purposes. Twelve gully influential factors were selected based on data availability and the literature review. The selected factors were related to lithology, geomorphology, soil, land cover, and topography (primary and secondary) settings. Analysis of factor importance using information gain ratio proved that out of 12 gully influential factors, eight were of more importance in developing gullies (the average merit was greater than zero). The most important factors and the training gully inventory map were used to generate three gully erosion susceptibility maps based on the three bivariate models used. For validation, the area under the operating characteristics curves for both success and prediction rates was used. The results indicated that the highest prediction rate of 82.9% was achieved using the information value technique. All the bivariate models had prediction rates greater than 80%, and thus they were regarded as very good estimators. The final conclusion was that the bivariate models offer advanced techniques for mapping gully erosion susceptibility.     

Indicator Variogram Models: Do We Have Much Choice?

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.     

A non-parametric bivariate entropy estimator for spatial processes

In this paper, entropy is presented as an alternative measure to characterize the bivariate distribution of a stationary spatial process. This non-parametric estimator attempts to quantify the concept of spatial ordering, and it provides a measure of how Gaussian the experimental bivariate distribution is. The concept of entropy is explained and the classical definition presented, along with some important results. In particular, the reader is reminded that, for a known mean and covariance, the bivariate Gaussian distribution maximizes entropy. A relative entropy estimator is introduced in order to measure departure of an experimental bivariate distribution from the bivariate Gaussian. Two case studies are presented as examples.     

A GIS-based approach for gully erosion susceptibility modelling using bivariate statistics methods in the Ourika watershed,Morocco

Gully erosion is an important environmental issue with severe impacts. This study aimed to characterize gully erosion susceptibility and assess the capability of information value (InfVal) and frequency ratio (FR) models for its spatial prediction in Ourika watershed of the High Atlas region of Morocco. These two bivariate statistical methods have been used for gully erosion susceptibility mapping by comparing each data layer of causative factor to the existing gully distribution. Weights to the gully causative factors are assigned based on gully density. Gullies have been mapped through field surveys and Google earth high-resolution images. Lithofacies, land use, slope gradient, length-slope, aspect, stream power index, topographical wetness index and plan curvature were considered predisposing factors to gullying. The digitized gullies were randomly split into two parts. Sixty-five percent (65%) of the mapped gullies were randomly selected as training set to build gully susceptibility models, while the remaining 35% cases were used as validation set for the models’ validation. The results showed that barren and sparse vegetation lands and slope gradient above 50% were very susceptible to gully erosion. The ROC curve was used for testing the accuracy of the mentioned models. The analysis confirms that the FR model (AUC 80.61%) shows a better accuracy than InfVal model (AUC 52.07%). The performance of the gully erosion susceptibility map constructed by FR model is greater than that of the map produced by InfVal model. The findings proved that GIS-based bivariate statistical methods such as frequency ratio model could be successfully applied in gully susceptibility mapping in Morocco mountainous regions and in other similar environments. The produced susceptibility map represents a useful tool for sustainable planning, conservation and protection of land from gully processes.     

年最大洪水两变量联合分布研究

采用Von Mises分布拟合年最大洪水发生时间的概率分布,采用皮尔逊Ⅲ型分布拟合年最大洪水量级的概率分布,选用能够较好反映年最大洪水发生时间和量级之间相关结构的Gumbel Archimedean Copula函数,建立两变量联合分布,并定义和分析条件频率、联合频率和两变量重现期.实例分析表明年最大洪水的两变量分布拟合较好,可挖掘更多信息,为洪水设计分析提供了一条新的途径.