Reproduction of secondary data in projection pursuit transformation

A challenge when working with multivariate data in a geostatistical context is that the data are rarely Gaussian. Multivariate distributions may include nonlinear features, clustering, long tails, functional boundaries, spikes, and heteroskedasticity. Multivariate transformations account for such features so that they are reproduced in geostatistical models. Projection pursuit as developed for high dimensional data exploration can also be used to transform a multivariate distribution into a multivariate Gaussian distribution with an identity covariance matrix. Its application within a geostatistical modeling context is called the projection pursuit multivariate transform (PPMT). An approach to incorporate exhaustive secondary variables in the PPMT is introduced. With this approach the PPMT can incorporate any number of secondary variables with any number of primary variables. A necessary alteration to the approach to make this numerically practical was the implementation of a continuous probability estimator that relies on Bernstein polynomials for the transformation that takes place in the projections. Stopping criteria were updated to incorporate a bootstrap t test that compares data sampled from a multivariate Gaussian distribution with the data undergoing transformation.     

Exploring a valid model for the variogram of an isotropic spatial process

The variogram is one of the most important tools in the assessment of spatial variability and a crucial parameter for kriging. It is widely known that an estimator for the variogram cannot be used as its representator in some contexts because of its lack of conditional semi negative definiteness. Consequently, once the variogram is estimated, a valid family must be chosen to fit an appropriate model. Under isotropy, this selection is carried out by eye from the observation of the variogram estimated curve. In this paper, a statistical methodology is proposed to explore a valid model for the variogram. The statistic for this approach is based on quadratic forms depending on smoothed random variables which gather the underlying spatial variation. The distribution of the test statistic is approximated by a shifted chi-square distribution. A simulation study is also carried out to check the power and size of the test. Reference bands, as a complementary graphical tool, are calculated. An example from the literature is used to illustrate the methodologies presented.     

Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.     

Impact of log-transmissivity variogram structure on groundwater flow and transport predictions

We analyze the impact of the choice of the variogram model adopted to characterize the spatial variability of natural log-transmissivity on the evaluation of leading (statistical) moments of hydraulic heads and contaminant travel times and trajectories within mildly (randomly) heterogeneous two-dimensional porous systems. The study is motivated by the fact that in several practical situations the differences between various variogram types and a typical noisy sample variogram are small enough to suggest that one would often have a hard time deciding which of the tested models provides the best fit. Likewise, choosing amongst a set of seemingly likely variogram models estimated by means of geostatistical inverse models of flow equations can be difficult due to lack of sensitivity of available model discrimination criteria. We tackle the problem within the framework of numerical Monte Carlo simulations for mean uniform and radial flow scenarios. The effect of three commonly used isotropic variogram models, i.e., Gaussian, Exponential and Spherical, is analyzed. Our analysis clearly shows that (ensemble) mean values of the quantities of interest are not considerably influenced by the variogram shape for the range of parameters examined. Contrariwise, prediction variances of the quantities examined are significantly affected by the choice of the variogram model of the log-transmissivity field. The spatial distribution of the largest/lowest values of the relative differences observed amongst the tested models depends on a combination of variogram shape and parameters and relative distance from internal sources and the outer domain boundary. Our findings suggest the need of developing robust techniques to discriminate amongst a set of seemingly equally likely alternative variogram models in order to provide reliable uncertainty estimates of state variables.     

Bayesian improver of a distribution

 An estimate of a distribution obtained from a sample by any method of classical statistics may be erroneous when the sample is not representative of the population. A subjective distribution elicited from an expert may be miscalibrated when information is scanty and experience limited. The Bayesian Improver of a Distribution (BID) exploits a coherence principle and improves, in the ex ante sense, an initial estimate of a continuous distribution by using (i) the known distribution of a related variate and (ii) information about the dependence structure between the two variates. The theory of BID is developed into an applied (ABID) procedure. The ABID estimator is applicable to any continuous, monotone likelihood ratio dependent variates with arbitrary, strictly increasing marginal distributions, parametric or nonparametric; it is analytic in form and easy to implement via statistical or judgmental methods; it converges to the true distribution, provided the initial estimator does, as the sample size n→∞; it outperforms the initial estimator in the expected Kolmogorov–Smirnov distance for all n; and it offers the greatest gains when n is small – precisely when improved estimates are needed most.     

Maximum-likelihood estimation of unsaturated hydraulic parameters

Maximum-likelihood estimators properly represent measurement error, thus provide a statistically sound basis for evaluating the adequacy of a model fit and for finding the multivariate parameter confidence region. We demonstrate the advantages of using maximum-likelihood estimators rather than simple least-squares estimators for the problem of finding unsaturated hydraulic parameters. Inversion of outflow data given independent retention data can be treated by an extension to a Bayesian estimator. As an example, we apply the methodology to retention and transient unsaturated outflow observations, both obtained on the same medium sand sample. We found the van Genuchten expression to be adequate for the retention data, as the best fit was within measurement error. The Cramer–Rao confidence bound described the true parameter uncertainty approximately. The Mualem–van Genuchten expression was, however, inadequate for our outflow observations, suggesting that the parameters (, n) may not always be equivalent in describing both retention and unsaturated conductivity.     

Sequential kriging and cokriging: Two powerful geostatistical approaches

A sequential linear estimator is developed in this study to progressively incorporate new or different spatial data sets into the estimation. It begins with a classical linear estimator (i.e., kriging or cokriging) to estimate means conditioned to a given observed data set. When an additional data set becomes available, the sequential estimator improves the previous estimate by using linearly weighted sums of differences between the new data set and previous estimates at sample locations. Like the classical linear estimator, the weights used in the sequential linear estimator are derived from a system of equations that contains covariances and cross-covariances between sample locations and the location where the estimate is to be made. However, the covariances and cross-covariances are conditioned upon the previous data sets. The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. It also provides a new way to incorporate additional information into a previous estimation.     

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.     

Estimation of extreme Component-wise Excess design realization: a hydrological application

The classic univariate risk measure in environmental sciences is the Return Period (RP). The RP is traditionally defined as “the average time elapsing between two successive realizations of a prescribed event”. The notion of design quantile related with RP is also of great importance. The design quantile represents the “value of the variable(s) characterizing the event associated with a given RP”. Since an individual risk may strongly be affected by the degree of dependence amongst all risks, the need for the provision of multivariate design quantiles has gained ground. In contrast to the univariate case, the design quantile definition in the multivariate setting presents certain difficulties. In particular, Salvadori, G., De Michele, C. and Durante F. define in the paper called “On the return period and design in a multivariate framework” (Hydrol Earth Syst Sci 15:3293–3305, 2011) the design realization as the vector that maximizes a weight function given that the risk vector belongs to a given critical layer of its joint multivariate distribution function. In this paper, we provide the explicit expression of the aforementioned multivariate risk measure in the Archimedean copula setting. Furthermore, this measure is estimated by using Extreme Value Theory techniques and the asymptotic normality of the proposed estimator is studied. The performance of our estimator is evaluated on simulated data. We conclude with an application on a real hydrological data-set.     

Inference for the truncated exponential distribution

The constructed estimator is introduced for the right truncation point of the truncated exponential distribution. The new estimator is most efficient in important ranges of truncation points for finite sample sizes. The introduced inverse mean squared error clearly indicates the good behaviour of the new estimator. The estimation of the scaling parameter is considered in all discussions and computations. The methods and models of the extreme value theory are not appropriate to estimate the truncation point because they work only in the case of very large sample sizes. Furthermore, a procedure for a first goodness-of-fit test is introduced. All this has been researched by extensive Monte Carlo simulations for different truncation points and sample sizes. Finally, the new inference methods are applied at the end for the random distribution of wildfire sizes and earthquake magnitudes.     

Expected probability weighted moment estimator for censored flood data

Two well-known methods for estimating statistical distributions in hydrology are the Method of Moments (MOMs) and the method of probability weighted moments (PWM). This paper is concerned with the case where a part of the sample is censored. One situation where this might occur is when systematic data (e.g. from gauges) are combined with historical data, since the latter are often only reported if they exceed a high threshold. For this problem, three previously derived estimators are the “B17B” estimator, which is a direct modification of MOM to allow for partial censoring; the “partial PWM estimator”, which similarly modifies PWM; and the “expected moments algorithm” estimator, which improves on B17B by replacing a sample adjustment of the censored-data moments with a population adjustment. The present paper proposes a similar modification to the PWM estimator, resulting in the “expected probability weighted moments (EPWM)” estimator. Simulation comparisons of these four estimators and also the maximum likelihood estimator show that the EPWM method is at least competitive with the other four and in many cases the best of the five estimators.     

Compositional Bayesian indicator estimation

Indicator kriging is widely used for mapping spatial binary variables and for estimating the global and local spatial distributions of variables in geosciences. For continuous random variables, indicator kriging gives an estimate of the cumulative distribution function, for a given threshold, which is then the estimate of a probability. Like any other kriging procedure, indicator kriging provides an estimation variance that, although not often used in applications, should be taken into account as it assesses the uncertainty of the estimate. An alternative approach to indicator estimation is proposed in this paper. In this alternative approach the complete probability density function of the indicator estimate is evaluated. The procedure is described in a Bayesian framework, using a multivariate Gaussian likelihood and an a priori distribution which are both combined according to Bayes theorem in order to obtain a posterior distribution for the indicator estimate. From this posterior distribution, point estimates, interval estimates and uncertainty measures can be obtained. Among the point estimates, the median of the posterior distribution is the maximum entropy estimate because there is a fifty-fifty chance of the unknown value of the estimate being larger or smaller than the median; that is, there is maximum uncertainty in the choice between two alternatives. Thus in some sense, the latter is an indicator estimator, alternative to the kriging estimator, that includes its own uncertainty. On the other hand, the mode of the posterior distribution estimator, assuming a uniform prior, is coincidental with the simple kriging estimator. Additionally, because the indicator estimate can be considered as a two-part composition which domain of definition is the simplex, the method is extended to compositional Bayesian indicator estimation. Bayesian indicator estimation and compositional Bayesian indicator estimation are illustrated with an environmental case study in which the probability of the content of a geochemical element in soil being over a particular threshold is of interest. The computer codes and its user guides are public domain and freely available.     

Assessment of regional air pollution distribution by point cumulative semivariogram method at Erzurum urban center, TURKEY

In the assessment of air quality, regional distribution and dispersion with distance are important, together with the variations of pollutants in time. On this occasion, the point cumulative semi-variogram (PCSV) method is used in order to find simply regional distribution of pollutants of Erzurum urban centre. This method is based simply on the summation of square differences in air pollutant concentrations between different sites. Monthly regional variation maps of Erzurum are constructed by finding radius of influence (for SO₂, from 1000 m to 3500 m and, for TSP, 1000–2000 m) and PCSV scattering diagram data at different levels by using monthly average sulphur dioxide (SO₂) and total suspended particulate (TSP) matter concentrations in 2001–2002 winter season. Consequently, the air pollution distribution of Erzurum is assessed.     

Mixture model based multivariate statistical analysis of multiply censored environmental data

Environmental data are commonly constrained by a detection limit (DL) because of the restriction of experimental apparatus. In particular due to the changes of experimental units or assay methods, the observed data are often cut off by more than one DL. Measurements below the DLs are typically replaced by an arbitrary value such as zeros, half of DLs, or DLs for convenience of analysis. However, this method is widely considered unreliable and prone to bias. In contrast, maximum likelihood estimation (MLE) method for censored data has been developed for better performance and statistical justification. However, the existing MLE methods seldom address the multivariate context of censored environmental data especially for water quality. This paper proposes using a mixture model to flexibly approximate the underlying distribution of the observed data due to its good approximation capability and generation mechanism. In particular, Gaussian mixture model (GMM) is mainly focused in this study. To cope with the censored data with multiple DLs, an expectation–maximization (EM) algorithm in a multivariate setting is developed. The proposed statistical analysis approach is verified from both the simulated data and real water quality data.     

基于地质统计方法与DEM的地震灾情空间插值研究

历次破坏性地震的震害调查和强震观测资料显示, 地形地貌对地震灾害有着显著的影响. 地震发生后, 为了能够及时、 准确地为地震救灾指挥提供灾情分布信息, 该文借鉴了地质统计学方法, 利用灾情速报人员上报的地震现场离散点灾情短信对灾区进行灾情空间模拟的同时, 将数学高程模型(DEM)中所包含的高程、 坡度等地形地貌信息作为影响因素引入协克里金(Co-Kriging)插值; 并以汶川M_S8.0地震灾情短信数据为例, 分别对确定性插值、 地质统计学插值结果与有无考虑坡度因素的地质统计学插值结果进行了交叉检验. 结果表明, 考虑坡度影响因素的协克里金插值在合适的模型和参数下取得了最优的灾情模拟效果. 该方法为地震应急期间进行较高精度的灾情模拟提供了一种新的可行思路.      

Fast modeling of cross-covariances in the LMC: A tool for data integration

Although modeling of cross-covariances by fitting the linear model of coregionalization (LMC) is considered a cumbersome task, cross-covariances are the key for integration of data for multiple attributes in environmental hydrology, aquifer and reservoir characterizations using multivariate geostatistics. This paper proposes a novel method of modeling cross-covariances in the linear model of coregionalization (LMC). The classic minimum/maximum autocorrelation factors (MAF) method is analyzed and found to be a good tool to discriminate the elementary nested structures of directional sample covariance matrices. Thus, separate modeling of the scalar sample covariance for each MAF factor may allow to obtain the complete LMC model for the original attributes after a back rotation of the diagonal model covariance matrix of directional factors. However, such a back rotation is not computable following the classic MAF formulation. This paper introduces an ambi-rotational minimum/maximum autocorrelation factors (AMAF) method that allows a back and forth double rotation of the directional diagonal model covariance matrix for factors. This approach provides a device for modeling of the full matrix of directional covariance and cross-covariance for the original attributes in the LMC without recurring to iterations. In this way, the use of multivariate geostatistics for data integration is allowed avoiding collocated approaches or rotation and modeling of data factor scores. The method is illustrated with an example for covariances for three attributes.     

A multivariate non‐parametric model for synthetic generation of daily streamflow

A p‐order multivariate kernel density model based on kernel density theory has been developed for synthetic generation of multivariate variables. It belongs to a kind of data‐driven approach and is able to avoid prior assumptions as to the form of probability distribution (normal or Pearson III) and the form of dependence (linear or non‐linear). The p‐order multivariate kernel density model is a non‐parametric method for synthesis of streamflow. The model is more flexible than conventional parametric models used in stochastic hydrology. The effectiveness and satisfactoriness of this model are illustrated through its application to the simultaneous synthetic generation of daily streamflow from Pingshan station and Yibin‐Pingshan region (Yi‐Ping region) of the Jinsha River in China. Copyright © 2007 John Wiley & Sons, Ltd.     

Semi-variograms for modelling the spatial pattern of landform and soil properties

Many geomorphic properties can be treated as spatially-dependent random variables. Some are second-order stationary, others appear to vary without bound. In these circumstances their variation is best described by the semi-variogram. In most instances the semi-variogram can be modelled by a simple mathematical function, which itself is bounded for a stationary variable and unbounded otherwise. The function must be conditional negative semi-definite to be permissible. More complex variation can be represented by combining two or more permissible models. Sample semi-variograms of several landform and soil properties illustrate the common types of semi-variogram. Their form and parameters are interpreted in physical terms.