共查询到20条相似文献，搜索用时 24 毫秒
1.
A Mathematical View of Weights-of-Evidence,Conditional Independence,and Logistic Regression in Terms of Markov Random Fields 总被引：2，自引：1，他引：1
Helmut Schaeben 《Mathematical Geosciences》2014,46(6):691-709
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. 相似文献
2.
3.
《Applied Geochemistry》2005,20(1):157-168
In monitoring a minor geochemical element in groundwater or soils, a background population of values below the instrumental detection limit is frequently present. When those values are found in the monitoring process, they are assigned to the detection limit which, in some cases, generates a probability mass in the probability density function of the variable at that value (the minimum value that can be detected). Such background values could distort both the estimation of the variable at nonsampled locations and the inference of the spatial structure of variability of the variable. Two important problems are the delineation of areas where the variable is above the detection limit and the estimation of the magnitude of the variables inside those areas. The importance of these issues in geochemical prospecting or in environmental sciences, in general related with contamination and environmental monitoring, is obvious. In this paper the authors describe the two-step procedure of indicator kriging and ordinary kriging and compare it with empirical maximum likelihood kriging. The first approach consists of using a binary indicator variable for estimating the probability of a location being above the detection limit, plus ordinary kriging conditional to the location being above the detection limit. An estimation variance, however, is not available for that estimator. Empirical maximum likelihood kriging, which was designed to deal with skew distributions, can also deal with an atom at the origin of the distribution. The method uses a Bayesian approach to kriging and gives intermittency in the form of a probability map, its estimates providing a realistic assessment of their estimation variance. The pros and cons of each method are discussed and illustrated using a large dataset of As concentration in groundwater. The results of the two methods are compared by cross-validation. 相似文献
4.
Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities. 相似文献
5.
Empirical Maximum Likelihood Kriging: The General Case 总被引：4，自引：0，他引：4
Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution. 相似文献
6.
Roland Froidevaux 《Mathematical Geology》1984,16(4):327-350
One of the tasks routinely carried out by geostatisticians is the evaluation of global mining reserves corresponding to a given cutoff grade and size of selective mining units. A long with these recovery figures, the geostatistician generally provides an assessment of the global estimation variance, which represents the precision of the overall average grade estimate, when no cutoff is applied. Such a global estimation variance is of limited interest for evaluating mining projects; what is required is the reliability of the estimate of recovered reserves or, in other words, the conditional estimation variance. Unfortunately, classical linear geostatistical methods fail to provide an easy way to estimate this variance. Through the use of simulated deposits (representing various types of regionalization)the present paper reviews and discusses the effects of changes in cutoff grade and selective mining unit size on the conditional estimation variance. It is shown that, when the cutoff grade is applied to a pointsupport (sample-size)distribution, the conditional estimation variance appears to be readily accessible by classical formulas, once the conditional semivariogram is known. However, the evaluation of the conditional estimation variance seems to be less straightforward for the general case when a cutoff is applied to the average grade distribution of selective mining units. Empirical approximation formulas for the conditional estimation variance are tentatively proposed, and their performance in the case of the simulated deposits is shown. The limitations of these approximations are discussed, and possible ways of formalizing the problem suggested. 相似文献
7.
受工程勘察成本及试验场地限制,可获得的试验数据通常有限,基于有限的试验数据难以准确估计岩土参数统计特征和边坡可靠度。贝叶斯方法可以融合有限的场地信息降低对岩土参数不确定性的估计进而提高边坡可靠度水平。但是,目前的贝叶斯更新研究大多假定参数先验概率分布为正态、对数正态和均匀分布,似然函数为多维正态分布,这种做法的合理性有待进一步验证。总结了岩土工程贝叶斯分析常用的参数先验概率分布及似然函数模型,以一个不排水黏土边坡为例,采用自适应贝叶斯更新方法系统探讨了参数先验概率分布和似然函数对空间变异边坡参数后验概率分布推断及可靠度更新的影响。计算结果表明：参数先验概率分布对空间变异边坡参数后验概率分布推断及可靠度更新均有一定的影响,选用对数正态和极值I型分布作为先验概率分布推断的参数后验概率分布离散性较小。选用Beta分布和极值I型分布获得的边坡可靠度计算结果分别偏于保守和危险,选用对数正态分布获得的边坡可靠度计算结果居中。相比之下,似然函数的影响更加显著。与其他类型似然函数相比,由多维联合正态分布构建的似然函数可在降低对岩土参数不确定性估计的同时,获得与场地信息更为吻合的计算结果。另外,构建似然函数时不同位置处测量误差之间的自相关性对边坡后验失效概率也具有一定的影响。 相似文献
8.
Although first-order reliability method is a common procedure for estimating failure probability, the formulas derived for bimodal bounds of system failure probability have not been widely used as expected in present reliability analyses. The reluctance for applying these formulas in practice may be partly due to the impression that the procedures to implement the system reliability theory are tedious. Among the methods for system reliability analysis, the approach suggested in Ditlevsen 1979 is considered here because it is a natural extension of the first-order reliability method commonly used for failure probability estimation corresponding to a single failure mode, and it can often provide reasonably narrow failure probability bounds. To facilitate wider practical application, this paper provides a short program code in the ubiquitous Excel spreadsheet platform for efficiently calculating the bounds for system failure probability. The procedure is illustrated for a semi-gravity retaining wall with two failure modes, a soil slope with two and eight failure modes, and a loaded beam with three failure modes. In addition, simple equations are provided to relate the correlated but unrotated equivalent standard normals of the Low and Tang 2007 FORM procedure with the uncorrelated but rotated equivalent standard normals of the classical FORM procedure. Also demonstrated are the need for investigating different permutations of failure modes in order to get the narrowest bounds for system failure probability, and the use of SORM reliability index for system reliability bounds in a case where the curvature of the limit state surface cannot be neglected. 相似文献
9.
In a probabilistic analysis of rock slope stability, the Monte Carlo simulation technique has been widely used to evaluate
the probability of slope failure. While the Monte Carlo simulation technique has many advantages, the technique requires complete
information of the random variables in stability analysis; however, in practice, it is difficult to obtain complete information
from a field investigation. The information on random variables is usually limited due to the restraints of sampling numbers.
This is why approximation methods have been proposed for reliability analyses. Approximation methods, such as the first-order
second-moment method and the point estimate method, require only the mean and standard deviation of the random variable; therefore,
it is easy to utilize when the information is limited. Usually, a single closed form of the formula for the evaluation of
the factor of safety is needed for an approximation method. However, the commonly used stability analysis method of wedge
failure is complicated and cumbersome and does not provide a simple equation for the evaluation of the factor of safety. Consequently,
the approximation method is not appropriate for wedge failure. In order to overcome this limitation, a simple equation, which
is obtained from the maximum likelihood estimation method for wedge failure, is utilized to calculate the probability of failure.
A simple equation for the direct estimation of the safety factors for wedge failure has been empirically derived from failed
and stable cases of slope, using the maximum likelihood estimation method. The developed technique has been applied to a practical
example, and the results from the developed technique were compared to the results from the Monte Carlo simulation technique. 相似文献
10.
基于液化侧向变形实用统计模型和地震概率模型,建立了可以考虑地震随机特征和土体性质不确定性的液化侧向变形超越概率模型框架,通过实际案例初步探讨了模型的有效性,并将超越概率模型与现有统计模型的预测结果进行了对比。分析结果表明,若液化侧向变形的条件概率满足正态分布,标准差在5%到20%期望值范围内变化时,对位移超越概率影响不大;若满足对数正态分布,标准差对超越概率有一定影响。实用统计模型只能预测指定地震水平下的液化侧向变形值,而超越概率模型考虑了指定时间内所有可能地震的发生概率,可以同时预测变形值及发生概率,更加适合用于区域性的地震液化灾害评估。 相似文献
11.
Gilles Bourgault 《Mathematical Geosciences》2014,46(7):841-868
A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space. 相似文献
12.
Two phenomena of similar objects with different spectra and different objects with similar spectrum often result in the difficulty of separation and identification of all types of geographical objects only using spectral information．Therefore,there is a need to incorporate spatial structural and spatial association properties of the surfaces of objects into image processing to improve the accuracy of classification of remotely sensed imagery．In the current article,a new method is proposed on the basis of the principle of multiple-point statistics for combining spectral information and spatial information for image classification．The method was validated by applying to a case study on road extraction based on Landsat TM taken over the Chinese YeHow River delta on August 8,1999． The classification results have shown that this new method provides overall better results than the traditional methods such as maximum likelihood classifier (MLC) 相似文献
13.
Eulogio Pardo-Igúzquiza 《Mathematical Geology》1998,30(1):95-108
In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters. 相似文献
14.
Two methods for obtaining information on the fractional composition of organic matter in natural waters (membrane filtration and oxythermography) are combined in this study. The method has a number of indisputable advantages, which distinguish it among the currently available methods of analytical chemistry. In earth science, the fractional composition of organic materials in natural waters is estimated by the concentration of organic carbon (Corg) in fractions. In our opinion, an indicator of chemical consumption of oxygen (CCO) for ecological estimation of water reservoir state is more informative, because this indicator carries specific ecological information on the necessary oxygen consumption for oxidation of pollutants coming into the natural water environment. Therefore, estimation of the CCO parameter in fractions provides real information on the availability of organic matter and the probable danger from its content in water. This is the principal novelty of the information obtained and our study. The method was tested in the study of variation of the fractional composition of water matter during passage of Volga water through the waterworks (upon removal of water from the storage reservoir through the dam of the Ivan’kovskaya hydro power plant in the Dubna area). Analogous results are almost absent in earth science literature. There is a lack of data on the influence of water release through the dam even with respect to the major-component chemical composition. Variation of the fractional composition of matter (both organic and inorganic) upon water release through a waterworks is not discussed in literature. Thus, this study is of fundamental character. 相似文献
15.
The Nu Expression for Probabilistic Data Integration 总被引：4，自引：0，他引：4
The general problem of data integration is expressed as that of combining probability distributions conditioned to each individual
datum or data event into a posterior probability for the unknown conditioned jointly to all data. Any such combination of
information requires taking into account data interaction for the specific event being assessed. The nu expression provides
an exact analytical representation of such a combination. This representation allows a clear and useful separation of the
two components of any data integration algorithm: individual data information content and data interaction, the latter being
different from data dependence. Any estimation workflow that fails to address data interaction is not only suboptimal, but
may result in severe bias. The nu expression reduces the possibly very complex joint data interaction to a single multiplicative
correction parameter ν
0, difficult to evaluate but whose exact analytical expression is given; availability of such an expression provides avenues
for its determination or approximation. The case ν
0=1 is more comprehensive than data conditional independence; it delivers a preliminary robust approximation in presence of
actual data interaction. An experiment where the exact results are known allows the results of the ν
0=1 approximation to be checked against the traditional estimators based on assumption of data independence. 相似文献
16.
建立基于模拟退火遗传算法(Sjmualted Annealing Genetic Algorithm,SAGA)的改进极大似然法,即将似然函数相反数求解极小值的表达式作为目标函数,依据矩法估计参数取值范围作为约束条件,然后应用SAGA进行参数估计.与常规极大似然法思路有本质不同,改进极大似然法通过遗传算法进行参数优化.通过蒙特卡罗试验,验证了改进极大似然法在参数估计和不同频率设计值估计两个方面均具有很好的准确性,与基于最大熵原理的方法效果相当,优于其他方法;同时该方法不受线型类型、参数数目和约束条件的限制;可以避免应用常规极大似然法时出现似然方程无解等情况;且求解过程简便快捷,使极大似然法在理论上和实际应用中都成为有效的方法. 相似文献
17.
Mine gas disaster prediction and prevention are based on gas content measurement, which results in initial stage loss when determining coal gas desorption contents in engineering applications. We propose a Bayesian probability statistical method in the coal gas desorption model on the basis of constrained prior information. First, we use a self-made coal sample gas desorption device to test initial stage gas desorption data of tectonic coal and undeformed coal. Second, we calculate the initial stage loss of different coal samples with the power exponential function parameters by using Bayesian probability statistics and least squares estimation. Results show that Bayesian probability statistics and least squares estimation can be used to obtain regression and desorption coefficients, thereby illustrating the Bayesian estimation method’s validity and reliability. Given that the Bayesian probability method can apply prior information to constrain the model’s posterior parameters, it provides results that are statistically significant in the initial stage loss of coal gas desorption by connecting observation data and prior information.
相似文献18.
Huifen Liu Peiyuan Lin 《Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards》2018,12(3):190-202
This paper presents a maximum likelihood estimation of the ultimate bond strength for soil nails in clays. Both uncensored and censored ultimate bond strength data for soil nails are collected from the literature. Based on the concept of maximum likelihood, a log-likelihood function is constructed for estimating the mean and coefficient of variation (COV) of the ultimate bond strength jointly using the two types of data. The mean and COV are determined as the pair that maximises the log-likelihood function. Two distribution models (normal and lognormal) are used for the estimation. A comparison of the relative competence between the two candidate distribution models that are adopted for describing the collected uncensored and censored data is performed using the Bayesian Information Criterion. Example designs of soil nail walls against internal pullout limit state of nails and overall stability limit state are provided to demonstrate the benefit of taking censored data into account for estimation of the ultimate bond strength of soil nails. 相似文献
19.
Hybrid Estimation of Semivariogram Parameters 总被引：1，自引：0，他引：1
Two widely used methods of semivariogram estimation are weighted least squares estimation and maximum likelihood estimation.
The former have certain computational advantages, whereas the latter are more statistically efficient. We introduce and study
a “hybrid” semivariogram estimation procedure that combines weighted least squares estimation of the range parameter with
maximum likelihood estimation of the sill (and nugget) assuming known range, in such a way that the sill-to-range ratio in
an exponential semivariogram is estimated consistently under an infill asymptotic regime. We show empirically that such a
procedure is nearly as efficient computationally, and more efficient statistically for some parameters, than weighted least
squares estimation of all of the semivariogram’s parameters. Furthermore, we demonstrate that standard plug-in (or empirical)
spatial predictors and prediction error variances, obtained by replacing the unknown semivariogram parameters with estimates
in expressions for the ordinary kriging predictor and kriging variance, respectively, perform better when hybrid estimates
are plugged in than when weighted least squares estimates are plugged in. In view of these results and the simplicity of computing
the hybrid estimates from weighted least squares estimates, we suggest that software that currently estimates the semivariogram
by weighted least squares methods be amended to include hybrid estimation as an option. 相似文献
20.
Parameter identification is one of the key elements in the construction of models in geosciences. However, inherent difficulties
such as the instability of ill-posed problems or the presence of multiple local optima may impede the execution of this task.
Regularization methods and Bayesian formulations, such as the maximum a posteriori estimation approach, have been used to
overcome those complications. Nevertheless, in some instances, a more in-depth analysis of the inverse problem is advisable
before obtaining estimates of the optimal parameters. The Markov Chain Monte Carlo (MCMC) methods used in Bayesian inference
have been applied in the last 10 years in several fields of geosciences such as hydrology, geophysics or reservoir engineering.
In the present paper, a compilation of basic tools for inference and a case study illustrating the practical application of
them are given. Firstly, an introduction to the Bayesian approach to the inverse problem is provided together with the most
common sampling algorithms with MCMC chains. Secondly, a series of estimators for quantities of interest, such as the marginal
densities or the normalization constant of the posterior distribution of the parameters, are reviewed. Those reduce the computational
cost significantly, using only the time needed to obtain a sample of the posterior probability density function. The use of
the information theory principles for the experimental design and for the ill-posedness diagnosis is also introduced. Finally,
a case study based on a highly instrumented well test found in the literature is presented. The results obtained are compared
with the ones computed by the maximum likelihood estimation approach. 相似文献