Importance of Skewness and Kurtosis Statistical Tests for Outlier Detection and Elimination in Evaluation of Geochemical Reference Materials

A common characteristic of concentration data compilations for geochemical reference materials (GRM) is a skewed frequency distribution because of aberrant analytical data. Rejection of outlying results usually is required to obtain a better estimate of mean concentration values. The present work describes the application of an approach, based on skewness and kurtosis statistical tests, to establish reliable concentration values in study of rare-earth element compilations for GRM. Frequency histograms of initial concentrations for these elements differ significantly from a normal distribution, as deduced from their skewness and kurtosis coefficients. The statistical procedure shows that rejection of outliers yields normal distributions which can be used to provide mean concentrations with smaller standard deviations for a large number of cases, although for many elements our %RSD are similar (within 5%) to literature values. This procedure has been applied to derive new concentration data for rare-earth elements in 26 GRM, which are compared with earlier compilations to show that the statistical procedure proposed here provides mean values with greater quality value. Although most present mean concentrations are similar (within 5%) to those reported in previous literature, some of them are significantly different (with differences of up to 40%) in a few GRM.     

Interpretation of moment measures derived from water-lain sediments

ABSTRACT Although mean size provides a useful value for characterizing sediments, it is an inadequate expression of the interaction between sediments and hydraulic regimes and many authors have explored the significance of other statistical moment measures in different parts of the particle size spectrum. Few publications, however, have described such relationships over a very wide range of particle sizes. Using a combined data base (of marine and lacustrine sediments) we have been able to demonstrate the following: The distributions of standard deviation (and entropy) values are repeated in similar form on either side of the sand size class, describing mixtures between sands and gravel and sands and clay. Skewness is modified by the hydraulic response of the sediment and a skewness divide is present at about 2.7 φ, where it coincides with the minimum shear velocity required to initiate particle motion. This divide may be used to distinguish between high and low energy regimes. Kurtosis, like standard deviation, is a measure of the mixing of end member populations but it is more sensitive to changes in the tails of distribution curves. The maximum positive peak of kurtosis, at about 2.5 φ, lies close to the skewness divide (in our data). By using skewness/kurtosis plots it is possible to define separate field distribution for sediments which, to a greater or lesser degree, appear to conform to hydraulic equilibrium. By using sediments which are close to equilibrium (and lie at or close to our skewness/kurtosis boundary curves, it should be possible to define hydraulic shear velocities, at the time of formation. Values may be established by empirical relationships or by defining the break-point (traction and intermittent suspension) between linear components of individual sample φ size/probability distribution plots. Such relationships should be enhanced by using hydraulic equivalent diameters instead of sieve diameters. In fine sediments, the use of fractional (silt: clay) ratios offers sensitivity at a level comparable to that of skewness/kurtosis plots and may be more easily computed.     

边坡可靠度分析的非侵入式随机有限元法

提出基于非侵入式随机有限元法的边坡可靠度分析方法,并编写计算程序NISFEM。采用有限元滑面应力法计算边坡安全系数,将Hermite随机多项式展开与SIGMA/W和SLOPE/W模块有机结合实现边坡可靠度非侵入式随机分析。根据随机多项式展开系数,给出边坡安全系数前4阶统计矩（均值、标准差、偏度和峰度）和Sobol指标解析表达式,并采用Sobol指标进行边坡可靠度参数敏感性分析。最后,以均质土坡可靠度问题为例,证明该方法在边坡可靠度分析中的有效性。结果表明,边坡可靠度分析的非侵入式随机有限元法能够有效地考虑边坡变形对边坡可靠度的影响,计算效率远远高于蒙特卡罗模拟方法（MCS）,是解决复杂边坡可靠度问题一种有效地分析手段;黏聚力和内摩擦角变异性对边坡安全系数前四阶统计矩具有明显的影响,重度变异性对安全系数前4阶统计矩几乎没有影响;抗剪强度参数间负相关性对边坡安全系数均值几乎没有影响,但对安全系数标准差、偏度和峰度均有明显的影响。此外,随着抗剪强度参数间负相关性的增加,边坡安全系数由近似正态分布逐渐变为明显的非正态分布。     

Standard reference water samples for rare earth element determinations

Standard reference water samples (SRWS) were collected from two mine sites, one near Ophir, CO, USA and the other near Redding, CA, USA. The samples were filtered, preserved, and analyzed for rare earth element (REE) concentrations (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) by inductively coupled plasma-mass spectrometry (ICP-MS). These two samples were acid mine waters with elevated concentrations of REEs (0.45–161 μg/l). Seventeen international laboratories participated in a ‘round-robin’ chemical analysis program, which made it possible to evaluate the data by robust statistical procedures that are insensitive to outliers. The resulting most probable values are reported. Ten to 15 of the participants also reported values for Ba, Y, and Sc. Field parameters, major ion, and other trace element concentrations, not subject to statistical evaluation, are provided.     

Relative Efficiency of Single-Outlier Discordancy Tests for Processing Geochemical Data on Reference Materials and Application to Instrumental Calibrations by a Weighted Least-Squares Linear Regression Model

Numerous studies report geochemical data on reference materials (RMs) processed by outlier-based methods that use univariate discordancy tests. However, the relative efficiency of the discordancy tests is not precisely known. We used an extensive geochemical database for thirty-five RMs from four countries (Canada, Japan, South Africa and USA) to empirically evaluate the performance of nine single-outlier tests with thirteen test variants. It appears that the kurtosis test (N15) is the most powerful test for detecting discordant outliers in such geochemical RM databases and is closely followed by the Grubbs type tests (N1 and N4) and the skewness test (N14). The Dixon-type tests (N7, N8, N9 and N10) as well as the Grubbs type test (N2) depicted smaller global relative efficiency criterion values for the detection of outlying observations in this extensive database. Upper discordant outliers were more common than the lower discordant outliers, implying that positively skewed inter-laboratory geochemical datasets are more frequent than negatively skewed ones and that the median, a robust central tendency indicator, is likely to be biased especially for small-sized samples. Our outlier-based procedure should be useful for objectively identifying discordant outliers in many fields of science and engineering and for interpreting them accordingly. After processing these databases by single-outlier discordancy tests and obtaining reliable estimates of central tendency and dispersion parameters of the geochemical data for the RMs in our database, we used these statistical data to apply a weighted least-squares linear regression (WLR) model for the major element determinations by X-ray fluorescence spectrometry and compared the WLR results with an ordinary least-squares linear regression model. An advantage in using our outlier procedure and the new concentration values and uncertainty estimates for these RMs was clearly established.     

An application of least-squares inverse analysis in kinetic interpretations of hydrous pyrolysis experiments

A least-squares inverse method is applied to the estimation of optimum kinetic parameters with statistical error bounds from concentration data obtained in isothermal hydrous pyrolysis experiments. The inverse method requires the specification of a data-parameter relationship (e.g., classical kinetic theory), the prior covariance matrices of data and parameter errors, as well as the prior central estimates of data and parameters. The reaction scheme considered is the common case of kerogen breakdown by Gaussian-weighted independent parallel first-order reactions and bitumen cracking by a single first-order reaction. The nonlinearity of the problem is reduced by a logarithmic transformation, which suggests a parameterization in terms of logarithmic concentrations, activation energies, and logarithmic Arrhenius factors. The linearized variance analysis is valid for the case studied, and the posterior covariance matrix reveals which parameters are constrained by the data. We find that the statistical errors in the average activation energy and the associated Arrhenius factor are strongly correlated. Hence, the parameters which determine the temperature dependence of the reaction rate have not been resolved independently. Furthermore, the kinetic results are very sensitive to the presence of a distribution of activation energies in kerogen breakdown. This distribution is not constrained by the data. As a consequence, neglecting the consideration of distributions of activation energies results in activation parameter values which are much too low. This is the major reason for the commonly encountered discrepancy between kinetic parameter values obtained from hydrous pyrolysis and micropyrolysis experiments, respectively.     

内蒙古白音郭勒地区地球化学异常提取方法对比研究

勘查地球化学的基础性问题是如何合理地提取地球化学异常下限,这是影响勘查地球化学在矿产勘查中效果的主要因素。利用SPSS软件,通过含量对数频率直方图,结合偏度、峰度检验法检验元素的含量分布型式,发现研究区的Cu、Ag、Pb、Zn元素不服从正态或对数正态分布,而呈现多峰态、高丛集的多重分形特征。采用传统的统计学方法、85%累计频率法、含量-总量多重分形法提取相关元素地球化学异常,并进行对比研究。研究结果表明,基于元素空间分布及含量分布规律的含量-总量多重分形方法,有效克服了传统方法的局限,比较客观地反映研究区的地球化学异常信息,是研究地球化学异常比较合理的方法。     

Optimal composite sample size selection, applications in geochemistry and remote sensing

Nearly all the data in exploration geochemistry and remote sensing represent composites. However, composites may arise implicitly or be created explicitly. Bearing in mind that a common exploration task is the classification of data as being above or below some predetermined threshold the size of the composite may be critical to the recognition of a relatively rare, subcomposite, anomalous event. Two approaches are developed, based on statistical, and cost-analytical considerations. The statistical model allows for spatial correlation in the data, of importance when sampling is undertaken continuously along a drill core or flight line. Tables are presented for optimal composite sample size selection based on both models. The procedure is illustrated by an example taken from a drilling program. In general, the cost-analytical model leads to smaller composites than the statistical model. When spatial independence may be assumed the cost-optimal composite sizes are almost always smaller than those suggested by the statistical approach.     

Multivariate normality in morphometric analysis

Data on some living (salamanders and grasshoppers) and fossil (Devonian brachiopods) animals are analyzed by means of recently developed methods for the large-scale treatment of multivariate normality. Multivariate nonnormality was found to exist in all situations, even if the univariate deviations in the skewness and kurtosis statistics proved to be without significance for the most part. The effect of logarithmically transforming the data appears to be a mixed blessing. Apart from the fact that the investigator is removed one step from the biological relationships in his data by carrying out a transformation of them, the betterment in the multivariate interconnections with respect to normality tends to be slight, despite the general improvement in the univariate values. The relationship between sample size and the multivariate normality measures b_1,p and b_2,p are studied empirically.     

基于Kurtosis-IHS的遥感影像融合

针对传统IHS遥感影像融合的光谱扭曲和易引入噪声问题,提出了一种基于Kurtosis-IHS的遥感影像融合方法：从多光谱图像强度分量与全色图像最优逼近的角度出发,迭代求解多光谱各波段图像融合系数,减少光谱失真;根据全色图像的峭度获取其细节分量,达到抑制噪声的目的;最后使用基于Kurtosis-IHS的融合公式得到融合结果.实验选取2组实际遥感数据,结果表明,融合方法在CC、ERGAS、RASE、RMSE及SID 5个光谱性能评价指标上均优于传统的IHS、PCA、DWT及NSCT融合方法,在第二组遥感影像融合测试中,Kurtosis-IHS方法比传统IHS方法的5个指标分别提高了41.80％、57.09％、57.11％、57.11％、49.74％;在含噪环境下,融合方法的融合结果PSNR值均达到最大.所提出的Kurtosis-IHS方法优于传统的IHS融合方法,在提高空间分辨率的同时,具有较好地保持光谱信息及有效抑制噪声的优点.     

Assessment of the amount of polar bears (Ursus maritimus) on the basis of perennial vessel counts

This article presents a method of assessment of the amount of polar bears on the basis of data on vessel counts. The results of 32 vessel transect counts in the southern part of the Kara Sea are used as the primary data. They are processed in an original program using separate extrapolation by sectors and summarizing of data of several different counts. In recent years, the calculated amount of bears in the studied region has varied from 3300 to 3100 individuals (3200, on average). The relative statistical error of estimations smoothly decreases from 8 to 7%. The confidence interval of the estimation at statistical error of 7% and probability level of 95% is 2700–3500 individuals.     

Computer program for presentation of grain-size data by the graphic method

A computer program is described for a Hewlett-Packard desk-top Calculator (Model 9820A)-Plotter (Model 9862A) which plots a histogram, frequency polygon and/or cumulative curve, as well as standard statistical parameters and percentages of gravel, sand, silt, and clay from raw weight data from a sediment grain-size analysis. The program utilizes the graphic method for calculating statistical parameters in contrast to other commonly used computer programs derived for the method of moments. This program is advantageous because (1) it is rapid, (2) it allows direct comparison with grain-size data in the literature which have been computed by the graphic method, (3) either a complete presentation of a distribution can be obtained on one sheet of paper or a series of cumulative curves or frequency polygons can be prepared on a single graph which is ready for publication photocopying, (4) automatic extrapolation of data points from a cumulative curve eliminates inconsistencies arising from manual extrapolation, and (5) the desk-top calculator is more convenient to use, eliminates punching of data cards, and can be operated by someone without a knowledge of computers and programming. Disadvantages of the program are: (1) some authors feel critical percentiles are more accurately read off cumulative curves plotted on a probability ordinate than on an arithmetic ordinate, (2) the cumulative curve is a series of straight-line segments, (3) the frequency polygon is not as accurate a representation of grain-size distribution as a frequency curve and (4) size class intervals for the histogram and frequency polygon are standardized at 0.5φ, even though gravel and mud fractions are analysed at whole φ intervals. For most studies, the advantages of the program outweigh the disadvantages.     

SIPVADE: A New Computer Programme with Seventeen Statistical Tests for Outlier Detection in Evaluation of International Geochemical Reference Materials and its Application to Whin Sill Dolerite WS-E from England and Soil-5 from Peru

A new computer programme was written in programming language TURBOC, which enables us to apply a procedure involving seventeen statistical tests (a total of sixty five single or multiple outlier versions of these tests) for outlier detection in univariate sample at a high confidence level of 99% (significance level α= 0.01). The outlying observations should be evaluated first for technical reasons and then rejected manually from the data base until no more outliers are detected and the final statistical parameters are computed from the remaining data. This programme has been used successfully to process two reference material data bases: WS-E from England and Soil-5 from Peru. The final mean values for WS-E are more reliable (characterized by smaller standard deviations and narrower confidence limits) than those obtained earlier using a different statistical approach. The application of a large number of statistical tests to Soil-5 also resulted in smaller standard deviation values for most elements than the method involving a limited number of such tests. For WS-E, some laboratories seem to have produced multiple data that were detected as statistical outliers. A close analysis of the distribution of outliers as a function of laboratory, country and analytical method leads to a technical justification for these outlying observations, probably in terms of inadequate QA/QC practices. Use of geochemical criteria indicates that the new mean values in WS-E might be closer to the "true" concentrations. This procedure of outlier detection and elimination is therefore recommended in the study of the existing RM.     

The correlation between the macroseismic attenuation trend and the geo-structural framework; The calabropeloritan arc as an example

After describing attempts at perfecting a methodology for studying isotropic and anisotropic macroseismic fields in previous works, the authors here try to identify the causes of anisotropy in the context of the “new basement tectonics”.The seismic data are taken both from reconstructions of the macroseismic fields of historic events, by means of a critical analysis of the data, and from macroseismic fields of recent events surveyed by the authors. These data are correlated to the structural framework obtained through recent neotectonic studies and the lineament distribution traced on satellite images and using the “shadow” method. Generally the direction of elongation of the mesoseismic area is closely dependent on the source parameters and can be associated with recent and present-day systems outlined by the latest neotectonic studies.The best correlation is observed, however, with the lineament pattern obtained using the “shadow” method: the domains of the lineaments associated with the preferential trend of the macroseismic field show, in the rose diagram of cumulative number, values of prevalence and kurtosis higher than average; in the cumulative lengths diagram, on the other hand, they show prevalence maxima and, in particular, kurtosis maxima which are all the higher the more the anisotropic trend of field is accentuated. Using the “Giant Griffith Cracks” model for the lineaments, it can be deduced that the swarms refer to fracture systems with greater vertical development generated during the most recent tectonic phases.Finally, from a study of the dynamic characteristics of the elastic waves, that are the main agents responsible for macroseismic effects, it can observed that the wavelength order of magnitude is comparable with that of the linear parameters in the “warp” formed by the “Giant Cracks”. It can, thus, be deduced that the strong absorption of energy can be determined by the fracture swarm when the wave propagation occurs orthogonally to the swarm.     

普通铅数据处理在PC-1500机上的实现

使用BASIC语言编制的程序,在PC—1500计算机上处理普通铅数据,可得到3种方法、5个年龄和源区特征值等,可绘制铅增长曲线、等时线并给出等时线方程。还可用本程序处理异常铅和Pb—Pb法年龄数据。程序采用多入口,模块化处理,具有操作简便、速度快、功能强的特点。最后本文以河北兴隆高板河铅—锌—硫多金属矿床为例,对普通铅测试数据,上机运算,得到了理想的效果。     

琼州海峡及周边海域沉积环境及近万年以来沉积演化

对琼州海峡及周边海域约600个表层样品进行粒度分析后发现,琼州海峡内沉积物以粗粒的含砾砂及砂为主,自海峡内向口外则由砂向粉砂过渡,最外侧分布有更细粒的粘土质粉砂,东口砂质沉积物边缘以弧形分布为主,西口呈指状分布.按峰态类型可将粒度频率曲线分布划分为海峡东单峰态、海峡东双峰态、海峡内单双峰态、海峡西单峰态及海峡西双峰态5个区域.采用GSTA模型对沉积物运移趋势分析后发现,沉积物在海峡内主要由南北两侧向中间运移,海峡东口沉积物具有比较明显的由口内向口外输运的趋势,海峡西部沉积物出海峡口后显示了向北输运的趋势,沉积物粒度特征分布及运移趋势说明海峡内及两侧三角洲内沉积物主要来自于海峡底部及南北两岸的潮流冲刷物.根据琼州海峡现代沉积环境,结合周边4个柱状沉积物粒度及14C测年分析,发现琼州海峡的最终形成大约开始在距今8 000 a前,由于海平面的上升,潮流作用塑造了海峡初期的地貌,直到距今约5 000 ~4 600 a前,琼州海峡一直处于快速发展阶段,潮流三角洲发育也最广,之后由于暖期的结束,海平面下降,随着潮流作用的减弱,琼州海峡趋于稳定状态,逐渐形成今天的海峡及口外水下三角洲地貌.      

浙江土壤地球化学基准值与环境背景值

以浙江省农业地质环境调查取得的区域地球化学资料(2002—2004年)为依据,遵循地球化学背景值的基本概念,根据成土母质类型将土壤地球化学基准值划分成13个统计单元,土壤环境背景值以浙北杭嘉湖与宁绍平原、浙东沿海温黄与温瑞平原、浙中金衢盆地区统计单元,在反复剔除异常数据后,获得了52种元素(氧化物)的平均值、标准离差和变异系数,为区域土壤环境质量标准的制订、土壤污染评价和治理修复提供了重要的地球化学依据。     

土的物理力学性指标统计程序(IBM—PC机)

在IBM—PC微机上编制土的物理力学性指标的统计计算程序是可行的。程序全部汉化,並采用人机对话形式。程序能自动形成数据文件,有自动检索功能,可迅速而准确地完成各土层的物理、力学指标的统计计算工作。     

基于三维结构面网络模拟的岩体变形模量确定方法

变形模量作为表征岩体变形特征的重要参数,是岩体工程设计的基础力学参数之一。在综合前人研究的基础上提出了基于三维结构面网络模拟确定岩体变形模量的方法,编制了三维结构面网络模拟及计算RQD值的程序。以西南山区某水电站坝址区岩体为实例,在概率统计各组优势结构面几何要素分布形式的基础上,采用Monte Carlo法生成三维结构面网络模型,据此获得岩体三维RQD值,利用RMR法进行质量评价和变形模量的确定,并与钻孔资料统计法和原位试验结果进行对比分析。结果表明:基于钻孔资料统计的估算结果与原位试验相对误差达31.62%,且其估算值偏低;基于三维结构面网络模拟的估算结果与原位试验相对误差仅为2.56%,估算结果精度较高。该方法考虑了岩体质量的空间各向异性,对岩体的质量评价和参数确定更为合理,研究成果为该工程的优化设计提供了合理的计算参数。      

岩体结构统计均质区的划分

本文介绍由Ｍｉｌｌｅｒ提出的基本概率统计理论的关联表分析，结合施密特投影图研究岩体统计均质区的划分方法。对该法进行了适当的修改，编写了相应的计算机程序。并对三峡永久船闸地区的岩体结构，进行了岩体结构统计均质区的划分，获得了良好的效果。