马氏距离与欧氏距离方法在地球化学异常处理中的对比

正确和有效地确定综合地球化学异常是地球化学异常处理中的重要内容。综合地球化学异常的确定主要是基于多元正态分布理论,是多变量统计方法的直接推广和应用,马氏距离就是一个在地球化学离群点识别及判别分析等方面应用广泛的综合指标。以往比较普遍使用的是以欧氏距离为基础的计算方法,近20多年来基于马氏距离计算方法的应用逐渐增多。本文以河北某地1∶5万地球化学数据处理为例,从理论上和应用效果上对两种方法进行了比较,马氏距离方法考虑到了不同元素之间的相互关系,利用马氏距离方法求得的综合地球化学异常范围集中、界线清楚、强度突出、与已知矿体的吻合程度高,较欧氏距离方法具有明显优势,并在实际中可行。     

多元地球化学异常识别的核马氏距离方法

地球化学数据满足多元正态分布时,马氏距离是一种有效识别多元地球化学异常的综合指标。然而,由于地质系统的复杂性、成矿作用的多期多阶段性以及控矿因素的多重性常常导致多元地球化异常临界面是非线性的和模糊的,用马氏距离定义的平滑超椭球面不能准确表示这种复杂曲面。核函数能够将地球化学样品集非线性变换至特征空间,背景样品的映像集合在特征空间中构成一种流型,异常样品的映像则零散分布于流型的边缘及外围。计算和比较样品映像到样品映像总体的核马氏距离,可以识别异常样品。把该方法应用于白山地区多元地球化学异常识别,用核马氏距离、马氏距离和主成分得分识别金-银、金-银-砷-铋-汞、金-银-铜-铅-锌-锑-钴、金-银-铜-铅-锌-砷-锑-铋-汞-钴4种组合模式的多元地球化学异常。研究结果表明：复合核函数马氏距离的多元地球化学异常识别效果优于其他方法。     

基于EXCEL的马氏距离计算方法

建立在多元正态分布理论基础上的马氏距离,考虑了均值、方差和协方差三个参数,在异常评价中具有基于一元正态分布理论的单指标异常评价方法不可比拟的优点。马氏距离的计算主要是矩阵运算,计算工作量较大。这里提出了利用M icrosoft Excel提供的函数,结合VB编程进行马氏距离的计算方法,编制了相应的计算程序,对栾川三川—赤土店地区水系沉积物测量数据进行了实际计算。结果表明,利用Excel进行马氏距离计算具有简单易用、方便快捷的特点,易于推广普及。     

地球化学综合异常的圈定及找矿效果

能否正确有效地圈定地球化学综合异常是地球化学数据处理的重要内容,对找矿效果起决定作用。传统方法多在单元素异常的基础上圈定综合异常,而忽略了异常可能由地层本身高背景富集引起。笔者引用马氏距离识别地球化学离群点的方法圈定综合异常。由于马氏距离是基于多元正态分布理论,是多变量统计方法的直接推广和应用,其全面考虑了元素均值、方差及元素间协方差3个参数,是直接针对样本的计算过程,同时也是一元方法的直接推广,所以其具有独特的异常识别功能。经各种不同比例尺和不同采样介质化探数据处理应用显示,利用马氏距离方法求得的地球化学综合异常具边界唯一、界线清楚,减少了人为圈定异常的影响;异常强度突出,指标单一,可作为评价异常的一个重要参数使用;发现矿体(矿化)的概率程度高等优势。该方法可在实际中推广应用。     

遥感图像像素级异常识别的一种方法

遥感图像异常识别是遥感应用领域一个颇受关注的研究课题,在军事目标识别和自然环境保护等许多领域都有潜在应用价值。不妨假设遥感图像背景像素分布于随空间位置缓慢变化的一系列高斯超椭球体内,异常像素则分布于超椭球体之外。在这种假设前提下,首先应用Weiszfeld方法估算遥感图像中一系列高斯超椭球体的重心和波段协方差矩阵;然后,计算各像素到对应的超椭球体重心的马氏距离,并用直方图法确定马氏距离的异常下限;最后,把马氏距离高于异常下限的像素作为异常像素识别出来。在GDAL遥感图像数据输入输出函数库基础上,用VC++语言开发了遥感图像像素级异常识别的算法程序;用美国亚特兰大TM图像进行了方法的应用实验研究。结果表明,该方法对遥感图像中的局部异常具有很好的识别效果。     

河南栾川叫河-大清沟地区地球化学异常特征

根据叫河-大清沟地区1:50000水系沉积物测量成果,在分析区内元素地球化学特征的基础上,利用R型因子分析和马氏距离对元素分布特征和规律进行了进一步的总结与探讨,系统阐述了测区内地层的地球化学特征、(衬值)异常特征、马氏距离异常特征,指明了寻找铅锌银多金属矿的有利层位和成矿元素主要富集地段,为区内地质找矿工作提供了地球化学依据.     

地球化学背景与异常划分的多元方法

本文引入稳健统计学原理,提出划分地球化学背景与异常的多元方法。该方法是常用一元方法的直接推广,它以多元正态分布理论为基础,以马氏距离为统计量,扩大了背景与异常概念的内涵,定义了临界面、背景域等新的概念,建立了多元数据的背景与异常划分准则,在异常识别、综合异常图示等方面有特殊功能。主要介绍简单背景假设下的一般数学原理、计算方法步骤及简单应用,不讨论复杂情况下的解决办法。     

河南栾川叫河—大清沟地区地球化学异常特征

根据叫河—大清沟地区1∶50000水系沉积物测量成果,在分析区内元素地球化学特征的基础上,利用R型因子分析和马氏距离对元素分布特征和规律进行了进一步的总结与探讨,系统阐述了测区内地层的地球化学特征、(衬值)异常特征、马氏距离异常特征,指明了寻找铅锌银多金属矿的有利层位和成矿元素主要富集地段,为区内地质找矿工作提供了地球化学依据。     

马氏距离综合异常评价方法在青海省门县西河坝地区1∶5万水系沉积物测量中的运用

马氏距离（Mahalanobis distance）是一种有效的计算样本集间相似度的方法。它不受量纲的影响,可以排除变量之间的相关性的干扰,且通常能夸大变化微小的变量的作用。由于化探数据信息量大,计算繁琐,该方法在实际生产应用较少。借助强大的数学运算软件Matlab,能实现快速计算马氏距离,结合Mapgis等相关的GIS平台,可以快速圈定综合异常并能很好的强化具一定找矿前景的弱小异常,为下一步异常查证起到快速有效的指导作用。依据青海省门源县西河坝地区1∶5万水系沉积物测量的数据成果,通过与传统综合异常圈定方法进行对比,马氏距离圈定综合异常具一定的优势。     

利用Χ~2分布确定多元地球化学异常的方法

与单元素的概率格纸图解法相似，利用Ｘ２分布可直观地识别多元离群样品和进行多重总体分布。在研究多元地球化学异常时，首先运用Ｘ２分布图确定正常总体，然后确定多元正常总体的平均值向量、斜方差矩阵和马氏距离的背景上限值，最后利用这些正常总体的参数计算全域的马氏距离并圈定多元地球化学异常。     

Probabilistic outlier detection for sparse multivariate geotechnical site investigation data using Bayesian learning

Various uncertainties arising during acquisition process of geoscience data may result in anomalous data instances(i.e.,outliers)that do not conform with the expected pattern of regular data instances.With sparse multivariate data obtained from geotechnical site investigation,it is impossible to identify outliers with certainty due to the distortion of statistics of geotechnical parameters caused by outliers and their associated statistical uncertainty resulted from data sparsity.This paper develops a probabilistic outlier detection method for sparse multivariate data obtained from geotechnical site investigation.The proposed approach quantifies the outlying probability of each data instance based on Mahalanobis distance and determines outliers as those data instances with outlying probabilities greater than 0.5.It tackles the distortion issue of statistics estimated from the dataset with outliers by a re-sampling technique and accounts,rationally,for the statistical uncertainty by Bayesian machine learning.Moreover,the proposed approach also suggests an exclusive method to determine outlying components of each outlier.The proposed approach is illustrated and verified using simulated and real-life dataset.It showed that the proposed approach properly identifies outliers among sparse multivariate data and their corresponding outlying components in a probabilistic manner.It can significantly reduce the masking effect(i.e.,missing some actual outliers due to the distortion of statistics by the outliers and statistical uncertainty).It also found that outliers among sparse multivariate data instances affect significantly the construction of multivariate distribution of geotechnical parameters for uncertainty quantification.This emphasizes the necessity of data cleaning process(e.g.,outlier detection)for uncertainty quantification based on geoscience data.     

Unmasking multivariate anomalous observations in exploration geochemical data from sheeted-vein tin mineralization near Emmaville, N.S.W., Australia

Previous interpretations of surface-rock geochemical data from the sheeted-vein tin mineralization in the Emmaville district have been carried out using classical statistics. These investigations revealed low-contrast geochemical patterns of 3 to 5 ppm Sn, supported by 80 to 160 ppm F, block-average contours defining four of the six known mineral occurrences. Principal component scores for the association dominated by F-Li-Rb have defined the same four mineral occurrences. For the prospecting of similar deposits it is highly desirable to improve the data processing techniques to achieve more acceptable geochemical contrasts between anomalous and background levels. Minimum volume ellipsoid (MVE) estimation, a high-breakdown method (capable of accommodating up to 50% outliers) recently developed in robust statistics is applied to a subset of the data from the northeastern part of the Emmaville district. The anomalies related to mineralization in this part of the district are not as well developed compared to those in the west. The data set used in this study consists of 133 observations with 6 elements, namely Cu, Li, Rb, F, As and Sn.The detection of multivariate outliers (anomalous observations) by Mahalanobis distance calculation was carried out on the surface rock geochemical data. The robust Mahalanobis distances computed from MVE estimates of location and scatter shows little variation over background areas but are sharply enhanced over mineralization. In contrast, the usual Mahalanobis distances either fail to indicate the presence of mineralization altogether, or, at best, respond with feebly enhanced values that do not satisfactorily indicate the presence of mineralization.Graphical display of results from classical RQ-PCA performs poorly, revealing only 6 weakly anomalous observations related to mineral occurrences. Several additional observations from these occurrences have also gone undetected. On the other hand, results from MVE-robust RQ-mode principal component analysis show that the background observations cluster tightly within the 95% tolerance ellipse while the anomalous observations (related to mineral occurrences) are greatly enhanced and the variables that characterize them are clearly indicated. Results are consistent with those of robust Mahalanobis distance procedure; both techniques indicate essentially the same observations as being anomalous.     

Multivariate Spatial Outlier Detection Using Robust Geographically Weighted Methods

Outlier detection is often a key task in a statistical analysis and helps guard against poor decision-making based on results that have been influenced by anomalous observations. For multivariate data sets, large Mahalanobis distances in raw data space or large Mahalanobis distances in principal components analysis, transformed data space, are routinely used to detect outliers. Detection in principal components analysis space can also utilise goodness of fit distances. For spatial applications, however, these global forms can only detect outliers in a non-spatial manner. This can result in false positive detections, such as when an observation’s spatial neighbours are similar, or false negative detections such as when its spatial neighbours are dissimilar. To avoid mis-classifications, we demonstrate that a local adaptation of various global methods can be used to detect multivariate spatial outliers. In particular, we account for local spatial effects via the use of geographically weighted data with either Mahalanobis distances or principal components analysis. Detection performance is assessed using simulated data as well as freshwater chemistry data collected over all of Great Britain. Results clearly show value in both geographically weighted methods to outlier detection.     

Some aspects of transformations of compositional data and the identification of outliers

The statistical analysis of compositional data is based on determining an appropriate transformation from the simplex to real space. Possible transfonnations and outliers strongly interact: parameters of transformations may be influenced particularly by outliers, and the result of goodness-of-fit tests will reflect their presence. Thus, the identification of outliers in compositional datasets and the selection of an appropriate transformation of the same data, are problems that cannot be separated. A robust method for outlier detection together with the likelihood of transformed data is presented as a first approach to solve those problems when the additive-logratio and multivariate Box-Cox transformations are used. Three examples illustrate the proposed methodology.     

Spatial modelling of hazardous elements at waste dumps using geostatistical approach: a case study Sarcheshmeh copper mine,Iran

Mining activities and resulting wastes can be considered as one of the most important sources of hazardous elements in the environment. Knowledge of the spatial distribution of toxic elements in waste dump systems is necessary to assess environmental hazard and strategy. To achieve this goal, this paper investigates spatial distribution of toxic elements using statistical and geostatistical analysis. A total of 58 soil samples were collected, and the amount of As, Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb and Zn was then determined at “Sarcheshmeh” copper mine waste dumps. In order to evaluate the presence of multivariate outliers, Mahalanobis distance technique (D ²) was applied and the multivariate outlier samples were removed. This resulted in an increase in correlation coefficient. To reduce dimension of data set, principal component analysis was applied and four principal components were determined which indicate 83.463% of the total variance of data set. Estimated PCs together with the toxic elements maps based on the ordinary kriging display aggregation of toxic elements in some parts, and validity of predictions was evaluated using the leave-one-out cross-validation method. The regression coefficients of estimated and observed values presented the reliability of the kriging estimates. Sequential Gaussian simulation method was applied for principal components due to similar results of estimated principal components and toxic elements. The results of simulation maps are almost identical to estimated outcomes.     

A Robust Multiquadric Method for Digital Elevation Model Construction

The multiquadric method (MQ) with high interpolation accuracy has been widely used for interpolating spatial data. However, MQ is an exact interpolation method, which is improper to interpolate noisy sampling data. Although the least squares MQ (LSMQ) has the ability to smooth out sampling errors, it is inherently not robust to outliers due to the least squares criterion in estimating the weights of sampling knots. In order to reduce the impact of outliers on the accuracy of digital elevation models (DEMs), a robust method of MQ (MQ-R) has been developed. MQ-R includes two independent procedures: knot selection and the solution of the system of linear equations. The two independent procedures were respectively achieved by the space-filling design and the least absolute deviation, both of which are very robust to outliers. Gaussian synthetic surface, which is subject to a series of errors with different distributions, was employed to compare the performance of MQ-R with that of LSMQ. Results indicate that LSMQ is seriously affected by outliers, whereas MQ-R performs well in resisting outliers, and can construct satisfactory surfaces even though the data are contaminated by severe outliers. A real-world example of DEM construction was employed to evaluate the robustness of MQ-R, LSMQ, and the classical interpolation methods including inverse distance weighting method, thin plate spline, and ANUDEM. Results showed that compared with the classical methods, MQ-R has the highest accuracy in terms of root mean square error. In conclusion, when sampling data is subject to outliers, MQ-R can be considered as an alternative method for DEM construction.     

Outlier Detection for Compositional Data Using Robust Methods

Outlier detection based on the Mahalanobis distance (MD) requires an appropriate transformation in case of compositional data. For the family of logratio transformations (additive, centered and isometric logratio transformation) it is shown that the MDs based on classical estimates are invariant to these transformations, and that the MDs based on affine equivariant estimators of location and covariance are the same for additive and isometric logratio transformation. Moreover, for 3-dimensional compositions the data structure can be visualized by contour lines. In higher dimension the MDs of closed and opened data give an impression of the multivariate data behavior.     

珠江水系沉积物重金属元素背景值估算及污染特征分析

估算水系沉积物的地球化学背景值和识别其异常对人为污染判别与环境风险评估非常重要。采集并分析了珠江58件水系沉积物样品,经分析检验,Al、Fe和Sc被选作参考元素,并对比了确定地球化学背景及识别异常值的方法。其中,基于最小截断二乘法的回归分析是定义地球化学背景的有效方法,它是一种对异常值不敏感的稳健统计方法,而基于局部富集因子的箱线图和回归诊断图更适用于识别异常值。珠江不同河段重金属污染存在差异,北江和河网区主要受As、Cd、Cu、Pb和Zn污染,东江主要受Cu、Cr和Ni污染,而西江几乎不存在重金属污染。水系沉积物的主要污染类型是点源污染,主要污染来源是采矿和电镀等相关的工业活动。