Isometric Logratio Transformations for Compositional Data Analysis

Geometry in the simplex has been developed in the last 15 years mainly based on the contributions due to J. Aitchison. The main goal was to develop analytical tools for the statistical analysis of compositional data. Our present aim is to get a further insight into some aspects of this geometry in order to clarify the way for more complex statistical approaches. This is done by way of orthonormal bases, which allow for a straightforward handling of geometric elements in the simplex. The transformation into real coordinates preserves all metric properties and is thus called isometric logratio transformation (ilr). An important result is the decomposition of the simplex, as a vector space, into orthogonal subspaces associated with nonoverlapping subcompositions. This gives the key to join compositions with different parts into a single composition by using a balancing element. The relationship between ilr transformations and the centered-logratio (clr) and additive-logratio (alr) transformations is also studied. Exponential growth or decay of mass is used to illustrate compositional linear processes, parallelism and orthogonality in the simplex.     

Logratios and Natural Laws in Compositional Data Analysis

The impossibility of interpreting correlations of raw compositional components and associated statistical methods has been clearly demonstrated over the last four decades and alternative statistical methodology developed. Despite this a return to the traditional use of raw components has been advocated recently and alternative methodology such as logratio analysis strongly criticized. This paper exposes the fallacies in this recent advocacy and demonstrates the constructive role that logratio analysis can play in geological compositional problems, in particular in the investigation of natural laws and in subcompositional investigations.     

On criteria for measures of compositional difference

Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.     

BLU Estimators and Compositional Data

One of the principal objections to the logratio approach for the statistical analysis of compositional data has been the absence of unbiasedness and minimum variance properties of some estimators: they seem not to be BLU estimator. Using a geometric approach, we introduce the concept of metric variance and of a compositional unbiased estimator, and we show that the closed geometric mean is a c-BLU estimator (compositional best linear unbiased estimator with respect to the geometry of the simplex) of the center of the distribution of a random composition. Thus, it satisfies analogous properties to the arithmetic mean as a BLU estimator of the expected value in real space. The geometric approach used gives real meaning to the concepts of measure of central tendency and measure of dispersion and opens up a new way of understanding the statistical analysis of compositional data.     

Groups of Parts and Their Balances in Compositional Data Analysis

Amalgamation of parts of a composition has been extensively used as a technique of analysis to achieve reduced dimension, as was discussed during the CoDaWork'03 meeting (Girona, Spain, 2003). It was shown to be a non-linear operation in the simplex that does not preserve distances under perturbation. The discussion motivated the introduction in the present paper of concepts such as group of parts, balance between groups, and sequential binary partition, which are intended to provide tools of compositional data analysis for dimension reduction. Key concepts underlying this development are the established tools of subcomposition, coordinates in an orthogonal basis of the simplex, balancing element and, in general, the Aitchison geometry in the simplex. Main new results are: a method to analyze grouped parts of a compositional vector through the adequate coordinates in an ad hoc orthonormal basis; and the study of balances of groups of parts (inter-group analysis) as an orthogonal projection similar to that used in standard subcompositional analysis (intra-group analysis). A simulated example compares results when testing equal centers of two populations using amalgamated parts and balances; it shows that, in certain circumstances, results from both analysis can disagree.     

New Perspectives on Water Chemistry and Compositional Data Analysis

Water chemistry is commonly investigated to determine the suitability of water for various uses. With increased knowledge of aqueous chemistry, it has become possible to interpret the evolutionary processes that determine water composition and quality. This paper presents procedures for exploring and modeling the environment using compositional data from water analysis, utilizing statistical tools in an appropriate sample space. Our procedures build on a methodology based on log-ratios initiated by John Aitchison in the early 1980's. They are not only useful for interpreting the structure of the data, but also for characterizing and modeling the influence of geochemical processes acting on the environment. The geochemistry of water samples collected from wells on Vulcano Island (one of the Aeolian Islands of the Italian province of Sicily) will be used to illustrate the techniques, although an exhaustive overview would require many different examples. Vulcano island is a quiescent volcanic area where mobilization of chemical species by weathering of volcanic rocks and input of gaseous components from fumarolic activity has produced environmental changes expressed in the composition of phreatic waters at the surface and in the shallow subsurface. Changes in the chemical composition of waters in unconfined aquifers of the northwestern part of the island around the active crater appear to be useful in understanding the natural processes at work.     

Compositional Data Analysis: Where Are We and Where Should We Be Heading?

We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations.     

Compositional Data Analysis of Geological Variability and Process: A Case Study

Developments in the statistical analysis of compositional data over the last two decades have made possible a much deeper exploration of the nature of variability and the possible processes associated with compositional data sets from many disciplines. In this paper, we concentrate on geochemical data. First, we explain how hypotheses of compositional variability may be formulated within the natural sample space, the unit simplex, including useful hypotheses of sub-compositional discrimination and specific perturbational change. Then we develop through standard methodology, such as generalised likelihood ratio tests, statistical tools to allow the systematic investigation of a lattice of such hypotheses. Some of these tests are simple adaptations of existing multivariate tests but others require special construction. We comment on the use of graphical methods in compositional data analysis and on the ordination of specimens. The recent development of the concept of compositional processes is then explained, together with the necessary tools for a staying-in-the-simplex approach, such as the singular value decomposition of a compositional data set. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major oxide and trace element compositions of metamorphosed limestones from the Grampian Highlands of Scotland. Finally, we discuss some unresolved problems in the statistical analysis of compositional processes.     

Understanding the spatial representativeness of air quality monitoring network and its application to PM2.5 in the mainland China

Air pollution has seriously endangered human health and the natural ecosystem during the last decades. Air quality monitoring stations (AQMS) have played a critical role in providing valuable data sets for recording regional air pollutants. The spatial representativeness of AQMS is a critical parameter when choosing the location of stations and assessing effects on the population to long-term exposure to air pollution. In this paper, we proposed a methodological framework for assessing the spatial representativeness of the regional air quality monitoring network and applied it to ground-based PM_2.5 observation in the mainland of China. Weighted multidimensional Euclidean distance between each pixel and the stations was used to determine the representativeness of the existing monitoring network. In addition, the K-means clustering method was adopted to improve the spatial representativeness of the existing AQMS. The results showed that there were obvious differences among the representative area of 1820 stations in the mainland of China. The monitoring stations could well represent the PM_2.5 spatial distribution of the entire region, and the effectively represented area (i.e. the area where the Euclidean distance between the pixels and the stations was lower than the average value) accounted for 67.32% of the total area and covered 93.12% of the population. Forty additional stations were identified in the Northwest, North China, and Northeast regions, which could improve the spatial representativeness by 14.31%.     

Statistical Analysis of Landslide Events in Central America and their Run-out Distance

Statistical analyses of landslide deposits from similar areas provide information on dynamics and rheology, and are the basis for empirical relationships for the prediction of future events. In Central America landslides represent an important threat in both volcanic and non-volcanic areas. Data, mainly from 348 landslides in Nicaragua, and 19 in other Central American countries have been analyzed to describe landslide characteristics and to search for possible correlations and empirical relationships. The mobility of a landslide, expressed as the ratio between height of fall (H) and run-out distance (L) as a function of the volume and height of fall; and the relationship between the height of fall and run-out distance were studied for rock falls, slides, debris flows and debris avalanches. The data show differences in run-out distance and landslide mobility among different types of landslides and between debris flows in volcanic and non-volcanic areas. The new Central American data add to and seem consistent with data published from other regions. Studies combining field observations and empirical relationships with laboratory studies and numerical simulations will help in the development of more reliable empirical equations for the prediction of landslide run-out, with applications to hazard zonation and design of optimal risk mitigation measures.     

模糊理论在遥感图像分类中的应用

利用2000年假彩色遥感图像,采用模糊C-均值法中的欧氏距离和马氏距离法对崇明东滩的遥感图像进行了处理。通过对白色覆盖物、未耕种土地、一号水稻田、水体和二号水稻田的分类结果表明,欧氏距离的聚类结果优于马氏距离。     

加权距离判别法在泥石流危险度评价中的应用

将加权距离判别法引用到泥石流危险度分类中,建立了泥石流危险度分类模型。该模型选用了流域面积、主沟长度、流域最大相对高差、流域切割密度、主沟床弯曲系数、人口密度、泥沙补给长度比、植被覆盖率、一次泥石流（可能）最大冲出量和泥石流发生频率10项指标为建模参数,运用熵值法对这10个指标进行赋权,用已经分类的泥石流沟作为训练样本进行学习,建立了相应的判别准则。将待判泥石流沟样本代入判别准则进行判别分类,分类结果与传统分类结果100%吻合,验证了该模型的分类性能良好。该方法可以在实际工程中进行推广。     

核概率距离聚类方法及应用

研究聚类分析新方法一直是统计学和机器学习研究领域普遍关注的课题。针对概率距离聚类算法不能解决非线性可分聚类问题的缺欠,笔者应用核函数理论将该模型拓展成为一种能够解决非线性可分聚类问题的统计模型,称为核概率距离聚类分析模型。研制出一种应用新模型进行遥感图像非监督分类研究的实施策略和可行算法;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核概率距离聚类分析算法程序;用ERDAS软件提供的一幅7波段491像素×440像素大小的TM图像进行新方法分类应用实验研究。对比了新模型和其原版本的TM遥感图像非监督分类效果,结果表明新模型的非监督分类效果优于原有的分类模型。     

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.     

Euclidean Distance Matrix Analysis (EDMA): Estimation of mean form and mean form difference

Euclidean Distance Matrix Analysis (EDMA) of form is a coordinate free approach to the analysis of form using landmark data. In this paper, the problem of estimation of mean form, variance-covariance matrix, and mean form difference under the Gaussian perturbation model is considered using EDMA. The suggested estimators are based on the method of moments. They are shown to be consistent, that is as the sample size increases these estimators approach the true parameters. They are also shown to be computationally very simple. A method to improve their efficiency is suggested. Estimation in the presence of missing data is studied. In addition, it is shown that the superimposition method of estimation leads to incorrect mean form and variance-covariance structure.     

欧氏距离法在电测深找水中应用的可行性探讨

探讨了欧氏距离公式在电测深找水定量对比分析中的应用,并对几组数据进行计算分析,获得了较佳效果。     

梯度K法在电测深找水中的应用

梯度K法以欧氏距离公式为基础，应用于电测深找水，它能够利用实测数据较准确地确定最佳井位和估计涌水量，并获得较佳的地质效果。     

Studying fractal geometry on submicron length scales by small-angle scattering

Recent studies have shown that internal surfaces of porous geological materials, such as rocks and lignite coals, can be described by fractals down to atomic length scales, In this paper, the basic properties of self-similar and self-affine fractals are reviewed and how fractal dimensions can be measured by small-angle scattering experiments are discussed.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

Dealing with Zeros and Missing Values in Compositional Data Sets Using Nonparametric Imputation

The statistical analysis of compositional data based on logratios of parts is not suitable when zeros are present in a data set. Nevertheless, if there is interest in using this modeling approach, several strategies have been published in the specialized literature which can be used. In particular, substitution or imputation strategies are available for rounded zeros. In this paper, existing nonparametric imputation methods—both for the additive and the multiplicative approach—are revised and essential properties of the last method are given. For missing values a generalization of the multiplicative approach is proposed.